Dominant and subordinate outside options alter help and eviction in a pay-to-stay negotiation model

Dominant and subordinate outside options alter help and eviction in a pay-to-stay negotiation model Abstract In several cooperatively breeding species, subordinates that do not help sufficiently are punished or evicted from the group by dominant individuals. The credibility of dominant eviction threats may vary with the social context beyond the group level: when subordinates can easily breed in a neighboring territory, dominants may be less able to demand help from subordinates. Further, dominant ability to enforce subordinate cooperation may be reduced when it is difficult to replace evicted subordinates or in small groups where each subordinate makes a large contribution to group productivity. Here, we develop a 2-player game theoretic model to examine how the social context influences subordinate help and the threshold of help at which dominants evict subordinates. In contrast to predictions, we found that dominants demand more help when dominants are less able to replace evicted subordinates, suggesting that dominants punish a dereliction of helping behavior more strongly when they are unable to compensate for the loss of an evicted subordinate. In single sealed-bid games, subordinates help less than the fitness costs they impose on dominants and help does not vary with subordinate breeding opportunities outside the group. However, when subordinates can plastically increase help in response to demanding dominants (akin to pay-to-stay dynamics), subordinates provide more help overall, but decrease their help as breeding opportunities outside of the group increase. Our results demonstrate the importance of incorporating negotiation into theoretical models of helping strategies and demonstrate that plasticity is a key mechanism underlying pay-to-stay mechanisms of cooperation. INTRODUCTION The evolution and stability of helping behavior in group-living species has received a great deal of attention in the last 50 years. Specifically, questions center around why subordinates help, since helping behavior is often energetically costly and can reduce survival, growth, and future reproduction (Dickinson and Hatchwell 2004; Clutton-Brock 2006; Biedermann et al. 2011). Kin selection has been suggested as the primary mechanism regulating the evolution of helping in group-living birds and mammals (Russell 2004). However, in many species, subordinates that are unrelated to dominants also provide help (Cockburn 1998; Clutton-Brock 2009; Riehl 2013; Taborsky 2016) or kinship reduces the amount of help provided by subordinates (Stiver et al. 2005; Doutrelant et al. 2011; Zöttl et al. 2013d), suggesting that kin selection is not a mechanism underlying subordinate helping behavior in all taxa or in all social contexts (Taborsky et al. 2016). Another suggested mechanism underlying the evolution of helping behavior is pay-to-stay (Gaston 1978; Kokko et al. 2002; Bergmüller et al. 2007), in which subordinates provide help as a means to compensate for the costs, primarily in terms of competition for reproduction and resources, that they inflict upon dominants by remaining on the territory. The predictions of pay-to-stay have been supported in cichlid fish (Taborsky 1985; Balshine-Earn et al. 1998; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008; Zöttl et al. 2013d; Fischer et al. 2014; Hellmann et al. 2015b), paper wasps (Grinsted and Field 2017), and fairy wrens (Mulder and Langmore 1993; Cockburn 1998) and may be a potential mechanism underlying cooperation in other species of birds (Cockburn 1998; Leighton and Meiden 2016), fishes (Wong et al. 2007), mole rats (Reeve 1992), and meerkats (MacLeod et al. 2013). Under the pay-to-stay theory, subordinates that do not help sufficiently are punished or evicted from the group by more dominant individuals. Although resource holding potential, or the ability of an individual to use force within a dyadic relationship, strongly influences the ability of dominants to credibly exercise threats, it is becoming increasingly clear that a dominant’s ability to punish subordinates is also context-dependent (Lewis 2002; Cant and Johnstone 2009; Cant 2011). By helping on their current territory, subordinates often gain direct benefits from group membership or the potential to become a breeder on a high-quality territory (Riehl 2013); however, if subordinates have many opportunities to breed outside of their current group, they may be less likely to tolerate high demands for help in their current group relative to subordinates who have few options to breed or survive outside of the group, because eviction may be less costly to subordinates who will likely survive and breed successfully after being evicted (Bergmüller et al. 2005; Grinsted and Field 2017). Conversely, because dominants often derive fitness benefits (e.g. increased offspring survival, reduced workload) from subordinate help (Dickinson and Hatchwell 2004; Russell 2004; Johnstone 2011; Zöttl et al. 2013b), dominants may have less leverage to enforce subordinate cooperation when replacing evicted subordinates is difficult or if each subordinate makes a large contribution to group productivity (Cant and Johnstone 2006; Johnstone and Bshary 2008). Consequently, the social context likely alters the credibility of dominant eviction threats and, thus, the ability of subordinates to reduce the amount of help they provide without facing punishment. There is some empirical evidence demonstrating that helping behavior and eviction dynamics are influenced by the social context. Subordinates increase helping behavior when outside competition for their position in the group is higher (Bruintjes and Taborsky 2008) and when neighboring groups are present compared to when they are absent (Hellmann and Hamilton 2014). Furthermore, when subordinates are experimentally prevented from helping, dominants evict subordinates more frequently when there are more neighboring groups in close proximity, when dominants are potentially better able to replace evicted subordinates with subordinates from neighboring groups (Hellmann et al. 2015b). Finally, dominant tolerance of subordinates is regulated by the need for subordinate help: subordinates are evicted when subordinate help is not needed (Taborsky 1985), but they are accepted into the group when dominant need for subordinate help is high (Taborsky 1985; Zöttl et al. 2013c). These results demonstrate that a competitive social environment alters both subordinate help and dominant eviction thresholds and that subordinate eviction is more likely in cases when dominants are less reliant on subordinate help. However, experimental evidence that helping behavior is regulated by the cost of punishment or eviction for dominants is limited (but see Mulder and Langmore 1993; Fischer et al. 2014; Hellmann et al. 2015b). This is in part because experiments examining eviction dynamics are difficult to conduct: the most effective threats (i.e. eviction) are rarely triggered because subordinates in stable groups use both submissive and/or helping behavior to prevent eviction (Bergmüller et al. 2005; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008). Therefore, theoretical models can generate valuable predictions regarding eviction and helping dynamics that can advance our understanding of how punishment can alter group dynamics, as well as allow us to generate predictions that can be tested empirically across a variety of taxa with different social systems. Here, we present a 2-player model in which we examine how the social context influences the degree of help provided by a single subordinate and the threshold of helping behavior at which a dominant is willing to evict that subordinate. Although a past model (Hamilton and Taborsky 2005) explored how subordinate outside options alter helping and eviction dynamics, here we expand this model by examining helping and eviction dynamics relative to outside options for dominants (i.e. the ability of dominants to recoup the lost help of evicted subordinates) in addition to outside options for subordinates (i.e. the ability of evicted subordinates to become dominant or survive outside of a group). Furthermore, past game theoretic models examining helping and reproductive dynamics among group members have been single sealed-bid games, in which dominants and subordinates cannot respond to each other’s strategy (e.g. Cant and Field 2001; Kokko et al. 2002; Hamilton and Taborsky 2005). However, as there is evidence demonstrating that negotiation and plasticity in strategies are highly important in determining the degree of cooperation among partners (Taylor et al. 2006; Hamilton 2013), particularly for cooperation enforced by pay-to-stay (Raihani et al. 2012; Quiñones et al. 2016), we expand on traditional single sealed-bid games to understand how plasticity in either the dominant or subordinate strategy alters helping and eviction dynamics. Model 1: helping and eviction dynamics without plasticity Our game theoretical model consists of 2 classes of players: dominant and subordinate group members who are unrelated. Therefore, subordinates receive no kin-selected benefits of helping and dominants suffer no indirect fitness costs of evicting a subordinate. For the purposes of this model, we assume that subordinates who leave the group voluntarily or are evicted from the group cannot join a new group after leaving their current group; in essence, leaving or eviction from a group results in the subordinate either becoming a dominant in a different group or dying. This assumption was included in part to simplify recursive equations, but it is a realistic assumption in many group-living species where subordinates are large, such that acceptance into a new group is difficult because subordinates that are close in size to dominants are a reproductive threat to dominants and tend to be in conflict with dominants (Hamilton et al. 2005; Wong et al. 2007).Both parties may have a shared incentive to resolve conflict via the benefits of remaining in a productive group. For subordinates, the benefits of remaining in the group are 2-fold: the mortality rate of living in a group (μg) is lower than the mortality rate of leaving the group or being evicted from the group (μe). Additionally, the likelihood of becoming a dominant when subordinates live in a group (ag) is higher than the likelihood of becoming a dominant if evicted from the group (ae). We make this assumption because subordinates in many group-living species inherit their current territory when the current dominants die (Buston 2004; Stiver et al. 2006; Cant et al. 2016; Clutton-Brock and Manser 2016). Consequently, when a subordinate is part of a group, it can both inherit its current territory as well as take over the territory of another group; however, if a subordinate is evicted, it loses the opportunity to inherit its current territory. We assume that all rates and probabilities (e.g. mortality rates, territory inheritance) are time-independent and therefore, do not increase over time. If a subordinate who remains in the group survives (with a probability 1 − μg), it can obtain a dominant breeding position (with a given probability ag) and gain the fitness benefits associated with dominance ( WD¯), which is determined by the average fitness of dominants in the population given population-level values for subordinate help and dominant eviction thresholds ( h^, p^; Table 1).If the subordinate does not become dominant, it gains the fitness associated with staying a subordinate in a group (WSS). In either case, the subordinate pays a cost of helping and remaining in the group. Therefore, the fitness of a subordinate who remains in the group (WSS) is: Table 1 A list of parameters used in models and how they alter helping behavior and eviction thresholds in the single sealed-bid (h, p) and in the model with plasticity (h# and p#, which are determined by baseline levels of helping (h0) and eviction (p0) as well as individual responsiveness (λ)) Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase The patterns reported for the pay-to-stay plasticity (λh < 0: subordinates increase help in response to demanding dominants) hold regardless of the value of dominant responsiveness, λp. When only dominants are plastic (λp < 0 and λh = 0), and when λh > 0, the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. View Large Table 1 A list of parameters used in models and how they alter helping behavior and eviction thresholds in the single sealed-bid (h, p) and in the model with plasticity (h# and p#, which are determined by baseline levels of helping (h0) and eviction (p0) as well as individual responsiveness (λ)) Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase The patterns reported for the pay-to-stay plasticity (λh < 0: subordinates increase help in response to demanding dominants) hold regardless of the value of dominant responsiveness, λp. When only dominants are plastic (λp < 0 and λh = 0), and when λh > 0, the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. View Large WSS(h|p^, h^)=(1−μg)[agWD (p^,h^)¯+ (1−ag)WSS(h|p^, h^)−hc−phrs] (1) where h is the amount of help provided by the subordinate, c is the cost of helping, and phrs represents a cost of subordinate resistance to dominant eviction attempts, such that as the difference between subordinate help (h) and dominant eviction thresholds (p) decreases (i.e. when subordinate help is closer to the eviction threshold), subordinates face a mounting cost to remaining in the group (e.g. increased aggression from the dominant). This equation rearranges to: WSS (h|p^,h^)= (1−mg)(agWD (p^,h^)¯−hc−phrs)ag+ mg−mgag If a subordinate gets evicted and survives (with a probability of 1 − μe), it can obtain a dominant breeding position (with a probability of ae) and gain the fitness benefits associated with dominance ( WD¯). Consequently, the fitness payoff to a subordinate who gets evicted (WSE) is: WSE(h|p^, h^)=(1−μe)(aeWD (p^, h^)¯− es) (2) where es is the cost of being evicted for the subordinate (e.g. physical injury, stress). This equation implicitly states that evicted subordinates who do not attain a dominant position will die. Therefore, the total fitness for a subordinate (WS) is: WS(h|p^,h^)=peWSE(h|p^,h^)+(1−pe)(WSS(h|p^,h^)) (3) which is determined by the probability and associated fitness of staying in a group or being evicted from the group. The probability function determining the likelihood of subordinate eviction is specified as pe=12+12*tanh((p−h)*k), where the probability of subordinate eviction increases as the level of helping behavior (h) falls below the threshold of eviction (p). Using this sigmoid probability function allows for the possibility that dominants do not have perfect information or may make errors by evicting subordinates that are helping sufficiently or retaining subordinates that are lazy (Hamilton and Taborsky 2005). At the point where h=p, the probability of eviction is 50%. The parameter k determines the steepness of the curve at the inflection point, such that higher values of k decrease the probability of getting evicted if h>p (see Supplementary Figures 1 and 2 for further information on k). Both h and p range from 0 to 1. For dominants, subordinate membership in the group raises the productivity of the group (G) by a measure proportional to the amount of help subordinates provide. Consequently, 1 unit of help has a larger impact on the productivity of a highly productive group with many offspring relative to a less productive group. Forms of help that increase survival of offspring and are shareable among offspring (e.g. defense of a brood against nest predators) might be expected to have this form because each offspring has its chance of surviving to reproduction increased by some amount. In addition to helping benefits of subordinates, subordinates are also costly: we assume subordinates impose general costs on dominants (s; e.g. competition for food, reproduction). Further, dominants face a cost of demanding help ( p*rd), such that as dominants demand more help, they face an increased cost of having subordinates remain in the group (i.e. a resistance cost to demanding help via aggression). Therefore, the fitness of a dominant whose subordinate stays in the group is: WDS(p|p^,h^)=(1−m)((1+h)G−s−p*rd) (4) where m is the mortality rate of the dominant. If the dominant evicts the subordinate, its fitness is dependent upon the cost of eviction for the dominant (ed) as well as the probability that it is able to regain the help lost by evicting a subordinate (σ). We assume that dominants can recoup this lost help via another subordinate already in the group (such that small groups will have a lower probability of compensating for this loss than larger groups) or via accepting another subordinate in the group from a floater population of subordinates. The expected helping behavior of the replacement subordinate is simply the population-level value of helping ( h^). If the dominant evicts the subordinate and does not replace the evicted subordinate (with probability 1 − σ), dominant fitness is determined by the probability that it survives (1 − m), the benefits it receives from being a dominant in a group without the subordinate (G), and the costs it paid to evict the subordinate (e.g. aggression; ed). If the dominant does replace the evicted subordinate, we assume that the dominant can also evict the replacement subordinate. Consequently, if the dominant replaces the evicted subordinate (with probability σ), its fitness is merely the average fitness of dominants in the population ( WD¯). Therefore, the fitness of the dominant that evicts subordinates is recursive, as it is dependent upon the average dominant fitness in the population, assuming the population average level of helping behavior h^: WDE (p|p^,h^)= WD(p^,h^)¯+(1−σ)(1−m)(G− ed) (5) Dominant fitness, in overall, is determined by the probability and associated fitness of accepting or evicting the subordinate from the group: WD(p|p^,h^)= peWDE(p|p^,h^)+(1−pe)(WDS(p|p^,h^)) (6) In this model, there are 2 targets of selection: h, the amount of help provided, is under selection for the subordinates, and p, the threshold of helping behavior at which subordinates get evicted, is under selection for dominants. We numerically solve for the equilibrium values of these variables (using the ode45 solver in Matlab R2014b) by finding the values of h and p that satisfy the following conditions: ∂WS∂h|h=h^=h*,p=p^=p*=0 ∂WD∂p|h=h^=h*,p=p^=p*=0 We ran the model across all biologically realistic scenarios, as some parameter space (e.g. when subordinate probability of mortality is very high or when subordinate probability of becoming dominant is very low) results in subordinate fitness nearing 0. We solved for the evolutionary stability of this value by evaluating whether h* and p* satisfy the following conditions: ∂2WS∂h2|h=h^=h*,p=p^=p*<0 ∂2WD∂p2|h=h^=h*,p=p^=p*<0 We also solved for convergence stability by evaluating whether both eigenvalues of the Jacobian matrix, J, had negative real parts, where: J= [ ∂2WS∂h2|h=h^=h*,p=p^=p* ∂2WS∂h∂p|h=h^=h*,p=p^=p*∂2WD∂p∂h|h=h^=h*,p=p^=p*∂2WD∂p2|h=h^=h*,p=p^=p*] We found that the system is a convergence stable strategy (CSS), but is technically not an evolutionarily stable strategy (ESS: ∂2WS∂h2|h=h^=h*,p=p^=p*<0 but ∂2WD∂p2|h=h^=h*,p=p^=p* is approximately 0). The system can drift slightly away from the fixed point, but remains very close to the fixed point because as soon as it drifts away, there is strong selection to return. Results: helping and eviction dynamics without plasticity Both helping behavior and the threshold of eviction are highest when the dominant’s probability of replacing the subordinate (σ) is 0, and both decrease as σ increases, until the threshold of eviction reaches 0 at moderate values of σ (Figure 1: λp and λh = 0). Probability of subordinate eviction remains near 0 at all values of σ (subordinate eviction is highly probable when the threshold of eviction exceeds subordinate helping behavior and unlikely when helping behavior exceeds the eviction threshold). Both helping behavior and the threshold of eviction increase as cost of the subordinate to the dominant (s) increases (Figure 2: λp and λh = 0). Despite the fact that subordinate help increases with increasing s, subordinates provide a net fitness loss to the dominant across almost all parameter space because s is nearly always greater than subordinate help (with the exception of when the group productivity is low; Supplementary Figure 3). As group productivity (G) increases, both the threshold of eviction and subordinate help decrease such that dominants are more demanding and subordinates are more helpful in less productive groups (Supplementary Figure 3). Figure 1 View largeDownload slide In the single sealed-bid model without plasticity (black), as well as negotiation models (blue and red; h# and p# shown), both subordinate help (solid line) and threshold of eviction (dashed line) decrease as the probability that the dominant can replace an evicted subordinate (σ) increases. Parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 1 View largeDownload slide In the single sealed-bid model without plasticity (black), as well as negotiation models (blue and red; h# and p# shown), both subordinate help (solid line) and threshold of eviction (dashed line) decrease as the probability that the dominant can replace an evicted subordinate (σ) increases. Parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 2 View largeDownload slide The level of subordinate helping (solid line) and threshold of eviction (dashed line) across all values of s, the cost of the subordinate to the dominant. Subordinate help and eviction threshold are intermediate in our single sealed-bid model without plasticity (black line) relative to models where λh < 0 and λp < 0 (blue line: h# and p# shown) and λh > 0 (red line: h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, σ = 0.1, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 2 View largeDownload slide The level of subordinate helping (solid line) and threshold of eviction (dashed line) across all values of s, the cost of the subordinate to the dominant. Subordinate help and eviction threshold are intermediate in our single sealed-bid model without plasticity (black line) relative to models where λh < 0 and λp < 0 (blue line: h# and p# shown) and λh > 0 (red line: h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, σ = 0.1, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Subordinate helping behavior is not altered by its opportunities to breed and survive outside of the group: helping behavior does not vary with the probability of a subordinate becoming dominant if it is evicted or remains in the group (ae and ag, respectively: Figure 3A; λp and λh = 0) nor does it vary with the mortality rate of evicted subordinates or subordinates in a group (μe and μg, respectively: Figure 3B; λp and λh = 0). Helping behavior and the threshold of eviction are relatively unchanging due to dominant mortality, m, or due to the cost of helping to the subordinate, c (Supplementary Figures 4 and 5). Subordinate help and the threshold of eviction decrease as it becomes costlier for dominants to evict subordinates (ed), although helping behavior and the threshold of eviction do not vary with the value of the immediate cost of eviction for subordinates (es; Supplementary Figure 6). Figure 3 View largeDownload slide Values of helping behavior and eviction threshold as they vary with subordinate mortality in the group (μg) and after eviction (μe) as well as the probability that a subordinate becomes dominant in the group (ag) and after eviction (ae). Subordinate opportunities to breed after eviction (μe, ae) do not alter helping and eviction dynamics in the single sealed-bid model (A, B), but when subordinates can plastically increase their help in response to demanding dominants (negotiation model with λh < 0 and λp < 0), subordinates provide less help as prospects outside of the group improve (lower μe, higher ae; C, D; h# and p# shown). Subordinate ability to become dominant if it remains within the group does not alter helping and eviction across the reported values of ag and μg in the single sealed-bid model (A, B), but in our model with plasticity, subordinates provide more help as prospects within the group improve (lower μg, higher ag; C, D; h# and p# shown) when λh < 0 and λp<0. For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, σ = 0.1, rs = 0, and rd = 0. ag and ae, as well as μg and μe, vary independently of each other. Figure 3 View largeDownload slide Values of helping behavior and eviction threshold as they vary with subordinate mortality in the group (μg) and after eviction (μe) as well as the probability that a subordinate becomes dominant in the group (ag) and after eviction (ae). Subordinate opportunities to breed after eviction (μe, ae) do not alter helping and eviction dynamics in the single sealed-bid model (A, B), but when subordinates can plastically increase their help in response to demanding dominants (negotiation model with λh < 0 and λp < 0), subordinates provide less help as prospects outside of the group improve (lower μe, higher ae; C, D; h# and p# shown). Subordinate ability to become dominant if it remains within the group does not alter helping and eviction across the reported values of ag and μg in the single sealed-bid model (A, B), but in our model with plasticity, subordinates provide more help as prospects within the group improve (lower μg, higher ag; C, D; h# and p# shown) when λh < 0 and λp<0. For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, σ = 0.1, rs = 0, and rd = 0. ag and ae, as well as μg and μe, vary independently of each other. When only subordinates pay resistance costs (rs: punishment and aggression when subordinate help nears the eviction threshold), the threshold of eviction decreases as resistance costs increase. Because the level of subordinate help remains stable, this results in a greater difference between helping behavior and the threshold of eviction when resistances costs are high (Figure 4A). In contrast, subordinate help and the threshold of eviction both decrease when only dominants pay resistance costs (rd: cost of enforcing high help) or when both dominants and subordinates pay resistance costs (Figure 4A). For all reported results above, the probability of eviction remains low across all parameter space. Model 2: helping and eviction dynamics with plasticity Here, we expand the previously described model by allowing both players to respond plastically to the strategy of their partner. In real-world systems, dominant aggression and subordinate helping/submission likely serve as signals indicating subordinate willingness to help and dominant threshold of eviction, such that dominants and subordinates have information about their partner’s behavior and willingness to disband the group (Cant and Johnstone 2009). We use a negotiation framework similar to that presented in Taylor et al. (2006), which incorporates an individual’s responsiveness (λ): the degree to which the dominant or subordinate responds to a change in the offer of its partner. During the interaction, one player chooses its strategy and its partner responds with its action, which is determined by the first player’s strategy and its partner’s responsiveness. The first player then responds to its partner’s strategy and this process repeats until both players converge on a final set of strategies h# and p#, which are determined by the set of equations: h#= h0−p0λh1−λpλh (7) p#= p0−h0λp1−λpλh (8) In this framework, the dominant chooses its baseline level of eviction (p0) and the subordinate chooses its baseline level of help (h0). Because the equation for h# includes h0−p0λh, negative values of λh results in subordinates providing more help as dominants increase their demand (at least when λp = 0; akin to pay-to-stay dynamics). Similarly, because the equation for p# includes p0−h0λp, negative values of λp results in dominants demanding relatively more help from helpful subordinates than from lazy subordinates (at least when λh = 0; akin to escalating demand). We do not solve for responsiveness of the dominant (λp) and the subordinate (λh); these are specified in the model as additional parameters, where convergence requires that |λpλh| < 1. Rather than solving for the equilibrium values of h and p as in Model 1, we substitute h and p with h# and p# and find the values of h# and p# by solving for the values of p0 and h0 that satisfy the following conditions: ∂WS∂h0|h0=h0^=h0*, p0=p0^=p0*=0 ∂WD∂p0|h0=h0^=h0*, p0=p0^=p0*=0 For most parameter space, the model converges on a solution regardless of the starting parameters; however, for a small subset of parameter space (notably, for some parameter combinations when λh > 0 and/or λp < 0) the model only converges on a stable solution when the starting parameters are either near 0 or near 1. There appears to be another, unstable equilibrium point at different starting values of p0 and h0 for this parameter space. The model results presented below represent the stable solutions. Here, we again solved for evolutionary and convergence stability as in Model 1. When λh > 0 (regardless of the value of λp), the system is not ESS, but is CSS; consequently, it will wander away from the fixed point and return. When λh < 0 (regardless of the value of λp), the system is both ESS and CSS and remains at the fixed point. This is similar to Hamilton (2013), which found that the system was not ESS unless there was flexibility in cooperative strategies. Results: helping and eviction dynamics with plasticity When both λh < 0 (subordinates respond to high dominant demands with more help) and λp < 0 (dominants demand more help from helpful subordinates), subordinate help and the threshold of eviction are generally higher than in the single sealed-bid model (Figures 1 and 2). For example, although subordinate costs to dominants (s) are higher than the level of helping behavior in the original model (providing a fitness loss to dominants), helping behavior is greater than subordinate costs to dominants when λh < 0 and λp < 0 (Figure 2). Further, the effects of multiple model parameters differ substantially from the single sealed-bid model. In the original single sealed-bid model, subordinate opportunities inside and outside the group did not alter helping behavior and eviction thresholds. However, when λh < 0 and λp < 0, both helping behavior and the threshold of eviction increase as subordinate prospects inside the group improve (i.e. mortality inside the group (μg) decreases and probability of becoming a dominant in the group (ag) increases; Figure 3C and D). Conversely, as subordinate prospects outside the group improve (i.e. mortality outside the group (μe) decreases and probability of becoming a dominant after eviction (ae) increases), both helping behavior and the threshold of eviction decrease (Figure 3C and D). However, when subordinate opportunities within the group are always better than subordinate opportunities outside the group (e.g. ae and μe vary as a proportion of ag and μg, such that inside and outside options improve together), helping and eviction thresholds increase as both subordinate prospects inside and outside the group improve. As dominants face mounting costs of enforcing high help (rd), subordinate help and eviction thresholds decrease in both the original model and the negotiation model (Figure 4B); however, this is not the case for subordinate costs of resisting dominant demand (rs). When λh < 0 and λp < 0, increased subordinate resistance costs actually result in increased subordinate help and eviction thresholds; this is the case when both dominants and subordinates have resistance costs as well as when only subordinates have resistance costs, although the slope of the increase is higher when only subordinates have resistance costs (Figure 4B). Helping behavior and the threshold of eviction decrease as the cost of helping increases (Supplementary Figure 3) and as dominant mortality increases (Supplementary Figure 5). Subordinate help and the threshold of eviction increase as the cost of subordinate eviction (es) increases (Supplementary Figure 6). Figure 4 View largeDownload slide Values of helping behavior and eviction threshold when dominants have costs of enforcing help (rd: blue), when subordinates have costs of resisting eviction (rs: red), or when both dominants and subordinates have resistance costs (black). Results are shown for the original model (A) and the negotiation model with λh < 0 and λp<0 (B; h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, and σ = 0.1. Figure 4 View largeDownload slide Values of helping behavior and eviction threshold when dominants have costs of enforcing help (rd: blue), when subordinates have costs of resisting eviction (rs: red), or when both dominants and subordinates have resistance costs (black). Results are shown for the original model (A) and the negotiation model with λh < 0 and λp<0 (B; h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, and σ = 0.1. When λh < 0 with λp = 0 or λp > 0, all of the aforementioned patterns hold and subordinate help and the threshold of eviction are generally higher than when λh < 0 and λp < 0. However, when only dominants are plastic (λp < 0 and λh = 0), the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. Similarly, when λh > 0 (subordinates help less when dominant demand is high), regardless of the value of λp, the patterns of all model parameters are similar to those in the model without plasticity and both subordinate help and the threshold of eviction are generally lower than the single sealed-bid model (Figures 1 and 2). DISCUSSION In these models, subordinates control the amount of help that they provide and dominants specify a threshold of eviction, such that groups will usually disband when subordinate help is below the threshold of eviction and usually stay together when helping is higher than the eviction threshold. Our model demonstrates that outside options for dominants and subordinates alter both helping and eviction dynamics. Specifically, we found that dominants demand more help and subordinates provide more help when dominants are less likely to recoup the lost help of evicted subordinates (σ), which is likely the case when groups are small (such that other subordinates cannot compensate for the evicted helper) or when groups have no neighbors (such that a group cannot recruit new helpers). In empirical systems, we would therefore expect dominants to more readily punish lazy subordinates in small groups or groups with no neighbors. These model results are in agreement with the results of Fischer et al. (2014), who experimentally prevented subordinates in the cooperatively breeding cichlid Neolamprologus pulcher from helping. They found that subordinates were more likely to be evicted from smaller groups compared to larger groups and that subordinates in small groups, but not large groups, increased help once they were allowed to help again. Collectively, this demonstrates that both dominants and subordinates respond to a dereliction of helping behavior more strongly when dominants are less able to compensate for the loss of an evicted subordinate. However, these model results and the results reported by Fischer et al. (2014) are in contrast to Kutsukake and Clutton-Brock (2008), who found that dominant meerkats are less aggressive to unrelated subordinates when groups are smaller. Further, Hellmann and Hamilton (2014) found that, when neighboring groups are present, subordinates provide more help rather than less help. Consequently, further empirical and theoretical work is needed to understand when and to what extent dominant eviction is altered by dominant dependence on current subordinates. Negotiation games, which allow individuals to adjust their strategy in response to their partner’s optimal strategy, can produce different dynamics than traditional games in which partners lack information about each other’s strategy (Taylor et al. 2006; Cant and Young 2013; Hamilton 2013; Quiñones et al. 2016). Consequently, we expanded our single sealed-bid model to incorporate plastic responses to partners in order to understand if negotiation between dominants and subordinates would produce helping behavior and eviction dynamics similar to those observed in empirical systems. We found that, when subordinates adopt a strike scenario (λh > 0), subordinates provide less help at equilibrium than they do in the original model, but that this point is not evolutionarily stable. However, subordinate strike strategies appear to be rare in group-living species, as evidenced by many studies demonstrating that subordinates actually increase helping behavior or submission in response to higher dominant demands (pay-to-stay dynamics in cooperative breeders: Mulder and Langmore 1993; Balshine-Earn et al. 1998; Cockburn 1998; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008; Fischer et al. 2014; Hellmann et al. 2015b). It is possible, however, that subordinate strike scenarios may exist in species where 1) dominants are willing to tolerate nonhelping subordinates because dominants derive other benefits (increased survival: Kokko and Johnstone 1999; assurance of future mate: Fricke and Fricke 1977) from their presence and/or 2) it is costly for dominants to evict subordinates, such that the cost of evicting subordinates (e.g. aggression) outweighs the cost of having a nonhelping subordinate. Further exploration, both theoretically and empirically, would be useful for understanding when and how frequently this strategy is likely to evolve and the dynamics of this strategy near the equilibrium point. When we modeled negotiation dynamics that are akin to pay-to-stay dynamics (λh < 0, where subordinates respond to high dominant demands with more help), we find that helping and eviction dynamics differ substantially from the single-bid model without plasticity. In the single-bid model, subordinate outside options do not strongly influence help or eviction. In our negotiation model, subordinate helping behavior and eviction threshold decrease as subordinate prospects outside the group improve whereas helping behavior and eviction threshold increase as subordinate prospects within the group improve. These results are in line with empirical observations that subordinate helpers reduce the amount of help they provide when opportunities to breed outside the group are likely (Bergmüller et al. 2005; Young et al. 2005; Tibbetts 2007; Zöttl et al. 2013a), because threats of leaving are more credible when alternative options outside of the group are profitable (Cant and Johnstone 2009). Similarly, these results are also in agreement with theoretical predictions that as subordinate benefits of the current association increase relative to other options, subordinates may have less ability to negotiate based on their outside option (Buston and Zink 2009; Cant and Johnstone 2009; Cant 2011) and dominants should be able to demand more help in exchange for continued partnership (Cant 2011). Consequently, our model demonstrates that pay-to-stay requires that subordinates can plastically increase their help in response to increasing dominant demand. Interestingly, when dominants adopt an escalating demand strategy in conjunction with subordinate pay-to-stay (λp < 0, λh < 0), subordinates provide less help than when subordinates alone were plastic in their strategy (λp = 0, λh < 0), although they still help more than in the original model without plasticity. Similarly, if dominants adopt an escalating demand strategy without subordinate plasticity (λp < 0, λh = 0), model dynamics are similar to those in the original model and subordinates provide less help than they did in the original model. It is unclear how frequent escalating demand strategies are in nature, but the fact that there is often constant aggression against subordinates in cooperatively breeding species, regardless of help provided, suggests that this strategy may persist because subordinates respond to threats of punishment (Taborsky 2016) and because there is consistent individual variation in the propensity to help. As there is much empirical evidence demonstrating that individuals are relatively consistent in their willingness to cooperate (Bergmüller et al. 2010; English et al. 2010; Le Vin et al. 2011), demanding more help from helpful subordinates, rather than from individuals who are consistently less willing to cooperate, may be a successful strategy for promoting subordinate cooperation. Consequently, future models incorporating heterogeneity in partner phenotype—in terms of dominant demand for help, subordinate willingness to help, and subordinate responsiveness to dominant threats—would elucidate the extent to which variation in cooperative tendencies underlie observed empirical observations. In our negotiation model with pay-to-stay dynamics, both helping behavior and eviction thresholds are altered by costs to dominants of maintaining a high threshold of eviction and subordinate costs (e.g. aggression) of providing help that is close to the dominant threshold of eviction. Namely, we found that when only dominants face resistance costs, dominants demand less help from subordinates. In contrast, when both dominants and subordinates face resistance costs, subordinates are more helpful than when resistance costs are absent. Given that it is likely that both dominants and subordinates would simultaneously incur costs of enforcing help and resisting eviction, namely the cost of giving and receiving aggression, it is likely that these resistance costs will largely increase subordinate cooperation in pay-to-stay systems. Similar to Hamilton and Taborsky (2005), we found that unrelated helpers rarely compensate for the cost that they impose on dominants (e.g. reproduction: Clutton-Brock et al. 2001 and Hellmann et al. 2015a; food: Wittig and Boesch 2003), as the benefits of subordinate help in our model were nearly always lower than subordinate costs to dominants in our single-bid model. However, in our negotiation model with pay-to-stay dynamics, we found that subordinates helped consistently more than the costs that they imposed on dominants. This suggests that pay-to-stay dynamics alone, irrespective of other benefits of helping, can enforce subordinate help in cooperatively breeding species and, remarkably, produce higher levels of cooperation than kin selection alone (Quiñones et al. 2016). However, given that male and female dominants can differ in the costs of having subordinates, notably in terms of reproductive competition (Heg et al. 2008), additional theoretical and empirical work would be useful to understand how pay-to-stay dynamics regulating helping behavior differ based on sex. Our results provide a strong argument for examining the role of plasticity and negotiation in the evolution of cooperation among group members. This is in agreement with Quiñones et al. (2016), who found that subordinate help evolves from negotiation strategies between unrelated partners. Our model with plasticity in subordinate strategy that mirrored pay-to-stay dynamics produces results consistent with what would be expected with previous theory (Cant and Johnstone 2009; Cant 2011) and with previous empirical results (Bergmüller et al. 2005; Young et al. 2005; Tibbetts 2007; Zöttl et al. 2013a). Given these results, we predict that plasticity in dominant or subordinate behavior reduces or increases, respectively, the amount of help subordinates provide in empirical systems. Further, we expect that subordinate outside options should only influence subordinate help in systems with pay-to-stay mechanisms of cooperation. Past laboratory and field experiments in which subordinates have been experimentally prevented from helping (Balshine-Earn et al. 1998; Fischer et al. 2014; Hellmann et al. 2015b) or where demands of help were experimentally varied (Taborsky 1985; Zöttl et al. 2013c) have been successful in triggering dominant punishment and measuring the threshold at which dominants are willing to punish or evict subordinates. Combining these experiments with manipulations of dominant punishment, perceived competition from outside the group, and variation in the ease of dominants to replace evicted subordinates are also expected to shed light on the extent to which subordinates and dominants alter their behavior based on the relative leverage that they have in a given social situation. FUNDING This work was funded by The Ohio State University. Data accessibility: Analyses reported in this article can be reproduced using the data provided by Hellmann and Hamilton (2018). 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Kinship reduces alloparental care in cooperative cichlids where helpers pay-to-stay . Nat Commun . 4 : 1341 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Behavioral Ecology Oxford University Press

Dominant and subordinate outside options alter help and eviction in a pay-to-stay negotiation model

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Oxford University Press
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© The Author(s) 2018. Published by Oxford University Press on behalf of the International Society for Behavioral Ecology. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
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Abstract

Abstract In several cooperatively breeding species, subordinates that do not help sufficiently are punished or evicted from the group by dominant individuals. The credibility of dominant eviction threats may vary with the social context beyond the group level: when subordinates can easily breed in a neighboring territory, dominants may be less able to demand help from subordinates. Further, dominant ability to enforce subordinate cooperation may be reduced when it is difficult to replace evicted subordinates or in small groups where each subordinate makes a large contribution to group productivity. Here, we develop a 2-player game theoretic model to examine how the social context influences subordinate help and the threshold of help at which dominants evict subordinates. In contrast to predictions, we found that dominants demand more help when dominants are less able to replace evicted subordinates, suggesting that dominants punish a dereliction of helping behavior more strongly when they are unable to compensate for the loss of an evicted subordinate. In single sealed-bid games, subordinates help less than the fitness costs they impose on dominants and help does not vary with subordinate breeding opportunities outside the group. However, when subordinates can plastically increase help in response to demanding dominants (akin to pay-to-stay dynamics), subordinates provide more help overall, but decrease their help as breeding opportunities outside of the group increase. Our results demonstrate the importance of incorporating negotiation into theoretical models of helping strategies and demonstrate that plasticity is a key mechanism underlying pay-to-stay mechanisms of cooperation. INTRODUCTION The evolution and stability of helping behavior in group-living species has received a great deal of attention in the last 50 years. Specifically, questions center around why subordinates help, since helping behavior is often energetically costly and can reduce survival, growth, and future reproduction (Dickinson and Hatchwell 2004; Clutton-Brock 2006; Biedermann et al. 2011). Kin selection has been suggested as the primary mechanism regulating the evolution of helping in group-living birds and mammals (Russell 2004). However, in many species, subordinates that are unrelated to dominants also provide help (Cockburn 1998; Clutton-Brock 2009; Riehl 2013; Taborsky 2016) or kinship reduces the amount of help provided by subordinates (Stiver et al. 2005; Doutrelant et al. 2011; Zöttl et al. 2013d), suggesting that kin selection is not a mechanism underlying subordinate helping behavior in all taxa or in all social contexts (Taborsky et al. 2016). Another suggested mechanism underlying the evolution of helping behavior is pay-to-stay (Gaston 1978; Kokko et al. 2002; Bergmüller et al. 2007), in which subordinates provide help as a means to compensate for the costs, primarily in terms of competition for reproduction and resources, that they inflict upon dominants by remaining on the territory. The predictions of pay-to-stay have been supported in cichlid fish (Taborsky 1985; Balshine-Earn et al. 1998; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008; Zöttl et al. 2013d; Fischer et al. 2014; Hellmann et al. 2015b), paper wasps (Grinsted and Field 2017), and fairy wrens (Mulder and Langmore 1993; Cockburn 1998) and may be a potential mechanism underlying cooperation in other species of birds (Cockburn 1998; Leighton and Meiden 2016), fishes (Wong et al. 2007), mole rats (Reeve 1992), and meerkats (MacLeod et al. 2013). Under the pay-to-stay theory, subordinates that do not help sufficiently are punished or evicted from the group by more dominant individuals. Although resource holding potential, or the ability of an individual to use force within a dyadic relationship, strongly influences the ability of dominants to credibly exercise threats, it is becoming increasingly clear that a dominant’s ability to punish subordinates is also context-dependent (Lewis 2002; Cant and Johnstone 2009; Cant 2011). By helping on their current territory, subordinates often gain direct benefits from group membership or the potential to become a breeder on a high-quality territory (Riehl 2013); however, if subordinates have many opportunities to breed outside of their current group, they may be less likely to tolerate high demands for help in their current group relative to subordinates who have few options to breed or survive outside of the group, because eviction may be less costly to subordinates who will likely survive and breed successfully after being evicted (Bergmüller et al. 2005; Grinsted and Field 2017). Conversely, because dominants often derive fitness benefits (e.g. increased offspring survival, reduced workload) from subordinate help (Dickinson and Hatchwell 2004; Russell 2004; Johnstone 2011; Zöttl et al. 2013b), dominants may have less leverage to enforce subordinate cooperation when replacing evicted subordinates is difficult or if each subordinate makes a large contribution to group productivity (Cant and Johnstone 2006; Johnstone and Bshary 2008). Consequently, the social context likely alters the credibility of dominant eviction threats and, thus, the ability of subordinates to reduce the amount of help they provide without facing punishment. There is some empirical evidence demonstrating that helping behavior and eviction dynamics are influenced by the social context. Subordinates increase helping behavior when outside competition for their position in the group is higher (Bruintjes and Taborsky 2008) and when neighboring groups are present compared to when they are absent (Hellmann and Hamilton 2014). Furthermore, when subordinates are experimentally prevented from helping, dominants evict subordinates more frequently when there are more neighboring groups in close proximity, when dominants are potentially better able to replace evicted subordinates with subordinates from neighboring groups (Hellmann et al. 2015b). Finally, dominant tolerance of subordinates is regulated by the need for subordinate help: subordinates are evicted when subordinate help is not needed (Taborsky 1985), but they are accepted into the group when dominant need for subordinate help is high (Taborsky 1985; Zöttl et al. 2013c). These results demonstrate that a competitive social environment alters both subordinate help and dominant eviction thresholds and that subordinate eviction is more likely in cases when dominants are less reliant on subordinate help. However, experimental evidence that helping behavior is regulated by the cost of punishment or eviction for dominants is limited (but see Mulder and Langmore 1993; Fischer et al. 2014; Hellmann et al. 2015b). This is in part because experiments examining eviction dynamics are difficult to conduct: the most effective threats (i.e. eviction) are rarely triggered because subordinates in stable groups use both submissive and/or helping behavior to prevent eviction (Bergmüller et al. 2005; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008). Therefore, theoretical models can generate valuable predictions regarding eviction and helping dynamics that can advance our understanding of how punishment can alter group dynamics, as well as allow us to generate predictions that can be tested empirically across a variety of taxa with different social systems. Here, we present a 2-player model in which we examine how the social context influences the degree of help provided by a single subordinate and the threshold of helping behavior at which a dominant is willing to evict that subordinate. Although a past model (Hamilton and Taborsky 2005) explored how subordinate outside options alter helping and eviction dynamics, here we expand this model by examining helping and eviction dynamics relative to outside options for dominants (i.e. the ability of dominants to recoup the lost help of evicted subordinates) in addition to outside options for subordinates (i.e. the ability of evicted subordinates to become dominant or survive outside of a group). Furthermore, past game theoretic models examining helping and reproductive dynamics among group members have been single sealed-bid games, in which dominants and subordinates cannot respond to each other’s strategy (e.g. Cant and Field 2001; Kokko et al. 2002; Hamilton and Taborsky 2005). However, as there is evidence demonstrating that negotiation and plasticity in strategies are highly important in determining the degree of cooperation among partners (Taylor et al. 2006; Hamilton 2013), particularly for cooperation enforced by pay-to-stay (Raihani et al. 2012; Quiñones et al. 2016), we expand on traditional single sealed-bid games to understand how plasticity in either the dominant or subordinate strategy alters helping and eviction dynamics. Model 1: helping and eviction dynamics without plasticity Our game theoretical model consists of 2 classes of players: dominant and subordinate group members who are unrelated. Therefore, subordinates receive no kin-selected benefits of helping and dominants suffer no indirect fitness costs of evicting a subordinate. For the purposes of this model, we assume that subordinates who leave the group voluntarily or are evicted from the group cannot join a new group after leaving their current group; in essence, leaving or eviction from a group results in the subordinate either becoming a dominant in a different group or dying. This assumption was included in part to simplify recursive equations, but it is a realistic assumption in many group-living species where subordinates are large, such that acceptance into a new group is difficult because subordinates that are close in size to dominants are a reproductive threat to dominants and tend to be in conflict with dominants (Hamilton et al. 2005; Wong et al. 2007).Both parties may have a shared incentive to resolve conflict via the benefits of remaining in a productive group. For subordinates, the benefits of remaining in the group are 2-fold: the mortality rate of living in a group (μg) is lower than the mortality rate of leaving the group or being evicted from the group (μe). Additionally, the likelihood of becoming a dominant when subordinates live in a group (ag) is higher than the likelihood of becoming a dominant if evicted from the group (ae). We make this assumption because subordinates in many group-living species inherit their current territory when the current dominants die (Buston 2004; Stiver et al. 2006; Cant et al. 2016; Clutton-Brock and Manser 2016). Consequently, when a subordinate is part of a group, it can both inherit its current territory as well as take over the territory of another group; however, if a subordinate is evicted, it loses the opportunity to inherit its current territory. We assume that all rates and probabilities (e.g. mortality rates, territory inheritance) are time-independent and therefore, do not increase over time. If a subordinate who remains in the group survives (with a probability 1 − μg), it can obtain a dominant breeding position (with a given probability ag) and gain the fitness benefits associated with dominance ( WD¯), which is determined by the average fitness of dominants in the population given population-level values for subordinate help and dominant eviction thresholds ( h^, p^; Table 1).If the subordinate does not become dominant, it gains the fitness associated with staying a subordinate in a group (WSS). In either case, the subordinate pays a cost of helping and remaining in the group. Therefore, the fitness of a subordinate who remains in the group (WSS) is: Table 1 A list of parameters used in models and how they alter helping behavior and eviction thresholds in the single sealed-bid (h, p) and in the model with plasticity (h# and p#, which are determined by baseline levels of helping (h0) and eviction (p0) as well as individual responsiveness (λ)) Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase The patterns reported for the pay-to-stay plasticity (λh < 0: subordinates increase help in response to demanding dominants) hold regardless of the value of dominant responsiveness, λp. When only dominants are plastic (λp < 0 and λh = 0), and when λh > 0, the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. View Large Table 1 A list of parameters used in models and how they alter helping behavior and eviction thresholds in the single sealed-bid (h, p) and in the model with plasticity (h# and p#, which are determined by baseline levels of helping (h0) and eviction (p0) as well as individual responsiveness (λ)) Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase Parameters Single sealed-bid model Pay-to-stay plasticity: λ h < 0 Opportunities to breed inside and outside the group σ Probability dominant replaces evicted subordinate As σ increases, h and p decrease As σ increases, h# and p# decrease μe Subordinate mortality outside group h and p relatively steady h# and p# decrease as subordinate prospects outside the group improve (lower μe, higher ae) ae Probability subordinate becomes dominant if evicted μg Subordinate mortality in group h and p relatively steady h# and p# increase as subordinate prospects inside the group improve (lower μg, higher ag) ag Probability subordinate becomes dominant in group Costs associated with subordinate presence in the group s Cost of subordinate to dominants h and p increase h# and p# increase c Cost of helping to subordinate h and p increase slightly h# and p# decrease rd Dominant resistance cost to having subordinates h and p decrease h# and p# increase or decrease depending on rs rs Subordinate resistance cost to remaining in group h decreases or is unchanging depending on rd, p decreases h# and p# increase Other ed Cost of eviction to dominant h and p decrease h# and p# decrease es Cost of eviction to subordinate h and p unchanging h# and p# increase m Dominant mortality h and p increase slightly h# and p# decrease k Steepness of eviction probability curve h unchanging, p increases slightly h# unchanging, p# increases slightly G Group productivity h and p decrease h# and p# decrease, then increase The patterns reported for the pay-to-stay plasticity (λh < 0: subordinates increase help in response to demanding dominants) hold regardless of the value of dominant responsiveness, λp. When only dominants are plastic (λp < 0 and λh = 0), and when λh > 0, the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. View Large WSS(h|p^, h^)=(1−μg)[agWD (p^,h^)¯+ (1−ag)WSS(h|p^, h^)−hc−phrs] (1) where h is the amount of help provided by the subordinate, c is the cost of helping, and phrs represents a cost of subordinate resistance to dominant eviction attempts, such that as the difference between subordinate help (h) and dominant eviction thresholds (p) decreases (i.e. when subordinate help is closer to the eviction threshold), subordinates face a mounting cost to remaining in the group (e.g. increased aggression from the dominant). This equation rearranges to: WSS (h|p^,h^)= (1−mg)(agWD (p^,h^)¯−hc−phrs)ag+ mg−mgag If a subordinate gets evicted and survives (with a probability of 1 − μe), it can obtain a dominant breeding position (with a probability of ae) and gain the fitness benefits associated with dominance ( WD¯). Consequently, the fitness payoff to a subordinate who gets evicted (WSE) is: WSE(h|p^, h^)=(1−μe)(aeWD (p^, h^)¯− es) (2) where es is the cost of being evicted for the subordinate (e.g. physical injury, stress). This equation implicitly states that evicted subordinates who do not attain a dominant position will die. Therefore, the total fitness for a subordinate (WS) is: WS(h|p^,h^)=peWSE(h|p^,h^)+(1−pe)(WSS(h|p^,h^)) (3) which is determined by the probability and associated fitness of staying in a group or being evicted from the group. The probability function determining the likelihood of subordinate eviction is specified as pe=12+12*tanh((p−h)*k), where the probability of subordinate eviction increases as the level of helping behavior (h) falls below the threshold of eviction (p). Using this sigmoid probability function allows for the possibility that dominants do not have perfect information or may make errors by evicting subordinates that are helping sufficiently or retaining subordinates that are lazy (Hamilton and Taborsky 2005). At the point where h=p, the probability of eviction is 50%. The parameter k determines the steepness of the curve at the inflection point, such that higher values of k decrease the probability of getting evicted if h>p (see Supplementary Figures 1 and 2 for further information on k). Both h and p range from 0 to 1. For dominants, subordinate membership in the group raises the productivity of the group (G) by a measure proportional to the amount of help subordinates provide. Consequently, 1 unit of help has a larger impact on the productivity of a highly productive group with many offspring relative to a less productive group. Forms of help that increase survival of offspring and are shareable among offspring (e.g. defense of a brood against nest predators) might be expected to have this form because each offspring has its chance of surviving to reproduction increased by some amount. In addition to helping benefits of subordinates, subordinates are also costly: we assume subordinates impose general costs on dominants (s; e.g. competition for food, reproduction). Further, dominants face a cost of demanding help ( p*rd), such that as dominants demand more help, they face an increased cost of having subordinates remain in the group (i.e. a resistance cost to demanding help via aggression). Therefore, the fitness of a dominant whose subordinate stays in the group is: WDS(p|p^,h^)=(1−m)((1+h)G−s−p*rd) (4) where m is the mortality rate of the dominant. If the dominant evicts the subordinate, its fitness is dependent upon the cost of eviction for the dominant (ed) as well as the probability that it is able to regain the help lost by evicting a subordinate (σ). We assume that dominants can recoup this lost help via another subordinate already in the group (such that small groups will have a lower probability of compensating for this loss than larger groups) or via accepting another subordinate in the group from a floater population of subordinates. The expected helping behavior of the replacement subordinate is simply the population-level value of helping ( h^). If the dominant evicts the subordinate and does not replace the evicted subordinate (with probability 1 − σ), dominant fitness is determined by the probability that it survives (1 − m), the benefits it receives from being a dominant in a group without the subordinate (G), and the costs it paid to evict the subordinate (e.g. aggression; ed). If the dominant does replace the evicted subordinate, we assume that the dominant can also evict the replacement subordinate. Consequently, if the dominant replaces the evicted subordinate (with probability σ), its fitness is merely the average fitness of dominants in the population ( WD¯). Therefore, the fitness of the dominant that evicts subordinates is recursive, as it is dependent upon the average dominant fitness in the population, assuming the population average level of helping behavior h^: WDE (p|p^,h^)= WD(p^,h^)¯+(1−σ)(1−m)(G− ed) (5) Dominant fitness, in overall, is determined by the probability and associated fitness of accepting or evicting the subordinate from the group: WD(p|p^,h^)= peWDE(p|p^,h^)+(1−pe)(WDS(p|p^,h^)) (6) In this model, there are 2 targets of selection: h, the amount of help provided, is under selection for the subordinates, and p, the threshold of helping behavior at which subordinates get evicted, is under selection for dominants. We numerically solve for the equilibrium values of these variables (using the ode45 solver in Matlab R2014b) by finding the values of h and p that satisfy the following conditions: ∂WS∂h|h=h^=h*,p=p^=p*=0 ∂WD∂p|h=h^=h*,p=p^=p*=0 We ran the model across all biologically realistic scenarios, as some parameter space (e.g. when subordinate probability of mortality is very high or when subordinate probability of becoming dominant is very low) results in subordinate fitness nearing 0. We solved for the evolutionary stability of this value by evaluating whether h* and p* satisfy the following conditions: ∂2WS∂h2|h=h^=h*,p=p^=p*<0 ∂2WD∂p2|h=h^=h*,p=p^=p*<0 We also solved for convergence stability by evaluating whether both eigenvalues of the Jacobian matrix, J, had negative real parts, where: J= [ ∂2WS∂h2|h=h^=h*,p=p^=p* ∂2WS∂h∂p|h=h^=h*,p=p^=p*∂2WD∂p∂h|h=h^=h*,p=p^=p*∂2WD∂p2|h=h^=h*,p=p^=p*] We found that the system is a convergence stable strategy (CSS), but is technically not an evolutionarily stable strategy (ESS: ∂2WS∂h2|h=h^=h*,p=p^=p*<0 but ∂2WD∂p2|h=h^=h*,p=p^=p* is approximately 0). The system can drift slightly away from the fixed point, but remains very close to the fixed point because as soon as it drifts away, there is strong selection to return. Results: helping and eviction dynamics without plasticity Both helping behavior and the threshold of eviction are highest when the dominant’s probability of replacing the subordinate (σ) is 0, and both decrease as σ increases, until the threshold of eviction reaches 0 at moderate values of σ (Figure 1: λp and λh = 0). Probability of subordinate eviction remains near 0 at all values of σ (subordinate eviction is highly probable when the threshold of eviction exceeds subordinate helping behavior and unlikely when helping behavior exceeds the eviction threshold). Both helping behavior and the threshold of eviction increase as cost of the subordinate to the dominant (s) increases (Figure 2: λp and λh = 0). Despite the fact that subordinate help increases with increasing s, subordinates provide a net fitness loss to the dominant across almost all parameter space because s is nearly always greater than subordinate help (with the exception of when the group productivity is low; Supplementary Figure 3). As group productivity (G) increases, both the threshold of eviction and subordinate help decrease such that dominants are more demanding and subordinates are more helpful in less productive groups (Supplementary Figure 3). Figure 1 View largeDownload slide In the single sealed-bid model without plasticity (black), as well as negotiation models (blue and red; h# and p# shown), both subordinate help (solid line) and threshold of eviction (dashed line) decrease as the probability that the dominant can replace an evicted subordinate (σ) increases. Parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 1 View largeDownload slide In the single sealed-bid model without plasticity (black), as well as negotiation models (blue and red; h# and p# shown), both subordinate help (solid line) and threshold of eviction (dashed line) decrease as the probability that the dominant can replace an evicted subordinate (σ) increases. Parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 2 View largeDownload slide The level of subordinate helping (solid line) and threshold of eviction (dashed line) across all values of s, the cost of the subordinate to the dominant. Subordinate help and eviction threshold are intermediate in our single sealed-bid model without plasticity (black line) relative to models where λh < 0 and λp < 0 (blue line: h# and p# shown) and λh > 0 (red line: h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, σ = 0.1, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Figure 2 View largeDownload slide The level of subordinate helping (solid line) and threshold of eviction (dashed line) across all values of s, the cost of the subordinate to the dominant. Subordinate help and eviction threshold are intermediate in our single sealed-bid model without plasticity (black line) relative to models where λh < 0 and λp < 0 (blue line: h# and p# shown) and λh > 0 (red line: h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, σ = 0.1, ag = 0.5, ae = 0.25, rs = 0, and rd = 0. Subordinate helping behavior is not altered by its opportunities to breed and survive outside of the group: helping behavior does not vary with the probability of a subordinate becoming dominant if it is evicted or remains in the group (ae and ag, respectively: Figure 3A; λp and λh = 0) nor does it vary with the mortality rate of evicted subordinates or subordinates in a group (μe and μg, respectively: Figure 3B; λp and λh = 0). Helping behavior and the threshold of eviction are relatively unchanging due to dominant mortality, m, or due to the cost of helping to the subordinate, c (Supplementary Figures 4 and 5). Subordinate help and the threshold of eviction decrease as it becomes costlier for dominants to evict subordinates (ed), although helping behavior and the threshold of eviction do not vary with the value of the immediate cost of eviction for subordinates (es; Supplementary Figure 6). Figure 3 View largeDownload slide Values of helping behavior and eviction threshold as they vary with subordinate mortality in the group (μg) and after eviction (μe) as well as the probability that a subordinate becomes dominant in the group (ag) and after eviction (ae). Subordinate opportunities to breed after eviction (μe, ae) do not alter helping and eviction dynamics in the single sealed-bid model (A, B), but when subordinates can plastically increase their help in response to demanding dominants (negotiation model with λh < 0 and λp < 0), subordinates provide less help as prospects outside of the group improve (lower μe, higher ae; C, D; h# and p# shown). Subordinate ability to become dominant if it remains within the group does not alter helping and eviction across the reported values of ag and μg in the single sealed-bid model (A, B), but in our model with plasticity, subordinates provide more help as prospects within the group improve (lower μg, higher ag; C, D; h# and p# shown) when λh < 0 and λp<0. For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, σ = 0.1, rs = 0, and rd = 0. ag and ae, as well as μg and μe, vary independently of each other. Figure 3 View largeDownload slide Values of helping behavior and eviction threshold as they vary with subordinate mortality in the group (μg) and after eviction (μe) as well as the probability that a subordinate becomes dominant in the group (ag) and after eviction (ae). Subordinate opportunities to breed after eviction (μe, ae) do not alter helping and eviction dynamics in the single sealed-bid model (A, B), but when subordinates can plastically increase their help in response to demanding dominants (negotiation model with λh < 0 and λp < 0), subordinates provide less help as prospects outside of the group improve (lower μe, higher ae; C, D; h# and p# shown). Subordinate ability to become dominant if it remains within the group does not alter helping and eviction across the reported values of ag and μg in the single sealed-bid model (A, B), but in our model with plasticity, subordinates provide more help as prospects within the group improve (lower μg, higher ag; C, D; h# and p# shown) when λh < 0 and λp<0. For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, σ = 0.1, rs = 0, and rd = 0. ag and ae, as well as μg and μe, vary independently of each other. When only subordinates pay resistance costs (rs: punishment and aggression when subordinate help nears the eviction threshold), the threshold of eviction decreases as resistance costs increase. Because the level of subordinate help remains stable, this results in a greater difference between helping behavior and the threshold of eviction when resistances costs are high (Figure 4A). In contrast, subordinate help and the threshold of eviction both decrease when only dominants pay resistance costs (rd: cost of enforcing high help) or when both dominants and subordinates pay resistance costs (Figure 4A). For all reported results above, the probability of eviction remains low across all parameter space. Model 2: helping and eviction dynamics with plasticity Here, we expand the previously described model by allowing both players to respond plastically to the strategy of their partner. In real-world systems, dominant aggression and subordinate helping/submission likely serve as signals indicating subordinate willingness to help and dominant threshold of eviction, such that dominants and subordinates have information about their partner’s behavior and willingness to disband the group (Cant and Johnstone 2009). We use a negotiation framework similar to that presented in Taylor et al. (2006), which incorporates an individual’s responsiveness (λ): the degree to which the dominant or subordinate responds to a change in the offer of its partner. During the interaction, one player chooses its strategy and its partner responds with its action, which is determined by the first player’s strategy and its partner’s responsiveness. The first player then responds to its partner’s strategy and this process repeats until both players converge on a final set of strategies h# and p#, which are determined by the set of equations: h#= h0−p0λh1−λpλh (7) p#= p0−h0λp1−λpλh (8) In this framework, the dominant chooses its baseline level of eviction (p0) and the subordinate chooses its baseline level of help (h0). Because the equation for h# includes h0−p0λh, negative values of λh results in subordinates providing more help as dominants increase their demand (at least when λp = 0; akin to pay-to-stay dynamics). Similarly, because the equation for p# includes p0−h0λp, negative values of λp results in dominants demanding relatively more help from helpful subordinates than from lazy subordinates (at least when λh = 0; akin to escalating demand). We do not solve for responsiveness of the dominant (λp) and the subordinate (λh); these are specified in the model as additional parameters, where convergence requires that |λpλh| < 1. Rather than solving for the equilibrium values of h and p as in Model 1, we substitute h and p with h# and p# and find the values of h# and p# by solving for the values of p0 and h0 that satisfy the following conditions: ∂WS∂h0|h0=h0^=h0*, p0=p0^=p0*=0 ∂WD∂p0|h0=h0^=h0*, p0=p0^=p0*=0 For most parameter space, the model converges on a solution regardless of the starting parameters; however, for a small subset of parameter space (notably, for some parameter combinations when λh > 0 and/or λp < 0) the model only converges on a stable solution when the starting parameters are either near 0 or near 1. There appears to be another, unstable equilibrium point at different starting values of p0 and h0 for this parameter space. The model results presented below represent the stable solutions. Here, we again solved for evolutionary and convergence stability as in Model 1. When λh > 0 (regardless of the value of λp), the system is not ESS, but is CSS; consequently, it will wander away from the fixed point and return. When λh < 0 (regardless of the value of λp), the system is both ESS and CSS and remains at the fixed point. This is similar to Hamilton (2013), which found that the system was not ESS unless there was flexibility in cooperative strategies. Results: helping and eviction dynamics with plasticity When both λh < 0 (subordinates respond to high dominant demands with more help) and λp < 0 (dominants demand more help from helpful subordinates), subordinate help and the threshold of eviction are generally higher than in the single sealed-bid model (Figures 1 and 2). For example, although subordinate costs to dominants (s) are higher than the level of helping behavior in the original model (providing a fitness loss to dominants), helping behavior is greater than subordinate costs to dominants when λh < 0 and λp < 0 (Figure 2). Further, the effects of multiple model parameters differ substantially from the single sealed-bid model. In the original single sealed-bid model, subordinate opportunities inside and outside the group did not alter helping behavior and eviction thresholds. However, when λh < 0 and λp < 0, both helping behavior and the threshold of eviction increase as subordinate prospects inside the group improve (i.e. mortality inside the group (μg) decreases and probability of becoming a dominant in the group (ag) increases; Figure 3C and D). Conversely, as subordinate prospects outside the group improve (i.e. mortality outside the group (μe) decreases and probability of becoming a dominant after eviction (ae) increases), both helping behavior and the threshold of eviction decrease (Figure 3C and D). However, when subordinate opportunities within the group are always better than subordinate opportunities outside the group (e.g. ae and μe vary as a proportion of ag and μg, such that inside and outside options improve together), helping and eviction thresholds increase as both subordinate prospects inside and outside the group improve. As dominants face mounting costs of enforcing high help (rd), subordinate help and eviction thresholds decrease in both the original model and the negotiation model (Figure 4B); however, this is not the case for subordinate costs of resisting dominant demand (rs). When λh < 0 and λp < 0, increased subordinate resistance costs actually result in increased subordinate help and eviction thresholds; this is the case when both dominants and subordinates have resistance costs as well as when only subordinates have resistance costs, although the slope of the increase is higher when only subordinates have resistance costs (Figure 4B). Helping behavior and the threshold of eviction decrease as the cost of helping increases (Supplementary Figure 3) and as dominant mortality increases (Supplementary Figure 5). Subordinate help and the threshold of eviction increase as the cost of subordinate eviction (es) increases (Supplementary Figure 6). Figure 4 View largeDownload slide Values of helping behavior and eviction threshold when dominants have costs of enforcing help (rd: blue), when subordinates have costs of resisting eviction (rs: red), or when both dominants and subordinates have resistance costs (black). Results are shown for the original model (A) and the negotiation model with λh < 0 and λp<0 (B; h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, and σ = 0.1. Figure 4 View largeDownload slide Values of helping behavior and eviction threshold when dominants have costs of enforcing help (rd: blue), when subordinates have costs of resisting eviction (rs: red), or when both dominants and subordinates have resistance costs (black). Results are shown for the original model (A) and the negotiation model with λh < 0 and λp<0 (B; h# and p# shown). For all panels, parameters that are not varied are G = 1, c = 0.1, m = 0.1, s = 0.4, k = 50, ed = 0.05, es = 0.05, μg = 0.2, μe = 0.4, ag = 0.5, ae = 0.25, and σ = 0.1. When λh < 0 with λp = 0 or λp > 0, all of the aforementioned patterns hold and subordinate help and the threshold of eviction are generally higher than when λh < 0 and λp < 0. However, when only dominants are plastic (λp < 0 and λh = 0), the patterns of all model parameters are similar to those in the model without plasticity and subordinate help and the threshold of eviction are generally lower than the single sealed-bid model. Similarly, when λh > 0 (subordinates help less when dominant demand is high), regardless of the value of λp, the patterns of all model parameters are similar to those in the model without plasticity and both subordinate help and the threshold of eviction are generally lower than the single sealed-bid model (Figures 1 and 2). DISCUSSION In these models, subordinates control the amount of help that they provide and dominants specify a threshold of eviction, such that groups will usually disband when subordinate help is below the threshold of eviction and usually stay together when helping is higher than the eviction threshold. Our model demonstrates that outside options for dominants and subordinates alter both helping and eviction dynamics. Specifically, we found that dominants demand more help and subordinates provide more help when dominants are less likely to recoup the lost help of evicted subordinates (σ), which is likely the case when groups are small (such that other subordinates cannot compensate for the evicted helper) or when groups have no neighbors (such that a group cannot recruit new helpers). In empirical systems, we would therefore expect dominants to more readily punish lazy subordinates in small groups or groups with no neighbors. These model results are in agreement with the results of Fischer et al. (2014), who experimentally prevented subordinates in the cooperatively breeding cichlid Neolamprologus pulcher from helping. They found that subordinates were more likely to be evicted from smaller groups compared to larger groups and that subordinates in small groups, but not large groups, increased help once they were allowed to help again. Collectively, this demonstrates that both dominants and subordinates respond to a dereliction of helping behavior more strongly when dominants are less able to compensate for the loss of an evicted subordinate. However, these model results and the results reported by Fischer et al. (2014) are in contrast to Kutsukake and Clutton-Brock (2008), who found that dominant meerkats are less aggressive to unrelated subordinates when groups are smaller. Further, Hellmann and Hamilton (2014) found that, when neighboring groups are present, subordinates provide more help rather than less help. Consequently, further empirical and theoretical work is needed to understand when and to what extent dominant eviction is altered by dominant dependence on current subordinates. Negotiation games, which allow individuals to adjust their strategy in response to their partner’s optimal strategy, can produce different dynamics than traditional games in which partners lack information about each other’s strategy (Taylor et al. 2006; Cant and Young 2013; Hamilton 2013; Quiñones et al. 2016). Consequently, we expanded our single sealed-bid model to incorporate plastic responses to partners in order to understand if negotiation between dominants and subordinates would produce helping behavior and eviction dynamics similar to those observed in empirical systems. We found that, when subordinates adopt a strike scenario (λh > 0), subordinates provide less help at equilibrium than they do in the original model, but that this point is not evolutionarily stable. However, subordinate strike strategies appear to be rare in group-living species, as evidenced by many studies demonstrating that subordinates actually increase helping behavior or submission in response to higher dominant demands (pay-to-stay dynamics in cooperative breeders: Mulder and Langmore 1993; Balshine-Earn et al. 1998; Cockburn 1998; Bergmüller and Taborsky 2005; Bruintjes and Taborsky 2008; Fischer et al. 2014; Hellmann et al. 2015b). It is possible, however, that subordinate strike scenarios may exist in species where 1) dominants are willing to tolerate nonhelping subordinates because dominants derive other benefits (increased survival: Kokko and Johnstone 1999; assurance of future mate: Fricke and Fricke 1977) from their presence and/or 2) it is costly for dominants to evict subordinates, such that the cost of evicting subordinates (e.g. aggression) outweighs the cost of having a nonhelping subordinate. Further exploration, both theoretically and empirically, would be useful for understanding when and how frequently this strategy is likely to evolve and the dynamics of this strategy near the equilibrium point. When we modeled negotiation dynamics that are akin to pay-to-stay dynamics (λh < 0, where subordinates respond to high dominant demands with more help), we find that helping and eviction dynamics differ substantially from the single-bid model without plasticity. In the single-bid model, subordinate outside options do not strongly influence help or eviction. In our negotiation model, subordinate helping behavior and eviction threshold decrease as subordinate prospects outside the group improve whereas helping behavior and eviction threshold increase as subordinate prospects within the group improve. These results are in line with empirical observations that subordinate helpers reduce the amount of help they provide when opportunities to breed outside the group are likely (Bergmüller et al. 2005; Young et al. 2005; Tibbetts 2007; Zöttl et al. 2013a), because threats of leaving are more credible when alternative options outside of the group are profitable (Cant and Johnstone 2009). Similarly, these results are also in agreement with theoretical predictions that as subordinate benefits of the current association increase relative to other options, subordinates may have less ability to negotiate based on their outside option (Buston and Zink 2009; Cant and Johnstone 2009; Cant 2011) and dominants should be able to demand more help in exchange for continued partnership (Cant 2011). Consequently, our model demonstrates that pay-to-stay requires that subordinates can plastically increase their help in response to increasing dominant demand. Interestingly, when dominants adopt an escalating demand strategy in conjunction with subordinate pay-to-stay (λp < 0, λh < 0), subordinates provide less help than when subordinates alone were plastic in their strategy (λp = 0, λh < 0), although they still help more than in the original model without plasticity. Similarly, if dominants adopt an escalating demand strategy without subordinate plasticity (λp < 0, λh = 0), model dynamics are similar to those in the original model and subordinates provide less help than they did in the original model. It is unclear how frequent escalating demand strategies are in nature, but the fact that there is often constant aggression against subordinates in cooperatively breeding species, regardless of help provided, suggests that this strategy may persist because subordinates respond to threats of punishment (Taborsky 2016) and because there is consistent individual variation in the propensity to help. As there is much empirical evidence demonstrating that individuals are relatively consistent in their willingness to cooperate (Bergmüller et al. 2010; English et al. 2010; Le Vin et al. 2011), demanding more help from helpful subordinates, rather than from individuals who are consistently less willing to cooperate, may be a successful strategy for promoting subordinate cooperation. Consequently, future models incorporating heterogeneity in partner phenotype—in terms of dominant demand for help, subordinate willingness to help, and subordinate responsiveness to dominant threats—would elucidate the extent to which variation in cooperative tendencies underlie observed empirical observations. In our negotiation model with pay-to-stay dynamics, both helping behavior and eviction thresholds are altered by costs to dominants of maintaining a high threshold of eviction and subordinate costs (e.g. aggression) of providing help that is close to the dominant threshold of eviction. Namely, we found that when only dominants face resistance costs, dominants demand less help from subordinates. In contrast, when both dominants and subordinates face resistance costs, subordinates are more helpful than when resistance costs are absent. Given that it is likely that both dominants and subordinates would simultaneously incur costs of enforcing help and resisting eviction, namely the cost of giving and receiving aggression, it is likely that these resistance costs will largely increase subordinate cooperation in pay-to-stay systems. Similar to Hamilton and Taborsky (2005), we found that unrelated helpers rarely compensate for the cost that they impose on dominants (e.g. reproduction: Clutton-Brock et al. 2001 and Hellmann et al. 2015a; food: Wittig and Boesch 2003), as the benefits of subordinate help in our model were nearly always lower than subordinate costs to dominants in our single-bid model. However, in our negotiation model with pay-to-stay dynamics, we found that subordinates helped consistently more than the costs that they imposed on dominants. This suggests that pay-to-stay dynamics alone, irrespective of other benefits of helping, can enforce subordinate help in cooperatively breeding species and, remarkably, produce higher levels of cooperation than kin selection alone (Quiñones et al. 2016). However, given that male and female dominants can differ in the costs of having subordinates, notably in terms of reproductive competition (Heg et al. 2008), additional theoretical and empirical work would be useful to understand how pay-to-stay dynamics regulating helping behavior differ based on sex. Our results provide a strong argument for examining the role of plasticity and negotiation in the evolution of cooperation among group members. This is in agreement with Quiñones et al. (2016), who found that subordinate help evolves from negotiation strategies between unrelated partners. Our model with plasticity in subordinate strategy that mirrored pay-to-stay dynamics produces results consistent with what would be expected with previous theory (Cant and Johnstone 2009; Cant 2011) and with previous empirical results (Bergmüller et al. 2005; Young et al. 2005; Tibbetts 2007; Zöttl et al. 2013a). Given these results, we predict that plasticity in dominant or subordinate behavior reduces or increases, respectively, the amount of help subordinates provide in empirical systems. Further, we expect that subordinate outside options should only influence subordinate help in systems with pay-to-stay mechanisms of cooperation. Past laboratory and field experiments in which subordinates have been experimentally prevented from helping (Balshine-Earn et al. 1998; Fischer et al. 2014; Hellmann et al. 2015b) or where demands of help were experimentally varied (Taborsky 1985; Zöttl et al. 2013c) have been successful in triggering dominant punishment and measuring the threshold at which dominants are willing to punish or evict subordinates. Combining these experiments with manipulations of dominant punishment, perceived competition from outside the group, and variation in the ease of dominants to replace evicted subordinates are also expected to shed light on the extent to which subordinates and dominants alter their behavior based on the relative leverage that they have in a given social situation. FUNDING This work was funded by The Ohio State University. Data accessibility: Analyses reported in this article can be reproduced using the data provided by Hellmann and Hamilton (2018). 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Behavioral EcologyOxford University Press

Published: Mar 31, 2018

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