Demography when history matters: construction and analysis of second-order matrix population models

Demography when history matters: construction and analysis of second-order matrix population models Theor Ecol (2018) 11:129–140 https://doi.org/10.1007/s12080-017-0353-0 ORIGINAL PAPER Demography when history matters: construction and analysis of second-order matrix population models 1 1 Charlotte de Vries · Hal Caswell Received: 28 July 2017 / Accepted: 24 October 2017 / Published online: 8 December 2017 © The Author(s) 2017. This article is an open access publication Abstract History matters when individual prior conditions Keywords Second-order matrix population model · contain important information about the fate of individ- Individual heterogeneity · Prior stage dependence · uals. We present a general framework for demographic Carry over effects · Sensitivity analysis models which incorporates the effects of history on popula- tion dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for con- Introduction structing the population projection matrix from information on current state dynamics as a function of prior condi- Every demographic analysis requires a classification of indi- tion. Three biologically motivated classes of prior condition viduals by age, size, developmental stage, physiological are included: prior stages, linear functions of current and condition, or some other variable. These variables describe prior stages, and equivalence classes of prior stages. Tak- individual states (i-states) such that the fate of an individual ing advantage of the matrix formulation of the model, we depends only on its current state and the environment (Metz show how to calculate sensitivity and elasticity of any demo- 1977; Caswell and John 1992; Metz and Diekmann 1986). graphic outcome. Prior condition effects are a source of This requires the state variable to capture all the aspects of inter-individual variation in vital rates, i.e., individual het- the individual’s history that are relevant to its future fate erogeneity. As an example, we construct and analyze a (Caswell et al. 1972; Caswell 2001). The task of the popu- second-order model of Lathyrus vernus, a long-lived herb. lation modeler is to find an i-state variable that successfully We present population growth rate, the stable population captures past history. This is not easy; apparently reasonable distribution, the reproductive value vector, and the elastic- and frequently used choices of i-states may fail to capture ity of λ to changes in the second-order transition rates. We all the relevant information about individual history. quantify the contribution of prior conditions to the total het- Confronted with this problem, the modeler might choose erogeneity in the stable population of Lathyrus using the a completely different i-state variable (as plant ecologists did when giving up age-classified demography in favor entropy of the stable distribution. of size-classified models), or might add a dimension to the state space (as in extending stage-classified models to Electronic supplementary material The online version of this include both age and stage). Sometimes, however, it might article (https://doi.org/10.1007/s12080-017-0353-0) contains sup- be difficult or impossible to measure the relevant current plementary material, which is available to authorized users. characteristic, but a proxy for that characteristic might be found in some function of the prior condition of an individ- Charlotte de Vries ual. For example, resource storage can influence vital rates c.devries@uva.nl in plants but resource storage can be difficult to measure. Hal Caswell However, reproduction in the prior year can be used as a h.caswell@uva.nl proxy for resource storage in species where reproduction in one year may deplete resource storage and reduce fertility in Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Amsterdam, The Netherlands the following year. If it is not possible to measure resource 130 Theor Ecol (2018) 11:129–140 storage, one might therefore incorporate prior reproductive to changes in parameters. As an example, we develop a status into the state variable to improve the model. model and calculate the elasticity of the population growth A variety of prior conditions which affect vital rates have rate of the herbaceous perennial plant Lathyrus vernus. been found empirically. Reynolds and Burke (2011) found that chestnuts with fast early growth died younger than Terminology Our discussion requires careful definitions of chestnuts with slow early growth. Warren et al. (2014) found terms in order to clarify the way that historical effects are that previous breeding success and current body condition incorporated. We will say that the life cycle is described in may be among the most important determinants of breeding terms of stages (e.g., size classes). The prior condition of an propensity in female lesser scaup. Rouan et al. (2009) found individual is some function of its stage at the prior time and that choice of next breeding site is affected by both current its stage at the current time, and thus incorporates histori- and prior breeding site in Branta canadensis.These exam- cal information. The combination of current stage and prior ples show that prior condition can be a source of individual condition serves as the individual state variable for the anal- variation in vital parameters, i.e., a source of heterogene- ysis. We give an overview of the terminology used in this ity. Some attempts have been made to include this source paper in Table 1. of heterogeneity into population models, see Pfister and The prior condition can be any arbitrary function of prior Wang (2005), Ehrlen ´ (2000), andRouanetal. (2009), but and current stage. Prior condition might be the stage at the a framework for incorporating general prior conditions into prior time, it might be defined by membership in a set of demographic models does not exist and will be presented in stages at the prior time, or it might be defined as the differ- this paper. ence between the current and the prior stages. Suppose for When constructing a structured population model, the i- example that stages are defined by size, in the hope that a state variables are used to classify individuals into states in size classification would be a satisfactory i-state variable. It a population vector n(t) whose entries give the densities of might turn out that historical effects require including size each state. The population vector is projected forward by a at the prior time in the i-state variable. Alternatively, the population projection matrix A i-state might require information only on membership in a class of sizes (e.g., larger than average or smaller than n(t + 1) = An(t ). (1) average); we will refer to these classes of prior stages as equivalence classes. Or, the i-state might require informa- The matrix A can be decomposed into a matrix U, contain- tion on the change in size between the previous time and the ing transition probabilities for existing individuals, and a current time, and individuals might be classified by whether matrix F, describing the generation of new individuals by they have grown, shrunk, or remained in the same size class. reproduction: Models based on prior stage are described in section “Full prior stage dependence,” models based on general func- A = U + F. (2) tions of prior and current stages are described in section If prior conditions influence present dynamics, the vector n “Prior condition models,” and models based on equivalence and the matrices U, F,and A must be modified to account classes of prior stage are described in section “Equivalence classes of prior stages.” The matrices, vectors, and mathe- for these influences. Our goal in this paper is to present a systematic method for constructing such models in which matical operations used in this paper are listed in Table 2. individuals are classified by current stage and (very gener- ally defined) prior condition. Because we are considering effects of individual condition at just one prior time, we Model construction refer to these as second-order matrix population models. We will present the demographic analysis of such models at the We will begin by constructing the model in which the prior level of the individual, the cohort, and the population, and condition is defined by full information on the prior stage, show how to carry out sensitivity analyses of model results which we refer to as full prior stage dependence. Table 1 Terminology used to distinguish the state of an individual, the stage of the life cycle, and the prior condition of the individual • The state of an individual is the information necessary to predict the response of an individual to its environment. • The stage of an individual refers to a biologically defined category, usually a life cycle stage, which is used to define a (possibly unsuccessful) state variable. • The condition of an individual is a flexible term that refers to some function of the prior stage and current stage of the individual; this historical information may be combined with the present stage to obtain a state variable based on current stage and prior condition. Theor Ecol (2018) 11:129–140 131 Table 2 Mathematical notation used in this paper Quantity Description Dimension U Prior stage dependent cohort projection matrix s(s + 1) × s(s + 1) U Matrix with transition rates (from current to future state) for individuals with prior stageis × s V Prior condition dependent cohort projection matrix sr × sr F Prior stage dependent fertility matrix s(s + 1) × s(s + 1) F Matrix with fertility rates (from current to future state) for individuals with prior stageis × s G Prior condition dependent fertility matrix sr × sr n ˜ Prior stage model population vector n ˜ =vec(N)s(s + 1) × 1 m ˜ Prior condition model population vector m ˜ =vec(M)sr × 1 C Matrix relating the prior stage dependent vector n ˜ to the prior condition dependent population vector m ˜ sr × s(s + 1) φ(i, j ) Matrix whose (i, j )th entry indicates the prior condition for an individual that makes an i → j transition s × s k k φ Matrix whose (i, j )th entry is one if φ(i, j ) = k and zero otherwise, in MATLAB notation φ (i, j ) = (φ == k) s × s I Identity matrix s × s 1 Vector of ones s × 1 e The ith unit vector, with a 1 in the ith entry and zeros elsewhere Various E A matrix with a 1 in the (i,j) position and zeros elsewhere Various ij ⊗ Kronecker product X(:,i) Column i of matrix X vecX The vec operator, which stacks the columns of an m × n matrix X into a mn × 1 vector Dimensions of vectors and matrices are given where relevant; s denotes the number of classes in the full second-order model and r denotes the number of classes in the reduced second-order model Full prior stage dependence where n is the number of individuals whose current ij stage is i and whose prior stage was j.Ehrlen ´ (2000) Creating a prior stage model requires transition and fertil- suggested that this array could be updated by multipli- cation with a three-dimensional matrix. However, matrix ity rates to be measured for each possible prior stage. Since newborns do not have a well-defined prior stage in general, multiplication can not be used directly to project such a two-dimensional array (this would require tensors). Instead, an extra prior stage must be added for newborns. If there are s current stages, we will label the prior condition of new- the two-dimensional array is transformed into a vector by stacking the columns on top of each other: borns as stage s + 1. Individuals are thus classified by their current stage 1, 2,...,s and their prior stage 1, 2,...,s +1. n ˜ (t ) = vecN(t ). (8) The transition and fertility rates, conditional on prior stage, are given by the matrices U and F : k k We use the tilde notation to denote vectors and matrices that relate to the prior condition model. The entries in the popu- U s × s Transitions among stages for individuals (3) lation vector n ˜ are now ordered first by prior stage and then with prior stage k = 1,...,s + 1, by current stage: F s × s Reproduction by individuals (4) with prior stage k = 1,...,s + 1. The entries of U , denoted by u , are defined as ij u = P j → i | prior stage = k , (5) ij f = E offspring of stage i | prior stage = k . (6) ij It is useful to think of the state of the population as described by a two-dimensional array N of size s × (s + 1) The vector n ˜ is projected by a transition matrix U and ˜ ˜ ˜ a fertility matrix F.The U and F matrices are constructed, respectively, from the set of matrices U and F .The transi- k k tion matrix U projects individuals into their next stage while keeping track of their prior stage. The matrix U is written in terms of block matrices corresponding to the blocks in n ˜ 132 Theor Ecol (2018) 11:129–140 in Eq. 9.In MATLAB notation, the transition matrix for a prior stages. We consider the case where this function can model with s = 2is be defined as a linear transformation of the full prior stage dependent model, where the population vector is written as m ˜ , defined by m ˜ (t ) = Cn ˜ (t ), (15) for some matrix C. An example of such a prior condition is having previously grown, shrunk, or stayed the same size. where U (:,i) refers to the ith column of the matrix U and k k The matrix C maps i-states defined by the combination 0 is a column vector of size s×1. To understand the structure of (current stage × prior stage) to i-states defined by the of U, consider the (1,1) block in the upper left corner. This combination of (current stage × prior condition). block projects individuals with prior stage 1 at t to prior Suppose there are r distinct prior conditions. The state of stage 1 at t +1. The only such individuals have current stage the population is now given by a s × rarray M 1 at time t. They are projected by column 1 of U . All other columnsofthe(1,1)blockof U are zero. The other blocks of U are filled similarly. The blocks in the last row of U are zero because transi- tions into the newborn stage are impossible; the matrix F will fill up this row block. In general, for any number of stages s, where m represents the number of individuals whose cur- s+1 s ij rent stage is i and whose prior condition is j.AsinEq. 8, U = E ⊗ e ⊗ U e , (11) ij j i the population vector m ˜ is j =1 i=1 where e has size s × 1and E has size (s + 1) × (s + 1) m ˜ (t) = vecM(t ). (17) i ij and has a one in position (i,j) and zeros elsewhere. The key to the construction of the prior condition model The fertility matrix F is constructed from the F . Because is to derive C from the rule defining the prior condition. To individuals are always born into stage s + 1, the F appear do so, define a matrix φ in the last row of blocks in F. For the case with s = 2, the fertility matrix is φ(i, j ) = prior condition for an individual that makes an j → i transition. (18) Forexample,ifthe r = 3 prior conditions are shrinking, stasis, and growth 1for i< j shrinking, In general, for any number of stages s, ⎨ φ(i, j ) = 2for i = j stasis, (19) s+1 3for i> j growth. F = E ⊗ F , (13) (s+1)k k k=1 Next, we define a set of matrices φ (i, j ),for k = 1,...,r, where E has size (s + 1) × (s + 1) and has a one in given by (s+1)k position (s + 1, k) and zeros everywhere else. 1if φ(i, j ) = k, Given a set of transition matrices U and fertility matrices φ (i, j ) = (20) 0 otherwise . ˜ ˜ F ,Eqs. 11 and 13 define the second-order matrices U and F, and the population projection matrix Given the matrices φ , the matrix C is given by ˜ ˜ ˜ A = U + F. (14) T k C = 1 ⊗ e ⊗ I diag vecφ , (21) k s s+1 These matrices are subject to all the usual demographic k=1 analyses, including population growth, population structure, see Appendix A. To project the new population vector m ˜ , and sensitivity analysis (see section “Case study”foran ˜ ˜ ˜ we replace the matrices U and F with new matrices V and example). G, respectively, so that Prior condition models ˜ ˜ ˜ ˜ m(t + 1) = V + G m(t ) (22) A more general second-order model structure allows transi- The matrix V describes transitions of extant individuals tions and fertility to depend on some function of current and between the different i-states of the prior condition model Theor Ecol (2018) 11:129–140 133 and the matrix G describes the production of new individu- the C matrix that transforms the full prior stage dependent als in the prior condition model. Substituting (15) into both population vector n ˜ to the equivalence class population vec- sides of Eq. 22 yields tor m ˜ . We transform the population matrix N in Eq. 7 into the equivalence class population matrix M in Eq. 16 by a ˜ ˜ ˜ ˜ C U + F n ˜ (t ) = V + G Cn ˜ (t ). (23) matrix B; ˜ ˜ Equation 23 is satisfied if V and G satisfy the following M = NB. (31) equations: The matrix B has size (s + 1) × r and its entries are ˜ ˜ VC = CU (24) GC = CF. (25) 1if stage i ∈ equivalence class j, B(i, j ) = (32) 0 otherwise. In general, the matrix C is not square and does not have an inverse. So we use the Moore-Penrose pseudo-inverse C Applying the vec operator to Eq. 31 gives ˜ ˜ (see Abadir and Magnus (2005)) to solve for V and G: † m ˜ (t ) = (B ⊗ I )n ˜ (t ), (33) ˜ ˜ V = CUC , (26) ˜ ˜ G = CFC . (27) where we have used the following result from Roth (1934) that vecABC = C ⊗ A vecB. Equation 33 is a special If C is square and non-singular, the pseudo-inverse is the case of Eq. 15 with ordinary inverse: † −1 C = C . (28) C = (B ⊗ I ). (34) If C is not square but has linearly independent rows (i.e., has Since the rows of B are orthogonal, the matrix B and the full row rank), matrix (B ⊗ I ) both have full row rank and therefore C † T T −1 C = C (CC ) . (29) can be calculated from Eq. 29. If C has rows of zeroes (this happens if some combina- tions of current stage and prior conditions are impossible), Sensitivity analysis then C will not have full rank. In this case, C is com- puted from the singular value decomposition, implemented Sensitivity analysis provides the effect of changes in any in MATLAB with the function pinv(C) and in R with the parameter on any model outcome. In general, these com- function Ginv(C). For example, in a size-classified model, putations require derivatives of scalar-, vector-, or matrix- it is impossible to be in the smallest size class and to have valued functions with respect to scalar-, vector-, or matrix- grown into it from a smaller size class. In such cases, C valued arguments. Matrix calculus is a formalism which and thus also V and G have rows and columns of zeros enables us to consistently calculate such derivates. For an corresponding to the impossible combinations. introduction to matrix calculus, see Abadir and Magnus Equations 26 and 27 define the prior condition matrices (2005); for details see Magnus and Neudecker (1988). Eco- ˜ ˜ V and G and the population projection matrix logical applications of sensitivity analysis appear in Caswell ˜ ˜ A = V + G. (30) (2007, 2009, 2012). Consider some scalar or vector output of the model, ξ, As in Eq. 14, the usual demographic results can be obtained ˜ ˜ ˜ which is computed from U and F, or from V and G.The from these matrices. ˜ ˜ ˜ ˜ matrices U, F, V,and G are in turn computed from the matrices U and F . Suppose U and F depend on a p × 1 i i i i Equivalence classes of prior stages vector of parameters θ, i.e., U = U [θ ],thenweusethe i i chain rule to write Equivalence classes of prior stages are a special case of functions of current and prior stage. Equivalence classes are ˜ ˜ dξ dξ dvecU dξ dvecF = + , (35) subsets of prior conditions that depend only on the prior ˜ ˜ dθ dθ dθ dvec U dvec F stage. For example, individuals in a size-classified model or might be categorized into two equivalence classes depend- ing on whether they were previously above or below some ˜ ˜ dξ dξ dvecV dξ dvecG = + . (36) threshold size. ˜ ˜ dθ dvec V dθ dvec G dθ The machinery described in the previous section, i.e., Eqs. 21, 26,and 27, can be used to write down the equiva- The first terms in Eqs. 35 and 36 capture effects through sur- lence class model. However, there is an easier way to find vival and transitions, and the second terms capture effects 134 Theor Ecol (2018) 11:129–140 through fertility. The elasticities, or proportional sensitivi- and G), and the dependence of U and F on the parameter θ. i i ties, are given by These items are specific to the question under consideration, so we provide an example in the next section. ξ dξ −1 = diag (ξ ) diag(θ ). (37) θ dθ ˜ ˜ ˜ ˜ The derivatives of U, F, V,and G with respect to θ Case study ˜ ˜ depend on how U and F depend on the U and F matrices. i i Differentiating (11), we obtain As an example, we apply the second-order formalism to a fully prior stage dependent model of a perennial plant. s+1 dvecU dvecU U We will construct the model and calculate the population = Q . (38) dθ dθ growth rate, the stable population vector, the reproductive j =1 value vector, and the elasticity of the population growth rate Each term in the summation captures the effect of θ through λ to proportional changes in the demographic rates. one of the U . The matrix Q is given by j Our analysis is based on a study by Ehrlen ´ (2000)of Lathyrus vernus, a long-lived herb native to forest mar- gins and woodlands in central and northern Europe and Q = I ⊗ K ⊗ I vec E ⊗ I 2 (E ⊗ I ) . s+1 s,s+1 s ij ii s Siberia. Ehrlen ´ classified individuals into seven stages: seed i=1 (SD), seedling (SL), very small (VS), small (SM), large (39) vegetative (VL), flowering (FL), and dormant (DO). Ehrlen Similarly, differentiating (13)gives constructed two matrix models: a first-order model with no historical effects and a second-order model. We construct s+1 dvecF dvecF our second-order model from the transition and fertility = Q , (40) dθ dθ rates calculated by Ehrlen ´ (2000) (their Table 2). i=1 We introduce a special prior stage for newborns; the where each term captures the effect of θ through one of the second-order model therefore has a total of 7 × 8 = 56 F . The matrix Q is given by states. We used Ehrlen ´ (2000)’s demographic rates to con- struct a transition matrix U and a fertility matrix F for each Q = I ⊗ K ⊗ I vec E ⊗ I . (41) i i s+1 s,s+1 s s+1,i i s of the eight prior stages. In Table 3, the transition matrix To calculate the derivatives of the prior condition matri- for individuals who were previously in stage SM is shown ˜ ˜ ces V and G, we use the chain rule to write as an example. The first two columns of this matrix are zero because small plants can not go back to being seeds ˜ ˜ ˜ dvecV dvecV dvecU = , (42) or seedlings. All eight transition and fertility matrices are in dθ dθ dvecU the Supplementary Material. We constructed the projection matrix, A, from the U ˜ ˜ ˜ i dvecG dvecG dvecF = . (43) and F matrices using Eqs. 11 and 13. The resulting 56 × dθ dθ dvecF 56 matrices are available in the Supplementary Material. Differentiating Eqs. 26 and 27, we obtain dvecV = C ⊗ C, (44) dvecU Table 3 The transition matrix for Lathyrus vernus individuals with prior stage small (SM) dvecG = C ⊗ C. (45) dvecF Thus, ˜ ˜ dvecV dvecU = C ⊗ C , (46) dθ dθ ˜ ˜ dvecG dvecF = C ⊗ C . (47) dθ dθ The columns in this matrix represent transitions out of the following Equations 38 and 40 are substituted into Eq. 35. current stages from left to right: seed (SD), seedling (SL), very small Equations 46 and 47 are substituted into Eq. 36. All that (VS), small (SM), vegetative large (VL), flowering (FL), and dormant ˜ ˜ ˜ (DO) remains is to calculate the dependence of ξ on U and F (or V Theor Ecol (2018) 11:129–140 135 The population growth rate λ is given by the dominant eigenvalue of A, λ = 0.985. (48) 0.25 0.2 This agrees with the value reported for the second-order model by Ehrlen ´ (2000). Ehrlen ´ also fitted a first-order 0.15 model to the same data and found a value just above one 0.1 (λ = 1.010). The stable population vector, w ˜ , and the repro- 0.05 ductive value vector, v ˜, which are displayed in Figs. 1 and 2, are the right and left eigenvectors of A, respectively. The entries of the stable population vector are denoted by w ˜ ij SD SL NB for individuals with current stage i and prior stage j,analo- VS DO SM FL gously to the entries of the population vector n ˜ in Eq. 9.The VL VL c SM entries of the marginal stable current stage distribution, w , FL VS DO SL Current stage are given by SD Prior stage w = w ˜ (49) ij 0.3 j =1 0.25 and are shown in Fig. 1b. Similarly, the entries of the marginal stable prior stage distribution, w , are given by 0.2 w = w ˜ (50) ij j 0.15 i=1 0.1 andareshowninFig. 1c. Individuals with the same current stage have different 0.05 vital rates if they differ in prior stage and this heterogeneity due to prior stage affects the population growth rate λ.To SD SL VS SM VL FL DO quantify the relative effect of the different prior stage depen- Marginal distribution over current stages dent transition matrices on the population growth rate, we calculate the elasticity of the population growth rate, λ,to changes in the U matrices 0.3 0.25 ,i = 1,..., 8. (52) vec U 0.2 Substituting λ for ξ in Eq. 37 yields 0.15 λ 1 dλ = diag (vecU ) . (53) 0.1 vec U λ dvec U i i 0.05 As shown in Caswell (2001), dλ 0 T T SD SL VS SM VL FL DO NB = w ⊗ v , (54) i i Marginal distribution over prior stages dvec U where w and v are the right and left eigenvectors of U , i i i Fig. 1 a Stable current stage × prior stage distribution. b Marginal respectively, scaled so that current stage distribution. c Marginal prior stage distribution. Notation: seed (SD), seedling (SL), very small (VS), small (SM), vegetative large v w = 1forall i, (55) (VL), flowering (FL), dormant (DO), and newborn prior (NB) 1 w = 1forall i. (56) We sum the entries of Eq. 53 to get the elasticity of λ demography of individuals with prior stages seed, seedling, to a proportional change in all of the entries of U .The or newborn have little effect on λ. Proportional changes in results are shown in Fig. 3. We note that there are large individuals who were small vegetative at the prior time have differences among prior stages. Proportional changes in the effects an order of magnitude larger. Fraction Fraction Fraction 136 Theor Ecol (2018) 11:129–140 intuition, tradition, practical limitations, and formal statisti- cal analyses. Even after a careful choice of i-state, it may happen that individual prior conditions contain important information about the fate of individuals. In such cases, 0.8 history matters, and the methods presented here solve the 0.6 problem of how to incorporate information about it into matrix models. 0.4 Why stop at second-order effects, what about the effect 0.2 of the prior condition at t − 2, t − 3, etc.? It is theo- retically possible to extend the framework presented here to include dependence on higher-order effects. However, if DO FL the i-state variable is such that third- or even fourth-order VL SD SM historical effects are important, it might better to recon- SL VS VS SM SL sider the choice of i-state variable instead of including ever VL Current stage FL SD DO increasing historical dependence. NB Prior stage Statistical tests for second-order effects in longitudinal data using log-linear models have been developed by Bishop et al. (1975) and Usher (1979). Time series of longitudinal data are needed to perform these tests as well as for the sub- 3.5 sequent estimation of a prior condition dependent model. Capture–mark–recapture analyses for prior stage dependent 2.5 models have been developed by Pradel (2005)and Cole et al. (2014). Incorporating individual history requires a decision about 1.5 what aspects of the prior condition are important. We have presented three biologically motivated cases: prior condi- tion as the prior stage, prior condition as an arbitrary linear 0.5 function of current and prior stages, and prior condition as SD SL VS SM VL FL DO an equivalence class of prior stages. In each case, the nec- Current stage essary information is a set of fertility and survival/transition matrices for each prior stage. The resulting models use 3.5 block-structured matrices to project a vector of stages within prior conditions. These matrices can be used to calculate all the usual demographic outcomes. Because the matri- 2.5 ces are carefully constructed from U and F , they can be i i subjected to sensitivity analysis. It is straightforward to cal- 1.5 culate the sensitivity and elasticity of any model outcome to any parameters affecting the vital rates. Prior condition effects are more than just a convenient 0.5 tool in constructing i-states for population models. They are a biologically real source of inter-individual variation. SD SL VS SM VL FL DO NB Prior stage The importance of individual variation in vital parameters to ecological processes has become increasingly evident in recent years (Bolnick et al. 2011; Valpine et al. 2014; Fig. 2 a Reproductive value vector for each current stage × prior Caswell 2014; Vindenes and Langangen 2015; Steiner and stage combination. b Marginal current stage reproductive value vector. c Marginal prior stage reproductive value vector. Notation: seed (SD), Tuljapurkar 2012; Cam et al. 2016). Ignoring individual seedling (SL), very small (VS), small (SM), vegetative large (VL), variation in vital parameters, also referred to as individ- flowering (FL), dormant (DO), and newborn prior (NB) ual heterogeneity, can have important consequences for the demographic outcomes and subsequent conclusions; see for Discussion example Vaupel et al. (1979), Rees et al. (2000), Fujiwara and Caswell (2001), Vindenes and Langangen (2015), and When does history matter? Population models are based Cam et al. (2016). In his study of Lathyrus,Ehrlen ´ (2000) on i-state variables chosen by some mixture of biological found that including heterogeneity due to prior stage had Reproductive value Marginal reproductive value Marginal reproductive value Theor Ecol (2018) 11:129–140 137 Fig. 3 The elasticity of λ to a 0.35 proportional change in all of the 0.3 0.25 0.2 0.15 0.1 0.05 Seed Seedling Very small Small Vegetative Flowering Dormant Newborn Prior stage only a small effect on λ, although it was sufficient to cause pose interesting challenges and is an open research problem the population to decrease rather than increase. for population ecology. How much heterogeneity is introduced by the prior con- dition effects in the Lathyrus example? Observations of Acknowledgements We thank Gregory Roth for helpful discussions individual plants can identify their current stage, but not and Johan Ehrlen ´ for providing the data for the Lathyrus case study. their prior condition. The stable structure, w ˜ (Fig. 1), shows the joint probability distribution over current and prior Funding information This work was supported by European Research Council Advanced Grant 322989. stages, and the amount of heterogeneity in the stable popu- lation can be calculated from the entropy of this distribution. The entropy of the joint current × prior stage distribution is Appendix A: Derivation of the matrix C H(p, c) =− w ln w = 2.61. (57) ij ij i,j In this appendix, we will derive the matrix C in Eq. 21,which transforms the fully prior stage dependent population vector, This measures the overall heterogeneity in the stable pop- n ˜, into the prior condition dependent population vector, m ˜ , ulation structure. The heterogeneity in the marginal current stage distribution, w [Eq. 49], is m ˜ (t ) = Cn ˜ (t ), (60) c c as in Eq. 15. Recall that we defined the matrix φ with H(c) =− w ln w = 1.69. (58) i i elements i=1 φ(i, j ) = prior condition for an individual that makes an This is the observable heterogeneity in current stage. The heterogeneity contributed by the unobservable prior stage, j → itransition, (61) taking into account the relationship between current and and we defined a set of indicator matrices φ ,for k = prior stage, is 1,...,r,given by H (p|c) = H(p, c) − H(c) = 0.92, (59) 1if φ(i, j ) = k, φ (i, j ) = (62) Khinchin (1957). Thus, in this example, the prior stage con- 0 otherwise . tributes about 35% of the total heterogeneity in the stable In Appendix B, we show a few examples of φ matrices. population. Recall furthermore the matrix In this paper, we have considered only linear models in constant environments. However, models could easily be constructed to include density effects, by making the U and F functions of density. Periodic or stochastic models could be constructed by making the U and/or the F appropriate i i functions of time. The analysis of the resulting models may Elasticity of to change in U i 138 Theor Ecol (2018) 11:129–140 where n is the number of individuals whose current stage For example, when s = 3, φ is ij is i and whose prior stage was j, and the matrix where m is the number of individuals whose current stage ij is i and whose prior condition was j. Finally, recall that the (B) When prior condition = equivalence classes of prior population vectors are given by stages,then n(t ) = vecN(t ), (65) φ(i, j ) = function(j ) (74) m ˜ (t ) = vecM(t ). (66) k For example, for s = 3 and for the case of the Using the above defined matrices φ and N, the matrix equivalence classes j = 1and j> 1, we get M can be written as r 1for j = 1 φ(i, j ) = (75) k T M = φ ◦ N 1 ⊗ e . (67) 2for j = 2, 3 s+1 k=1 The first term in the product is a matrix containing the den- sities of individuals in all entries of N that correspond to prior condition k, and zeros elsewhere. The second term in the product adds the densities across each row of the matrix φ ◦ N and puts these elements in the kth column of M. Taking the vec of both sides of this equation gives T k m ˜ = 1 ⊗ e ⊗ I vec φ ◦ N , (68) k s s+1 k=1 (C) When prior condition = shrinkage, stasis, or growth, then T k = 1 ⊗ e ⊗ I diag vecφ n, (69) k s s+1 k=1 1for i< j shrinkage T φ(i, j ) = 2for i = j stasis (76) wherewehaveusedvecABC = C ⊗ A vecB (Roth 3for i> j growth 1934)and vec (A ◦ B) = diag (vecA) vec (B). Comparison with Eq. 60 shows that For example, for s = 3, φ is T k C = 1 ⊗ e ⊗ I diag vecφ . (70) k s s+1 k=1 Appendix B: Examples of φ and C matrices B.1 Examples of φ matrices In this appendix, we show a few examples of the matrix φ. (D) General more complicated growth conditions. For (A) When prior condition = prior stage,then example, suppose the prior condition reflects the amount of growth, so that φ(i, j ) = j for all i. (71) φ(i, j ) = i − j + s. (78) Theor Ecol (2018) 11:129–140 139 ⎛ ⎞ For s = 4, this results in the following φ ⎝ ⎠ φ = , (81) ⎛ ⎞ ⎝ ⎠ φ = 100 . (82) Substituting these into Eq. 70, and letting the sum range from 1 to 3 only, gives the matrix C as B.2 An example C matrix We consider the example in Eqs. 76 and 77 with three size classes and three growth classes corresponding to growth, shrinkage, and stasis. For simplicity, we consider a cohort model with no fertility and therefore no special prior condi- The transition matrix for the prior condition model, V, tion for newborns. The matrices φ are is given by Eq. 26 and requires the pseudo-inverse of the ⎛ ⎞ matrix C, which can be found using pinv(C) in MATLAB ⎝ ⎠ φ = 001 , (80) or Ginv(C) in R. Substitution of the pseudo-inverse of C into Eq. 26 and a few lines of algebra result in This matrix captures the complicated dependence of transi- References tions on both current stage and prior condition. 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Demography when history matters: construction and analysis of second-order matrix population models

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Theor Ecol (2018) 11:129–140 https://doi.org/10.1007/s12080-017-0353-0 ORIGINAL PAPER Demography when history matters: construction and analysis of second-order matrix population models 1 1 Charlotte de Vries · Hal Caswell Received: 28 July 2017 / Accepted: 24 October 2017 / Published online: 8 December 2017 © The Author(s) 2017. This article is an open access publication Abstract History matters when individual prior conditions Keywords Second-order matrix population model · contain important information about the fate of individ- Individual heterogeneity · Prior stage dependence · uals. We present a general framework for demographic Carry over effects · Sensitivity analysis models which incorporates the effects of history on popula- tion dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for con- Introduction structing the population projection matrix from information on current state dynamics as a function of prior condi- Every demographic analysis requires a classification of indi- tion. Three biologically motivated classes of prior condition viduals by age, size, developmental stage, physiological are included: prior stages, linear functions of current and condition, or some other variable. These variables describe prior stages, and equivalence classes of prior stages. Tak- individual states (i-states) such that the fate of an individual ing advantage of the matrix formulation of the model, we depends only on its current state and the environment (Metz show how to calculate sensitivity and elasticity of any demo- 1977; Caswell and John 1992; Metz and Diekmann 1986). graphic outcome. Prior condition effects are a source of This requires the state variable to capture all the aspects of inter-individual variation in vital rates, i.e., individual het- the individual’s history that are relevant to its future fate erogeneity. As an example, we construct and analyze a (Caswell et al. 1972; Caswell 2001). The task of the popu- second-order model of Lathyrus vernus, a long-lived herb. lation modeler is to find an i-state variable that successfully We present population growth rate, the stable population captures past history. This is not easy; apparently reasonable distribution, the reproductive value vector, and the elastic- and frequently used choices of i-states may fail to capture ity of λ to changes in the second-order transition rates. We all the relevant information about individual history. quantify the contribution of prior conditions to the total het- Confronted with this problem, the modeler might choose erogeneity in the stable population of Lathyrus using the a completely different i-state variable (as plant ecologists did when giving up age-classified demography in favor entropy of the stable distribution. of size-classified models), or might add a dimension to the state space (as in extending stage-classified models to Electronic supplementary material The online version of this include both age and stage). Sometimes, however, it might article (https://doi.org/10.1007/s12080-017-0353-0) contains sup- be difficult or impossible to measure the relevant current plementary material, which is available to authorized users. characteristic, but a proxy for that characteristic might be found in some function of the prior condition of an individ- Charlotte de Vries ual. For example, resource storage can influence vital rates c.devries@uva.nl in plants but resource storage can be difficult to measure. Hal Caswell However, reproduction in the prior year can be used as a h.caswell@uva.nl proxy for resource storage in species where reproduction in one year may deplete resource storage and reduce fertility in Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Amsterdam, The Netherlands the following year. If it is not possible to measure resource 130 Theor Ecol (2018) 11:129–140 storage, one might therefore incorporate prior reproductive to changes in parameters. As an example, we develop a status into the state variable to improve the model. model and calculate the elasticity of the population growth A variety of prior conditions which affect vital rates have rate of the herbaceous perennial plant Lathyrus vernus. been found empirically. Reynolds and Burke (2011) found that chestnuts with fast early growth died younger than Terminology Our discussion requires careful definitions of chestnuts with slow early growth. Warren et al. (2014) found terms in order to clarify the way that historical effects are that previous breeding success and current body condition incorporated. We will say that the life cycle is described in may be among the most important determinants of breeding terms of stages (e.g., size classes). The prior condition of an propensity in female lesser scaup. Rouan et al. (2009) found individual is some function of its stage at the prior time and that choice of next breeding site is affected by both current its stage at the current time, and thus incorporates histori- and prior breeding site in Branta canadensis.These exam- cal information. The combination of current stage and prior ples show that prior condition can be a source of individual condition serves as the individual state variable for the anal- variation in vital parameters, i.e., a source of heterogene- ysis. We give an overview of the terminology used in this ity. Some attempts have been made to include this source paper in Table 1. of heterogeneity into population models, see Pfister and The prior condition can be any arbitrary function of prior Wang (2005), Ehrlen ´ (2000), andRouanetal. (2009), but and current stage. Prior condition might be the stage at the a framework for incorporating general prior conditions into prior time, it might be defined by membership in a set of demographic models does not exist and will be presented in stages at the prior time, or it might be defined as the differ- this paper. ence between the current and the prior stages. Suppose for When constructing a structured population model, the i- example that stages are defined by size, in the hope that a state variables are used to classify individuals into states in size classification would be a satisfactory i-state variable. It a population vector n(t) whose entries give the densities of might turn out that historical effects require including size each state. The population vector is projected forward by a at the prior time in the i-state variable. Alternatively, the population projection matrix A i-state might require information only on membership in a class of sizes (e.g., larger than average or smaller than n(t + 1) = An(t ). (1) average); we will refer to these classes of prior stages as equivalence classes. Or, the i-state might require informa- The matrix A can be decomposed into a matrix U, contain- tion on the change in size between the previous time and the ing transition probabilities for existing individuals, and a current time, and individuals might be classified by whether matrix F, describing the generation of new individuals by they have grown, shrunk, or remained in the same size class. reproduction: Models based on prior stage are described in section “Full prior stage dependence,” models based on general func- A = U + F. (2) tions of prior and current stages are described in section If prior conditions influence present dynamics, the vector n “Prior condition models,” and models based on equivalence and the matrices U, F,and A must be modified to account classes of prior stage are described in section “Equivalence classes of prior stages.” The matrices, vectors, and mathe- for these influences. Our goal in this paper is to present a systematic method for constructing such models in which matical operations used in this paper are listed in Table 2. individuals are classified by current stage and (very gener- ally defined) prior condition. Because we are considering effects of individual condition at just one prior time, we Model construction refer to these as second-order matrix population models. We will present the demographic analysis of such models at the We will begin by constructing the model in which the prior level of the individual, the cohort, and the population, and condition is defined by full information on the prior stage, show how to carry out sensitivity analyses of model results which we refer to as full prior stage dependence. Table 1 Terminology used to distinguish the state of an individual, the stage of the life cycle, and the prior condition of the individual • The state of an individual is the information necessary to predict the response of an individual to its environment. • The stage of an individual refers to a biologically defined category, usually a life cycle stage, which is used to define a (possibly unsuccessful) state variable. • The condition of an individual is a flexible term that refers to some function of the prior stage and current stage of the individual; this historical information may be combined with the present stage to obtain a state variable based on current stage and prior condition. Theor Ecol (2018) 11:129–140 131 Table 2 Mathematical notation used in this paper Quantity Description Dimension U Prior stage dependent cohort projection matrix s(s + 1) × s(s + 1) U Matrix with transition rates (from current to future state) for individuals with prior stageis × s V Prior condition dependent cohort projection matrix sr × sr F Prior stage dependent fertility matrix s(s + 1) × s(s + 1) F Matrix with fertility rates (from current to future state) for individuals with prior stageis × s G Prior condition dependent fertility matrix sr × sr n ˜ Prior stage model population vector n ˜ =vec(N)s(s + 1) × 1 m ˜ Prior condition model population vector m ˜ =vec(M)sr × 1 C Matrix relating the prior stage dependent vector n ˜ to the prior condition dependent population vector m ˜ sr × s(s + 1) φ(i, j ) Matrix whose (i, j )th entry indicates the prior condition for an individual that makes an i → j transition s × s k k φ Matrix whose (i, j )th entry is one if φ(i, j ) = k and zero otherwise, in MATLAB notation φ (i, j ) = (φ == k) s × s I Identity matrix s × s 1 Vector of ones s × 1 e The ith unit vector, with a 1 in the ith entry and zeros elsewhere Various E A matrix with a 1 in the (i,j) position and zeros elsewhere Various ij ⊗ Kronecker product X(:,i) Column i of matrix X vecX The vec operator, which stacks the columns of an m × n matrix X into a mn × 1 vector Dimensions of vectors and matrices are given where relevant; s denotes the number of classes in the full second-order model and r denotes the number of classes in the reduced second-order model Full prior stage dependence where n is the number of individuals whose current ij stage is i and whose prior stage was j.Ehrlen ´ (2000) Creating a prior stage model requires transition and fertil- suggested that this array could be updated by multipli- cation with a three-dimensional matrix. However, matrix ity rates to be measured for each possible prior stage. Since newborns do not have a well-defined prior stage in general, multiplication can not be used directly to project such a two-dimensional array (this would require tensors). Instead, an extra prior stage must be added for newborns. If there are s current stages, we will label the prior condition of new- the two-dimensional array is transformed into a vector by stacking the columns on top of each other: borns as stage s + 1. Individuals are thus classified by their current stage 1, 2,...,s and their prior stage 1, 2,...,s +1. n ˜ (t ) = vecN(t ). (8) The transition and fertility rates, conditional on prior stage, are given by the matrices U and F : k k We use the tilde notation to denote vectors and matrices that relate to the prior condition model. The entries in the popu- U s × s Transitions among stages for individuals (3) lation vector n ˜ are now ordered first by prior stage and then with prior stage k = 1,...,s + 1, by current stage: F s × s Reproduction by individuals (4) with prior stage k = 1,...,s + 1. The entries of U , denoted by u , are defined as ij u = P j → i | prior stage = k , (5) ij f = E offspring of stage i | prior stage = k . (6) ij It is useful to think of the state of the population as described by a two-dimensional array N of size s × (s + 1) The vector n ˜ is projected by a transition matrix U and ˜ ˜ ˜ a fertility matrix F.The U and F matrices are constructed, respectively, from the set of matrices U and F .The transi- k k tion matrix U projects individuals into their next stage while keeping track of their prior stage. The matrix U is written in terms of block matrices corresponding to the blocks in n ˜ 132 Theor Ecol (2018) 11:129–140 in Eq. 9.In MATLAB notation, the transition matrix for a prior stages. We consider the case where this function can model with s = 2is be defined as a linear transformation of the full prior stage dependent model, where the population vector is written as m ˜ , defined by m ˜ (t ) = Cn ˜ (t ), (15) for some matrix C. An example of such a prior condition is having previously grown, shrunk, or stayed the same size. where U (:,i) refers to the ith column of the matrix U and k k The matrix C maps i-states defined by the combination 0 is a column vector of size s×1. To understand the structure of (current stage × prior stage) to i-states defined by the of U, consider the (1,1) block in the upper left corner. This combination of (current stage × prior condition). block projects individuals with prior stage 1 at t to prior Suppose there are r distinct prior conditions. The state of stage 1 at t +1. The only such individuals have current stage the population is now given by a s × rarray M 1 at time t. They are projected by column 1 of U . All other columnsofthe(1,1)blockof U are zero. The other blocks of U are filled similarly. The blocks in the last row of U are zero because transi- tions into the newborn stage are impossible; the matrix F will fill up this row block. In general, for any number of stages s, where m represents the number of individuals whose cur- s+1 s ij rent stage is i and whose prior condition is j.AsinEq. 8, U = E ⊗ e ⊗ U e , (11) ij j i the population vector m ˜ is j =1 i=1 where e has size s × 1and E has size (s + 1) × (s + 1) m ˜ (t) = vecM(t ). (17) i ij and has a one in position (i,j) and zeros elsewhere. The key to the construction of the prior condition model The fertility matrix F is constructed from the F . Because is to derive C from the rule defining the prior condition. To individuals are always born into stage s + 1, the F appear do so, define a matrix φ in the last row of blocks in F. For the case with s = 2, the fertility matrix is φ(i, j ) = prior condition for an individual that makes an j → i transition. (18) Forexample,ifthe r = 3 prior conditions are shrinking, stasis, and growth 1for i< j shrinking, In general, for any number of stages s, ⎨ φ(i, j ) = 2for i = j stasis, (19) s+1 3for i> j growth. F = E ⊗ F , (13) (s+1)k k k=1 Next, we define a set of matrices φ (i, j ),for k = 1,...,r, where E has size (s + 1) × (s + 1) and has a one in given by (s+1)k position (s + 1, k) and zeros everywhere else. 1if φ(i, j ) = k, Given a set of transition matrices U and fertility matrices φ (i, j ) = (20) 0 otherwise . ˜ ˜ F ,Eqs. 11 and 13 define the second-order matrices U and F, and the population projection matrix Given the matrices φ , the matrix C is given by ˜ ˜ ˜ A = U + F. (14) T k C = 1 ⊗ e ⊗ I diag vecφ , (21) k s s+1 These matrices are subject to all the usual demographic k=1 analyses, including population growth, population structure, see Appendix A. To project the new population vector m ˜ , and sensitivity analysis (see section “Case study”foran ˜ ˜ ˜ we replace the matrices U and F with new matrices V and example). G, respectively, so that Prior condition models ˜ ˜ ˜ ˜ m(t + 1) = V + G m(t ) (22) A more general second-order model structure allows transi- The matrix V describes transitions of extant individuals tions and fertility to depend on some function of current and between the different i-states of the prior condition model Theor Ecol (2018) 11:129–140 133 and the matrix G describes the production of new individu- the C matrix that transforms the full prior stage dependent als in the prior condition model. Substituting (15) into both population vector n ˜ to the equivalence class population vec- sides of Eq. 22 yields tor m ˜ . We transform the population matrix N in Eq. 7 into the equivalence class population matrix M in Eq. 16 by a ˜ ˜ ˜ ˜ C U + F n ˜ (t ) = V + G Cn ˜ (t ). (23) matrix B; ˜ ˜ Equation 23 is satisfied if V and G satisfy the following M = NB. (31) equations: The matrix B has size (s + 1) × r and its entries are ˜ ˜ VC = CU (24) GC = CF. (25) 1if stage i ∈ equivalence class j, B(i, j ) = (32) 0 otherwise. In general, the matrix C is not square and does not have an inverse. So we use the Moore-Penrose pseudo-inverse C Applying the vec operator to Eq. 31 gives ˜ ˜ (see Abadir and Magnus (2005)) to solve for V and G: † m ˜ (t ) = (B ⊗ I )n ˜ (t ), (33) ˜ ˜ V = CUC , (26) ˜ ˜ G = CFC . (27) where we have used the following result from Roth (1934) that vecABC = C ⊗ A vecB. Equation 33 is a special If C is square and non-singular, the pseudo-inverse is the case of Eq. 15 with ordinary inverse: † −1 C = C . (28) C = (B ⊗ I ). (34) If C is not square but has linearly independent rows (i.e., has Since the rows of B are orthogonal, the matrix B and the full row rank), matrix (B ⊗ I ) both have full row rank and therefore C † T T −1 C = C (CC ) . (29) can be calculated from Eq. 29. If C has rows of zeroes (this happens if some combina- tions of current stage and prior conditions are impossible), Sensitivity analysis then C will not have full rank. In this case, C is com- puted from the singular value decomposition, implemented Sensitivity analysis provides the effect of changes in any in MATLAB with the function pinv(C) and in R with the parameter on any model outcome. In general, these com- function Ginv(C). For example, in a size-classified model, putations require derivatives of scalar-, vector-, or matrix- it is impossible to be in the smallest size class and to have valued functions with respect to scalar-, vector-, or matrix- grown into it from a smaller size class. In such cases, C valued arguments. Matrix calculus is a formalism which and thus also V and G have rows and columns of zeros enables us to consistently calculate such derivates. For an corresponding to the impossible combinations. introduction to matrix calculus, see Abadir and Magnus Equations 26 and 27 define the prior condition matrices (2005); for details see Magnus and Neudecker (1988). Eco- ˜ ˜ V and G and the population projection matrix logical applications of sensitivity analysis appear in Caswell ˜ ˜ A = V + G. (30) (2007, 2009, 2012). Consider some scalar or vector output of the model, ξ, As in Eq. 14, the usual demographic results can be obtained ˜ ˜ ˜ which is computed from U and F, or from V and G.The from these matrices. ˜ ˜ ˜ ˜ matrices U, F, V,and G are in turn computed from the matrices U and F . Suppose U and F depend on a p × 1 i i i i Equivalence classes of prior stages vector of parameters θ, i.e., U = U [θ ],thenweusethe i i chain rule to write Equivalence classes of prior stages are a special case of functions of current and prior stage. Equivalence classes are ˜ ˜ dξ dξ dvecU dξ dvecF = + , (35) subsets of prior conditions that depend only on the prior ˜ ˜ dθ dθ dθ dvec U dvec F stage. For example, individuals in a size-classified model or might be categorized into two equivalence classes depend- ing on whether they were previously above or below some ˜ ˜ dξ dξ dvecV dξ dvecG = + . (36) threshold size. ˜ ˜ dθ dvec V dθ dvec G dθ The machinery described in the previous section, i.e., Eqs. 21, 26,and 27, can be used to write down the equiva- The first terms in Eqs. 35 and 36 capture effects through sur- lence class model. However, there is an easier way to find vival and transitions, and the second terms capture effects 134 Theor Ecol (2018) 11:129–140 through fertility. The elasticities, or proportional sensitivi- and G), and the dependence of U and F on the parameter θ. i i ties, are given by These items are specific to the question under consideration, so we provide an example in the next section. ξ dξ −1 = diag (ξ ) diag(θ ). (37) θ dθ ˜ ˜ ˜ ˜ The derivatives of U, F, V,and G with respect to θ Case study ˜ ˜ depend on how U and F depend on the U and F matrices. i i Differentiating (11), we obtain As an example, we apply the second-order formalism to a fully prior stage dependent model of a perennial plant. s+1 dvecU dvecU U We will construct the model and calculate the population = Q . (38) dθ dθ growth rate, the stable population vector, the reproductive j =1 value vector, and the elasticity of the population growth rate Each term in the summation captures the effect of θ through λ to proportional changes in the demographic rates. one of the U . The matrix Q is given by j Our analysis is based on a study by Ehrlen ´ (2000)of Lathyrus vernus, a long-lived herb native to forest mar- gins and woodlands in central and northern Europe and Q = I ⊗ K ⊗ I vec E ⊗ I 2 (E ⊗ I ) . s+1 s,s+1 s ij ii s Siberia. Ehrlen ´ classified individuals into seven stages: seed i=1 (SD), seedling (SL), very small (VS), small (SM), large (39) vegetative (VL), flowering (FL), and dormant (DO). Ehrlen Similarly, differentiating (13)gives constructed two matrix models: a first-order model with no historical effects and a second-order model. We construct s+1 dvecF dvecF our second-order model from the transition and fertility = Q , (40) dθ dθ rates calculated by Ehrlen ´ (2000) (their Table 2). i=1 We introduce a special prior stage for newborns; the where each term captures the effect of θ through one of the second-order model therefore has a total of 7 × 8 = 56 F . The matrix Q is given by states. We used Ehrlen ´ (2000)’s demographic rates to con- struct a transition matrix U and a fertility matrix F for each Q = I ⊗ K ⊗ I vec E ⊗ I . (41) i i s+1 s,s+1 s s+1,i i s of the eight prior stages. In Table 3, the transition matrix To calculate the derivatives of the prior condition matri- for individuals who were previously in stage SM is shown ˜ ˜ ces V and G, we use the chain rule to write as an example. The first two columns of this matrix are zero because small plants can not go back to being seeds ˜ ˜ ˜ dvecV dvecV dvecU = , (42) or seedlings. All eight transition and fertility matrices are in dθ dθ dvecU the Supplementary Material. We constructed the projection matrix, A, from the U ˜ ˜ ˜ i dvecG dvecG dvecF = . (43) and F matrices using Eqs. 11 and 13. The resulting 56 × dθ dθ dvecF 56 matrices are available in the Supplementary Material. Differentiating Eqs. 26 and 27, we obtain dvecV = C ⊗ C, (44) dvecU Table 3 The transition matrix for Lathyrus vernus individuals with prior stage small (SM) dvecG = C ⊗ C. (45) dvecF Thus, ˜ ˜ dvecV dvecU = C ⊗ C , (46) dθ dθ ˜ ˜ dvecG dvecF = C ⊗ C . (47) dθ dθ The columns in this matrix represent transitions out of the following Equations 38 and 40 are substituted into Eq. 35. current stages from left to right: seed (SD), seedling (SL), very small Equations 46 and 47 are substituted into Eq. 36. All that (VS), small (SM), vegetative large (VL), flowering (FL), and dormant ˜ ˜ ˜ (DO) remains is to calculate the dependence of ξ on U and F (or V Theor Ecol (2018) 11:129–140 135 The population growth rate λ is given by the dominant eigenvalue of A, λ = 0.985. (48) 0.25 0.2 This agrees with the value reported for the second-order model by Ehrlen ´ (2000). Ehrlen ´ also fitted a first-order 0.15 model to the same data and found a value just above one 0.1 (λ = 1.010). The stable population vector, w ˜ , and the repro- 0.05 ductive value vector, v ˜, which are displayed in Figs. 1 and 2, are the right and left eigenvectors of A, respectively. The entries of the stable population vector are denoted by w ˜ ij SD SL NB for individuals with current stage i and prior stage j,analo- VS DO SM FL gously to the entries of the population vector n ˜ in Eq. 9.The VL VL c SM entries of the marginal stable current stage distribution, w , FL VS DO SL Current stage are given by SD Prior stage w = w ˜ (49) ij 0.3 j =1 0.25 and are shown in Fig. 1b. Similarly, the entries of the marginal stable prior stage distribution, w , are given by 0.2 w = w ˜ (50) ij j 0.15 i=1 0.1 andareshowninFig. 1c. Individuals with the same current stage have different 0.05 vital rates if they differ in prior stage and this heterogeneity due to prior stage affects the population growth rate λ.To SD SL VS SM VL FL DO quantify the relative effect of the different prior stage depen- Marginal distribution over current stages dent transition matrices on the population growth rate, we calculate the elasticity of the population growth rate, λ,to changes in the U matrices 0.3 0.25 ,i = 1,..., 8. (52) vec U 0.2 Substituting λ for ξ in Eq. 37 yields 0.15 λ 1 dλ = diag (vecU ) . (53) 0.1 vec U λ dvec U i i 0.05 As shown in Caswell (2001), dλ 0 T T SD SL VS SM VL FL DO NB = w ⊗ v , (54) i i Marginal distribution over prior stages dvec U where w and v are the right and left eigenvectors of U , i i i Fig. 1 a Stable current stage × prior stage distribution. b Marginal respectively, scaled so that current stage distribution. c Marginal prior stage distribution. Notation: seed (SD), seedling (SL), very small (VS), small (SM), vegetative large v w = 1forall i, (55) (VL), flowering (FL), dormant (DO), and newborn prior (NB) 1 w = 1forall i. (56) We sum the entries of Eq. 53 to get the elasticity of λ demography of individuals with prior stages seed, seedling, to a proportional change in all of the entries of U .The or newborn have little effect on λ. Proportional changes in results are shown in Fig. 3. We note that there are large individuals who were small vegetative at the prior time have differences among prior stages. Proportional changes in the effects an order of magnitude larger. Fraction Fraction Fraction 136 Theor Ecol (2018) 11:129–140 intuition, tradition, practical limitations, and formal statisti- cal analyses. Even after a careful choice of i-state, it may happen that individual prior conditions contain important information about the fate of individuals. In such cases, 0.8 history matters, and the methods presented here solve the 0.6 problem of how to incorporate information about it into matrix models. 0.4 Why stop at second-order effects, what about the effect 0.2 of the prior condition at t − 2, t − 3, etc.? It is theo- retically possible to extend the framework presented here to include dependence on higher-order effects. However, if DO FL the i-state variable is such that third- or even fourth-order VL SD SM historical effects are important, it might better to recon- SL VS VS SM SL sider the choice of i-state variable instead of including ever VL Current stage FL SD DO increasing historical dependence. NB Prior stage Statistical tests for second-order effects in longitudinal data using log-linear models have been developed by Bishop et al. (1975) and Usher (1979). Time series of longitudinal data are needed to perform these tests as well as for the sub- 3.5 sequent estimation of a prior condition dependent model. Capture–mark–recapture analyses for prior stage dependent 2.5 models have been developed by Pradel (2005)and Cole et al. (2014). Incorporating individual history requires a decision about 1.5 what aspects of the prior condition are important. We have presented three biologically motivated cases: prior condi- tion as the prior stage, prior condition as an arbitrary linear 0.5 function of current and prior stages, and prior condition as SD SL VS SM VL FL DO an equivalence class of prior stages. In each case, the nec- Current stage essary information is a set of fertility and survival/transition matrices for each prior stage. The resulting models use 3.5 block-structured matrices to project a vector of stages within prior conditions. These matrices can be used to calculate all the usual demographic outcomes. Because the matri- 2.5 ces are carefully constructed from U and F , they can be i i subjected to sensitivity analysis. It is straightforward to cal- 1.5 culate the sensitivity and elasticity of any model outcome to any parameters affecting the vital rates. Prior condition effects are more than just a convenient 0.5 tool in constructing i-states for population models. They are a biologically real source of inter-individual variation. SD SL VS SM VL FL DO NB Prior stage The importance of individual variation in vital parameters to ecological processes has become increasingly evident in recent years (Bolnick et al. 2011; Valpine et al. 2014; Fig. 2 a Reproductive value vector for each current stage × prior Caswell 2014; Vindenes and Langangen 2015; Steiner and stage combination. b Marginal current stage reproductive value vector. c Marginal prior stage reproductive value vector. Notation: seed (SD), Tuljapurkar 2012; Cam et al. 2016). Ignoring individual seedling (SL), very small (VS), small (SM), vegetative large (VL), variation in vital parameters, also referred to as individ- flowering (FL), dormant (DO), and newborn prior (NB) ual heterogeneity, can have important consequences for the demographic outcomes and subsequent conclusions; see for Discussion example Vaupel et al. (1979), Rees et al. (2000), Fujiwara and Caswell (2001), Vindenes and Langangen (2015), and When does history matter? Population models are based Cam et al. (2016). In his study of Lathyrus,Ehrlen ´ (2000) on i-state variables chosen by some mixture of biological found that including heterogeneity due to prior stage had Reproductive value Marginal reproductive value Marginal reproductive value Theor Ecol (2018) 11:129–140 137 Fig. 3 The elasticity of λ to a 0.35 proportional change in all of the 0.3 0.25 0.2 0.15 0.1 0.05 Seed Seedling Very small Small Vegetative Flowering Dormant Newborn Prior stage only a small effect on λ, although it was sufficient to cause pose interesting challenges and is an open research problem the population to decrease rather than increase. for population ecology. How much heterogeneity is introduced by the prior con- dition effects in the Lathyrus example? Observations of Acknowledgements We thank Gregory Roth for helpful discussions individual plants can identify their current stage, but not and Johan Ehrlen ´ for providing the data for the Lathyrus case study. their prior condition. The stable structure, w ˜ (Fig. 1), shows the joint probability distribution over current and prior Funding information This work was supported by European Research Council Advanced Grant 322989. stages, and the amount of heterogeneity in the stable popu- lation can be calculated from the entropy of this distribution. The entropy of the joint current × prior stage distribution is Appendix A: Derivation of the matrix C H(p, c) =− w ln w = 2.61. (57) ij ij i,j In this appendix, we will derive the matrix C in Eq. 21,which transforms the fully prior stage dependent population vector, This measures the overall heterogeneity in the stable pop- n ˜, into the prior condition dependent population vector, m ˜ , ulation structure. The heterogeneity in the marginal current stage distribution, w [Eq. 49], is m ˜ (t ) = Cn ˜ (t ), (60) c c as in Eq. 15. Recall that we defined the matrix φ with H(c) =− w ln w = 1.69. (58) i i elements i=1 φ(i, j ) = prior condition for an individual that makes an This is the observable heterogeneity in current stage. The heterogeneity contributed by the unobservable prior stage, j → itransition, (61) taking into account the relationship between current and and we defined a set of indicator matrices φ ,for k = prior stage, is 1,...,r,given by H (p|c) = H(p, c) − H(c) = 0.92, (59) 1if φ(i, j ) = k, φ (i, j ) = (62) Khinchin (1957). Thus, in this example, the prior stage con- 0 otherwise . tributes about 35% of the total heterogeneity in the stable In Appendix B, we show a few examples of φ matrices. population. Recall furthermore the matrix In this paper, we have considered only linear models in constant environments. However, models could easily be constructed to include density effects, by making the U and F functions of density. Periodic or stochastic models could be constructed by making the U and/or the F appropriate i i functions of time. The analysis of the resulting models may Elasticity of to change in U i 138 Theor Ecol (2018) 11:129–140 where n is the number of individuals whose current stage For example, when s = 3, φ is ij is i and whose prior stage was j, and the matrix where m is the number of individuals whose current stage ij is i and whose prior condition was j. Finally, recall that the (B) When prior condition = equivalence classes of prior population vectors are given by stages,then n(t ) = vecN(t ), (65) φ(i, j ) = function(j ) (74) m ˜ (t ) = vecM(t ). (66) k For example, for s = 3 and for the case of the Using the above defined matrices φ and N, the matrix equivalence classes j = 1and j> 1, we get M can be written as r 1for j = 1 φ(i, j ) = (75) k T M = φ ◦ N 1 ⊗ e . (67) 2for j = 2, 3 s+1 k=1 The first term in the product is a matrix containing the den- sities of individuals in all entries of N that correspond to prior condition k, and zeros elsewhere. The second term in the product adds the densities across each row of the matrix φ ◦ N and puts these elements in the kth column of M. Taking the vec of both sides of this equation gives T k m ˜ = 1 ⊗ e ⊗ I vec φ ◦ N , (68) k s s+1 k=1 (C) When prior condition = shrinkage, stasis, or growth, then T k = 1 ⊗ e ⊗ I diag vecφ n, (69) k s s+1 k=1 1for i< j shrinkage T φ(i, j ) = 2for i = j stasis (76) wherewehaveusedvecABC = C ⊗ A vecB (Roth 3for i> j growth 1934)and vec (A ◦ B) = diag (vecA) vec (B). Comparison with Eq. 60 shows that For example, for s = 3, φ is T k C = 1 ⊗ e ⊗ I diag vecφ . (70) k s s+1 k=1 Appendix B: Examples of φ and C matrices B.1 Examples of φ matrices In this appendix, we show a few examples of the matrix φ. (D) General more complicated growth conditions. For (A) When prior condition = prior stage,then example, suppose the prior condition reflects the amount of growth, so that φ(i, j ) = j for all i. (71) φ(i, j ) = i − j + s. (78) Theor Ecol (2018) 11:129–140 139 ⎛ ⎞ For s = 4, this results in the following φ ⎝ ⎠ φ = , (81) ⎛ ⎞ ⎝ ⎠ φ = 100 . (82) Substituting these into Eq. 70, and letting the sum range from 1 to 3 only, gives the matrix C as B.2 An example C matrix We consider the example in Eqs. 76 and 77 with three size classes and three growth classes corresponding to growth, shrinkage, and stasis. For simplicity, we consider a cohort model with no fertility and therefore no special prior condi- The transition matrix for the prior condition model, V, tion for newborns. The matrices φ are is given by Eq. 26 and requires the pseudo-inverse of the ⎛ ⎞ matrix C, which can be found using pinv(C) in MATLAB ⎝ ⎠ φ = 001 , (80) or Ginv(C) in R. Substitution of the pseudo-inverse of C into Eq. 26 and a few lines of algebra result in This matrix captures the complicated dependence of transi- References tions on both current stage and prior condition. 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Theoretical EcologySpringer Journals

Published: Dec 8, 2017

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