In our increasingly unstable and unpredictable world, population dynamics rarely settle uniformly to long-term behaviour. However, projecting period-by-period through the preceding fluctuations is more data-intensive and analytically involved than evaluating at equilibrium. To efficiently model populations and best inform policy, we require pragmatic suggestions as to when it is necessary to incorporate short-term transient dynamics and their effect on eventual projected population size. To estimate this need for matrix population modelling, we adopt a linear algebraic quantity known as non-normality. Matrix non-normality is distinct from normality in the Gaussian sense, and indicates the amplificatory potential of the population projection matrix given a particular population vector. In this paper, we compare and contrast three well-regarded metrics of non-normality, which were calculated for over 1000 age-structured human population projection matrices from 42 European countries in the period 1960 to 2014. Non-normality increased over time, mirroring the indices of transient dynamics that peaked around the millennium. By standardising the matrices to focus on transient dynamics and not changes in the asymp- totic growth rate, we show that the damping ratio is an uninformative predictor of whether a population is prone to transient booms or busts in its size. These analyses suggest that population ecology approaches to inferring transient dynamics have too often relied on suboptimal analytical tools focussed on an initial population vector rather than the capacity of the life cycle to amplify or dampen transient fluctuations. Finally, we introduce the engineering technique of pseudospectra analysis to population ecology, which, like matrix non-normality, provides a more complete description of the transient fluctuations than the damping ratio. Pseudospectra analysis could further support non-normality assessment to enable a greater under- standing of when we might expect transient phases to impact eventual population dynamics. Keywords Damping ratio · Europe · Eurostat · Human demography · Population projection matrix · Pseudospectra * Alex Nicol-Harper Introduction alex.nh13@gmail.com Ocean and Earth Science, National Oceanography Our world is in constant flux, so populations are never at Centre, University of Southampton Waterfront Campus, equilibrium. Population dynamics are altered by ongoing Southampton, UK and abrupt processes, both immediately and over longer Environment and Sustainability Institute, College timescales, diverting trajectories from the paths they would of Engineering, Mathematics and Physical Sciences, otherwise follow. Transient fluctuations such as baby booms University of Exeter in Cornwall, Penryn Campus, Cornwall, UK dampen away, leaving population size modified by a process known as momentum (Keyfitz 1971; Espenshade and Tan- Department of Social Statistics and Demography, ESRC Centre for Population Change, Southampton Statistical nen 2015)—or more generally and formally, inertia. Inertia Sciences Research Institute, University of Southampton, occurs when unstable population structures cause eventual Southampton, UK population size to be larger or smaller than if projected from Centre for Ecology and Conservation, College of Life a stable initial stage structure; momentum is the special case and Environmental Sciences, University of Exeter for stationary populations with zero growth (Koons et al. in Cornwall, Penryn Campus, Cornwall, UK Vol.:(0123456789) 1 3 186 Population Ecology (2018) 60:185–196 2007). Given the importance of population projections to (vital rates), to project population structures over time. The national and global development policies (UN 2015), we ‘eigendecomposition’ of a matrix determines the spectrum need a better understanding of how transients affect popu- (set of eigenvalues) and ‘natural directions’ (set of eigenvec- lation dynamics in the short- and long-term (Osotimehin tors) of a matrix, and is used to analyse the model: for PPMs, 2011), and how responses are shaped by environmental and the dominant eigenvalue gives the asymptotic growth rate, social factors at a range of spatial scales (Hastings 2004; and its associated right and left eigenvectors determine the Harper 2013). stable (st)age structure and (st)age-specific reproductive val- Although equilibrium approximations are useful in the ues, respectively. Subdominant eigendata pertain to transient absence of complete population knowledge at each point responses, with decreasing influence over time following dis- in time (Caswell 2000), there is increasing recognition that turbance from the stable (st)age structure (Caswell 2001). systems are dynamic entities for which short-term transient The classical metric of the duration of this decreasing effects must also be considered as fundamental aspects of influence is the damping ratio, which is calculated as the ratio ecological dynamics (Hastings 2004; Ezard et al. 2010; of the dominant eigenvalue divided by the absolute value of Stott et al. 2010), explaining approximately half of the vari- the subdominant eigenvalue (Caswell 2001). As a measure of ation in growth rates in comparative studies of plants (Ellis ‘intrinsic population resilience’ to transient deviations (with and Crone 2013; McDonald et al. 2016). This is especially a higher value suggesting a shorter recovery time), the damp- important when shorter timescales are of greater applied rel- ing ratio has been shown to be useful in comparative demog- evance (Hastings 2004; Ezard et al. 2010), or when repeated raphy (Stott et al. 2011). However, it is methodologically lim- disturbances prevent populations from settling to equilib- ited, because rather than bounding the duration of transient rium behaviour (Townley and Hodgson 2008; Tremblay dynamics, it actually measures the asymptotic rate at which et al. 2015). In human populations, gradual demographic transients decay. As such, it correlates weakly with conver- transitions (from high to low rates of mortality and fertility) gence times of realistic population projections (Stott et al. are a major driver of transient phenomena (Blue and Espen- 2011) because transient dynamics are not determined solely shade 2011), over and above abrupt disturbances such as by the largest two eigenvalues, as the damping ratio assumes, wars and pandemics. In deterministic models—as used here but rather by the whole set. Figure 1 shows an eigenvalue for conceptual clarity (see Ezard et al. 2010)—transients can spectrum for a PPM for Bulgaria in 2014, demonstrating that be considered deterministic responses to stochastic events many of the lower eigenvalues can have magnitudes similar (Stott et al. 2010). This allows setting of bounds, which to the subdominant one—highlighting how much informa- “help to create an envelope of possible future population tion for predicting transient dynamics is lost when focusing scenarios around the mean, long-term prediction” (Townley solely on the damping ratio. More integrative measures of and Hodgson 2008, p. 1836), aiding in the incorporation of eigenvalue variation have the potential to increase the accu- at least some aspects of uncertainty into near-term estimates racy of transient dynamic predictions (cf. Crone et al. 2013). for a given population structure. In population ecology, transients are the result of an ini- We know that transients occur when disturbances desta- tial population vector being propagated through a population bilise population structure, causing deviation from the pro- projection matrix. The focus of efforts into transient fluctua- portional composition that balances different groups’ varying tions has most often centred on how the population structure contributions to population growth or decline (Townley and at a given point in time differs from the stable age distribu- Hodgson 2008). Precise predictions of transient dynamics tion [reviewed by Williams et al. (2011)]. As individuals at require detailed and frequent updating of population structures, different developmental (st)ages have different mortality and which is typically data-intensive, as it requires making spe- fertility rates, the discrepancy between observed and sta- cific, fine-grained assumptions about the future (Townley et al. ble population structures causes the aggregated population 2007). In long-lived organisms with age-dependent schedules growth rate to change despite constant demographic rates of maturation and reproduction, such as modern humans Homo (Koons et al. 2005; Ezard et al. 2010; Stott et al. 2011). This sapiens, structuring is by age: stable age structure is deter- focus on population structures represents a single side of the mined by the age-structured life table (Caswell 2001). Given same coin—a given initial condition can have very different that transient analysis “produce[s] output which is compli- transient dynamics depending on the matrix through which cated, and difficult to define succinctly” (Yearsley 2004, p. it is projected. This leads to asking whether there are prop- 245), it would be useful to have diagnostic tools to indicate if erties of the PPM that can indicate a system’s propensity to it is desirable to perform further analyses on transients. exhibit amplificatory dynamics. Asymptotic and transient behaviour can be disentangled in matrix population modelling (Caswell 2001). Population For readers unfamiliar with eigenvalues and eigenvectors, we rec- projection matrices (PPMs) are built using (st)age-specific ommend the following webpage: http://setos a.io/ev/eigen vecto rs-and- rates of reproduction and transition between life cycle stages eigen value s/. 1 3 Population Ecology (2018) 60:185–196 187 1960 to 2014 to build over 1000 PPMs of country–year combinations. After showing that non-normality has gener- ally increased in these PPMs over time, we use multivariate analyses to highlight the dependencies among the facets of matrix non-normality and classical ecological population dynamic metrics. Our three non-normality metrics correlate well with transient indices, but not with the damping ratio. These patterns are best drawn out through an important dis- tinction between non-normality for the system as a whole, combining asymptotic and transient dynamics, and that for the scaled system, when asymptotic growth rate is factored out. Finally, we also introduce to population ecology the technique of pseudospectra analysis (Trefethen and Embree 2005), originally derived from applications in fluid dynam- ics (Trefethen et al. 1993), which should prove helpful in the incorporation of non-normality assessment into matrix population modelling. Fig. 1 Eigenvalue spectrum for Bulgaria in 2014. Numbers corre- Methods spond to eigenvalues ordered by magnitude, which is calculated as the length of the vector joining each point to the origin (shown in red). Eigenvalues 13–18 lie on the origin. Note the similarity in mag- Data nitude of, say, the 4th eigenvalue to that of the 2nd We used the Eurostat database (http://ec.europa.eu/eur ost at) It has long been recognised within mathematics that tran- to collect secondary data on age-specific female population sient dynamics depend on a matrix characteristic known as sizes, births and deaths, for the 45 European countries with ‘normality’ (Elsner and Paardekooper 1987; Trefethen and complete population data for any subset of years 1960–2014 Embree 2005). If a matrix is normal its properties are fully (range 3–55 years, 6 complete sets, mean 28 years). The determined by eigendata (Trefethen and Embree 2005), the variables are provided in single-year age classes, up to the set of basis values and vectors that describe the core prop- oldest age recorded or an arbitrary ‘x years and over’ cat- erties of the system. While undoubtedly valuable (Caswell egory. Following standard human demography protocols 2001; Hodgson et al. 2006; Crone et al. 2011), eigendata (e.g., Keyfitz and Flieger 1968 , 1971, 1990; Wiśniowski are an asymptotic description and therefore cannot capture et al. 2016), we aggregated into 18 5-year bins, up to all dynamical aspects of populations’ short- and medium- ‘85 years and over’. Total births are available separated by term trajectories as determined by asymmetric, non-normal babies’ sex from 2007 only, so we estimated female births PPMs. Transient effects are limited in normal systems, but by taking the ‘sex ratio at birth’ values for the relevant can be substantial (Trefethen 1997) and potentially domi- countries and years from the World Bank Databank (http:// nating (Townley et al. 2007) in non-normal ones. A key datab ank.world bank.org/data/repor ts.aspx?sourc e=gende challenge then is to find and understand simple measures of r-statis tics), and calculating t heir grand mean. We removed non-normality that might predict and explain links between 121 country–year combinations that had five or more con- matrix asymmetry and transient dynamics in population secutive zero deaths across single-year classes—including ecology and evolutionary demography. all data for Andorra, Liechtenstein, and San Marino—since Here, we apply non-normality metrics to PPMs. Human this is either suggestive of inaccurate data collection and/or populations are particularly susceptible to transients as a curation, or related to impractical small population counts. result of culture and geopolitics inducing strong cohort This left 1,120 country–year combinations from 42 coun- effects (Ezard et al. 2010), in addition to long lifespan tries for matrix construction. Note that all available years (Koons et al. 2005, 2007). Momentum will dominate long- were used, so PPMs could overlap in their timeframes; for term population dynamics in Africa and Asia due to high example, where data were available for both 2001 and 2002, uncertainty and variability in fertility and mortality rates there would be a matrix using 2001 data projecting to 2006, (Azose et al. 2016), and can be expected to account for over and another from 2002 to 2007. half of all population growth in developing countries from 1995 to 2100 (Bongaarts 1994). We used Eurostat data for 1 3 188 Population Ecology (2018) 60:185–196 Note that Stott et al. (2011) differentially name positive Matrices and negative transient indices, such that a negative value of our ‘case-specific reactivity’ would correspond to ‘case- For each available country–year combination, we projected the observed population at year 0 to year 5, by premultiply- specific first timestep attenuation’ in their treatment. ing the initial population vector, n , by its corresponding PPM, A: i.e., n = An The timestep is 5 years due to the Non‑normality t+5 t. data being aggregated into 5-year bins; an individual which is 0–4 years old at year 0 will be 5–9 years old after projec- Elsner and Paardekooper (1987) reviewed matrix non-nor- tion. The initial population vectors had 18 entries represent- mality and presented four main metrics, one intuitive defi- ing the observed population structure across the 5-year age nition (distance from the set of normal matrices) and three bins; the PPMs were of dimension 18 × 18. Each matrix was pragmatic implementable suggestions (Table 1). All three generated via the following approximations for each bin: metrics have their foundations in AA* rather than just A, and tackle the discrepancy between AA* and A*A to reveal 5 × deaths • survival i.e., progression = 1 − the asymmetry of A. The Henrici metric uses the Frobenius population size norm of A*A, while the Frobenius and Ruhe metrics use – included along the matrix subdiagonal, for bins 0–4 the eigendata of A*A, also known as the singular value to 80–84 decomposition of A, as previously introduced to evolution- ary biology (Townley and Ezard 2013). The singular value 5 × deaths • 85 + survival i.e., stasis = 1 − population size decomposition of A is the eigendecomposition of AA*, yielding an alternative set of basis values and vectors. If – included in the final entry of the matrix diagonal � �� � 5 × births A is symmetric and normal, the singular value and eigen- • fertility = survival(maternal) population size decompositions are the same. With increasing asymmetry � � survival(0 − 4) (following the birth-flow approxima- of the PPM, the singular value and eigen-decompositions diverge. tion of Morris and Doak 2002) In order to isolate transient effects from the overall sys- – included along the top row of the matrix. tem, we present the results obtained by using standardised ( 𝐀 ) matrices in addition to raw ones (A); scaling by λ Note that negative survival values, which arose when removes differences in dynamics that result from populations quintupled deaths exceeded population size, were replaced increasing or decreasing (Koons et al. 2005; Townley and with zero. Additionally, survival was calculated separately Hodgson 2008; Stott et al. 2011). While Elsner and Paarde- for infants under 1 year and children aged 1–4 years—since kooper (1987) additionally present alternative versions of deaths are much higher in the former stage—and then com- the Frobenius and Henrici metrics using the spectral rather bined as follows: than Frobenius norm, we chose to limit our analyses to the survival(0 − 4)=(0.2 × survival (0))+(0.8 × survival (1 − 4)). Frobenius norm only, since it simplifies the interpretation of the Henrici metric (see Trefethen and Embree 2005, pp. For each matrix we computed: 444–445; Table 1). To visualise non-normality over time, we generated generalised additive mixed models (GAMMs) • eigenvalues, λ , using base R’s eigen() function, with year as a smoothed fixed effect, controlling for country • damping ratio = (Caswell 2001), as a random effect. We used the ‘gamm4’ package (Wood • case-specific reactivity, the relative population size after and Scheipl 2016), fitted with family ‘Gamma’ and the one projection interval, standardised for λ , = ‖𝐀𝐧 ‖ 1 0 1 ‘identity’ link function. (Stott et al. 2011) where ‖‖ is the one-norm (the sum of the modulus of the entries) of a vector, 𝐀 = 𝐀 ∕ , and Multivariate analyses n is the initial population structure scaled such that it sums to 1 (giving the proportions of the population in Linear correlations were calculated using Spearman’s each 5-year age bin), rank correlations; those presented were significant with inertia, the relative population size after the transient P < 0.05 and are given to 2 decimal places. Principal com- period (here defined as 100 timesteps i.e., 500 years) ponent analysis was used to assess relationships among = ‖ ‖ where ‖‖ is the one-norm (sum) of a vector 100 1 1 metrics. This was conducted (using base R’s prcomp() and 𝐧 = 𝐀 𝐧 , 100 0 function) for both raw and standardised matrices, with various non-normality metrics, discussed below. scaled and centred non-normality metrics and a range of relevant variables (see Table 2 for justifications and 1 3 Population Ecology (2018) 60:185–196 189 Table 1 Non-normality metrics Non-normality metric Formula Code in R Explanation Frobenius * * > sqrt(norm((Conj(t(A))%*%A) − (A%*%C One of the main conditions defining matrix ‖ − ‖ * * onj(t(A))), type=‘F’)) normality is the equality A A = AA ; this metric Or provides a measure of non-normality by quantify- > sqrt(norm((Conj(t(Re(Â)))%*%Re(Â)) − ing the discrepancy between A and A (Re(Â)%*%Conj(t(Re(Â)))), type=“F”)) Henrici > Re(sqrt(norm(A, type=“F”)^2 − sum(abs This metric considers all eigenvalues of matrix A, 2 n 2 ‖‖ − � � F k=1 (eigen(A)$values)^2))) and is in fact a rearrangement of the Frobenius Or norm of A A. It quantifies non-normality since > Re(sqrt(norm(Re(Â), type=“F”)^2 − sum “A is normal if and only if [formula] = 0” (Hen- (abs(eigen(Re(Â))$values)^2))) rici 1962, p. 27) Ruhe > max(svd(A)$d − abs(eigen(A)$values)) Maximum difference between singular value and max − k k k Or associated absolute eigenvalue: close to normal > max(svd(Re(Â))$d − abs(eigen(Re(Â))$ if similar, increasingly non-normal with distance. values)) The singular value decomposition is the eigende- composition of AA , yielding an alternative set of basis values and vectors A is a matrix ( 𝐀 if standardised); A is the conjugate transpose of A; is a scalar magnitude; ‖‖ is a matrix norm—subscript F specifies the Frobenius norm: (Σ Σ |a | ) (a is a matrix entry, where i denotes row and j denotes column); λ is the kth eigenvalue (ordered by decreasing j i ij ij k magnitude) of total n (n = matrix dimension); σ is the kth singular value (ordered corresponding to eigenvalues) (Elsner and Paardekooper 1987; Henrici 1962; Ruhe 1975) Table 2 Variables used in the principal component analysis Variable Justification Definition Year Transient dynamics were expected to change over time N/A Asymptotic growth rate, λ A component of total population growth rate The rate at which the population would grow or decline in A key matrix output the absence of transient dynamics The numerator of the damping ratio The dominant eigenvalue of a PPM Damping ratio A metric originally formulated to measure the duration The dominant eigenvalue divided by the absolute value of of transient impact the subdominant eigenvalue (which can be a complex number)—see “Matrices” Reactivity An index of short-term transient impact Relative population size, after scaling out the asymptotic growth rate, in the first timestep—see “Matrices” Inertia An index of long-term transient impact Relative population size, after scaling out the asymptotic growth rate, after 100 timesteps—see “Matrices” Frobenius non-normality Metric under consideration See Table 1 Henrici non-normality Metric under consideration See Table 1 Ruhe non-normality Metric under consideration See Table 1 definitions). We generated biplots from the informative Results principal components—defined as those with eigenvalues exceeding 1, after ‘conservative’ bias correction using Figure 2 shows how non-normality in European human the 95th percentile in parallel analysis (Peres-Neto et al. populations has increased over time. The top row illus- 2005) using the ‘paran’ package (Dinno 2012). We list trates non-normality of the whole system: raw matrices loadings that exceeded 10% of each axis, in the order of describe both asymptotic and transient dynamics. In that decreasing importance. context: the Frobenius metric changed little over the time The statistical software ‘R’ (version 3.3.2, R Develop- period; the Henrici metric increased up to a plateau begin- ment Core Team 2016) was used for all analyses and fig- ning around 1990 (with low outliers including Portugal ures, along with the ‘R ColorBrewer’ package (Neuwirth 1960–1975, enlarged on the figure and examined below); 2014) for the latter. the Ruhe metric showed an almost flat relationship. 1 3 190 Population Ecology (2018) 60:185–196 Fig. 2 Non-normality over time, bc measured by three different met- rics—from left to right: Frobe- nius, Henrici, Ruhe. Top row shows raw matrices, and bottom row standardised ones. Points are coloured by country. Solid lines are GAMM estimates, with dashed lines showing their 95% confidence intervals de Scaling focuses on transient effects by factoring out the Figure 3 shows that both λ and the damping ratio effect of λ . This increased the mean value of all non- decreased over time. The transient indices of reactivity and normality metrics, by almost 4% each. Comparing the inertia were strongly correlated with one another (ρ = 0.93), bottom and top rows of Fig. 2 shows that the shape of the both peaking around 1995. Furthermore, values for both estimated GAMM curves also changed, in terms of inter- exceeded 1 for over 97% of matrices, revealing a propensity cept, slope, and variance patterns. This can be coarsely for amplifying transient growth rather than decline, with the explained by a systematic change in λ : the annual mean latter being restricted to prior to 1971 for reactivity and 1991 dropped below 1 in 1975 and remained so for the rest for inertia. The transient indices were positively correlated of the time period (see Fig. 3a). The Frobenius metric with all scaled non-normality metrics (ρ > 0.52). (Fig. 2a, d) shows how higher λ values before 1975 were Of the three non-normality metrics, Henrici changed the pushing the curve up, while lower values afterwards pulled least with matrix standardisation; the scaled and unscaled it down. The overall effect resulted in similar increases versions were correlated at ρ = 0.82. Never theless, Fig. 4 over time across the scaled non-normality metrics (which shows that scaling still altered the Henrici metric’s relation- were pairwise correlated with one another at ρ > 0.78). ships with ecological measures of population dynamics. It However, in contrast to the smooth increase in the scaled decreased the strength of the relationship between non-nor- Frobenius metric, the Henrici and Ruhe metrics both show mality and damping ratio (Fig. 4a, c), such that there was a peak around 2000—close to that of the transient indices only a slight correlation with the effect of λ removed; this is (see Fig. 3c, d)—and appear to plateau by the end of the unsurprising given λ is the numerator of the damping ratio. time series. In contrast, scaling increased the strength of the relationship Even where scaling did not change the overall pattern, between non-normality and reactivity (Fig. 4b, d), such that as with the Henrici metric, increased variance allows an high values of scaled non-normality were a good predictor improved visualisation of dynamics. Additionally, outli- of strong immediate transient growth. ers tended to become less distinct, although corresponding Principal component analysis allowed more in-depth country–years are still distinguishable as bounds on vari- investigation of interrelationships among the variables, visu- ation in top and bottom rows of Fig. 2. The outlying line ally represented as biplots in Fig. 5. Using the unscaled non- of Portugal 1960–1975 on the plots of the Henrici metric normality metrics, the two significant principal components (enlarged points in Fig. 2b, e) corresponds to matrices with explained 72% of the variance. The first principal compo- very low old-age survivals and zero 85 + stasis. nent loaded onto λ (negatively), and transient indices, the 1 3 Population Ecology (2018) 60:185–196 191 Fig. 3 Ecological measures of ab population dynamics over time. Points are coloured by country. Asymptotic growth rate is the dominant eigenvalue, with the dashed line showing λ = 1 i.e., no population change—above the line is population growth; below, decline. For definitions of the other measures, refer to “Methods”. For reactivity and inertia, the dashed lines divide the plot into transient growth (> 1) and decline (< 1) Fig. 4 The Henrici non-nor- a b mality metric, unscaled (top) and scaled (bottom), against damping ratio (left) and reactiv- ity (right). Points are coloured by country. Simple Spearman’s rank correlation coefficients are given above each plot as a visualisation aid c d 1 3 192 Population Ecology (2018) 60:185–196 a b 1100 1100 1003 1070 1038 1003 989 1117 N.Frob 1054 1087 1038 969 1108 955 965 1023 897 933 894 1013 933 919 930 979 1046 1078 961 1096 1092 823 882 967 897 1002 1037 1031 1000 996 945 1068 1062 856 844 1035 1101 860 819 932 860 926 1098 1066 1118 1117 946 980 909 1115 1118 786 895 823 889 1014 1047 1107 910 1079 11091071 1088 9891087 1105 1054 1023 1092 756 858 851 808 1055 1075 955 1068 821 1021 1085 1043 1002 760 1108 1109 1115 1088 1024 7907771077 872 9871039 1008 1024 967 980 1037 1014 996 1098 1055 1106 725 835 1120 814 975 990 9461031 1062 1079 873 1045 36 919 932 961 1047 990 1058 834 910 790 873 1013 1085 1090 1104 760 788 781 747 1004 1091 882 895 926 979 1046 1078 1105 1027 953 939 956 889 945 800 1021 956 1116 1086 1007800 799 1095 year 844669 699 851 10961075 920 9931042 1074 758 695 738 768 821 858 636 729 917 903 971 920 729 63911011091 425 448 974 966 665 699 707 769 7511010 1059 835 788 769 1008 883 1097 1116 808 909 1035 814 1011 year 404471 505 958 938608 669 1076 716 10941028 866 965 819 856 1000 1066 1043 987 845 902 556 727 639 1001 1033 448894777612975 922 583 647 739 677 720 963 1064 1065 883 404 1007 974 1104872 903 953 917 10591022 497 479 531 1036 1067 906 994 935 930758739 988 1053 865 1061 931 928 9641034 1011 1102 1042 7475871039 906 1028 383 885 690 1103 998 899 829 3831120 1074 1107 781 939N.Ruhe 918 809 1111 648697 612 660 1113929 977 845 902 1004 778 455 621 667 678 708 620686 893 869 936 972959 1022 1053 3745 425 1027 834 751 1097 1016 431638 632 891 999 1005 1086 54 938648 869971 959 363 828 587 900 862 10401072 1058 497 1090 541 621 678 523 1001 618 716 708 799 866 994954 881 843 1072 568 541 1030 610 618593 1044 656 795 880988 471786727 5941071 481 847 594 523 1051 1083 941 918 765 809 993 363 1077 697 768 1036 829 935 1081 944 516 764 507 794 604 855 1073 1111 28 62 756 9661045 1067 6601095 972 8991102 733 857 410 533 558 1041 778 954 71 636 490 695 738 610 677686 657 549 1040 887 924 880 748 717 1049 N.Hen 457 810 585 995 N.Frob 958 865 485 720 566 690 548600862 629 601 687 942 703 490 566 853 579 9241016 20 12 453 568 665 667 558 585 632552 579 449 343 433389 904 552 567 628 436725 343 419 441 562 507 707 533 1094 982 849 807 776 673 737 548 1019 657887 982 825 881 7341049 53 80 398 608 464 583 6381010 656 936 405 1005 765 576704 441 464 779 820 783 527 600863 843 842 1081 847 556 931 516 764647 604 795 498 825 812 734 398 908 4191057 1093 985816 791 748 922479 531 433 379 963 737 575 900 524 324 10181012951 575 540 629 826 717 505 828 481 457 928998 628 426 1065912 948 590 643 412 501 905 761 849 1106 374 733 1064 527472 791 892746 563 615 871749 773 867463 940 489 515901 937 1009 601 807 1110 704 92 477 885703 620 1033 593964 1034 384 780 715 743 962369 428 549 576 812687 lambda1 431 794 412 891 364 999 344 761842 166 152 350 379 391 984 1032 407 451 976 818 1006 912 948776 39 673 779 358 977 837 875 792 1110685 lambda1 314 927 997 1112 440 475 405 426 449827 864 524 892 973 674 166 113 455 324304 646 893 929 501 750 817 331 394 477 537 619 712 1026 1063 915 4181099498 780 730 942 123 30 181 511 619 941 567 863 826 981 674 37 335 304 384 1020 472 746 47 350 331 503 391 475325 7191080 806 453 1119 388 358682 983386 787 804 397364 753 670 981 700 1048 792 644 806 152 8571030 286 428 451816 730 700 689 854 655 31 415 414 512 499950 833 949 890 718830 840 802 640 750837 875 247 590 643 10761044 407 783 540640 305 802 771762 784 591 28 45 54 23 181 348 437 368 550 577 602 630 371646349 722 793 817 616 1015 1080 31 348 944 388563529 489 515 855 754 1048564 538 370 354393 409 473 529525 688 947 56 409 810 749 371 1113 319 418 440 581753 287644 1015 40 276 310 375 485 435 460486 6581050 366 992968 878 986 1069 741 854 1114 785 775 715 310 329 368 534 615 1061554 369 386 270 853 397 710 634 1114 676 731 62 71 194 260 292 329562 914921662 952 868 771 64370 260 137415 537 1119 718 351 722 741 659 701 775 20 511 728 351 757 650 710 762 14 215 688 1103 1083 300650680 526631 578 603 714 627 12 80 316 482 347 436503 1089 459 554 286 852 378 692 344 911 689 719 7023 40 276314 316 297 908 927658 1051 2541009 338 692 1069 745 4830 534 346 357 1084 51 130 292 157 437410 512 904 486 1089349 281 378 968 255 271 911 1084 785 15 7 147 199 337 297508 373 970 934 798 10171082 796 325 680 7541052 591 685 22 135 337 149499 577 602 550 389 949237 249 366 265 905 1099 670 818 500 1052755 513 654 684574 599 39 484 330 784 48 206 74202 335280 354 375 460 630 1056 430 222 1012 1019 293 3461020 947 671 22 130 157 171264 3531056 698 327 623 338 763 6 87171 243 199 347 482 871 525 962 867698 940976 606 901 623 238 416 103 57 47 68 243 228280 420 308 328 510 581 731 564 655 68 143 264 414 144820 1032 559 833 997 454 787 1073 427 318 223 551395487 547 92 65 75 8597 206 186 319 307 815 745 N.Hen 7 194 83 228 186 73 308 396 377139 934 126 586 277 218 311 1041 327 330 406 596 543 438 461 522 81 1 340 56 143 390399 211 356 258 396 274 334 367 270 309305 831 57 147160 99 77 211393 394 473 163 153 890 207 233 1112 332 463 827 317 496 93 114 146 16187 241 586430 913 606 318 596 732 631 755 15 180 360173 174 145176 274 328 104 417 508 115387261 1063 182 898864 466 570 208 474 299 613 836 724 470 88 321 113 189 134302 173 188 213 417 377 313 559 668 332877884 896702 659701 714 220 109 159 188 213241 258 95 420 82 105435 131 1093 244 728 167 995 798 830 309868 986 376 560626 125 220 160 360 9977 221 215 159 226 290 442 813 254 634 287 726 916874 641 N.Ruhe 164 90151205321 226 94 367 262 970 445 743 773 983757 195 815 796 357 937 450 385 952 896 518874 588 663 723 72 51109 202145 4451025 222 237 300 570859 526 672 538 179 193 759788 133 76390 201 119 245 439459712 852 187 668 307 662291 1006 365 859 63 138 107201 144 611 454652 311 273 289291 317 165 6585112 134 14634 230 290 334 682178172 443 973 345 196 759 822 424598 55 146 64174132 283 439 374 2951060 850 387261 210 225 240 293 257 271 696 642 551 684 627 114 235 100 120 302 132 162 86 373 484510 804 840 1050 183 275 641 403 535 546 573 38 46 133 110 175 120 1029 468 991 277 450 255299 846 578 603 671 654 1 102 161 148 283 248 313 252 158 1025 200 1017 878 1082 782 763 616 831 492382 521 694 693 150 180 205100 236 131 465 536 474822 676 513 110 107 124 177 356 232116468 140 229 992 294 913 273 212 289 421 288298 478 342 296 362483 447 886 848 805 235 74 76190 83119 232 248 198207 406 427 543 782 735 500 438 81 67 399353 442465 224 278 239 951 985 611272 696702 242 923 355 916 879 801 21 151 193 34 66 230 139491 888 281 238 797461487 724 203 295 127 491 240 256 517 542 726 793672 259 797 36 164 29 91 102 121162 177 73137 118 245 104 253960 185 759 836 189 121 98 217 1060214 536400 168 735 326 732 458 469 841664 165 136 148 231 115126 957153 170244249 265 223 789 766 614 138 175 221118250 210 198 225 991257 361 154 227 789 614 423 434 413495774 770 reac 112 191203 216 158 187898 561589 298 416 613 103 55 192 91 136 108 231 411 1029 284915494380 642 752315 336 767 179 86 6795 217 105 82 94262 925 466 385 923 839 275 395879 588 599 93 204 66 219251 59129 156 268 185921 322 341 306 381 766 402649679 635 246156 200 626 694 574 72 125 340 190234 60236253 282 925569 352 617 595 861 392 124 129 279 167 233 182 443 838 876 637 518 560 6346 18 263 279 984 850 652 312 359 736 504 13 18 98 149 142 269 263 218195212208 365 492 666 801 reac 191 266 142170408 561 645 372721 740 5123 251 108116 411 285408 229 172 421 355 447 841598 150 50 209 462 957 285 197 888 637 666 589622 509 446709 252278 462 542 345 259 705 376 767 663 547 38 216 267 432 339592 811 607 69 267 224 303517 361 242752509535573 470 1018 950 877 884 323565 520545 582 127 239 569 196 664 69 42 246 269 303 705 675 691 135 192 89 341 494323 478 483 424805 546723 522 29 5 84 96 1057 184 838 846 803 333 572 706 653555 204 21960 234 79250 122 247 59 163 140 227 770 496 13 89 58 401914 301320 876 539 452 609 625 530 53 84 209322 214 256 400 306 183 326 617 943978 403 79 1026 488 514 268432 284 178 272288 595 296382 649 886 521 17 9 61 122 111 456 429 683 713 101 111 294 342 362 803 434 336 458 679 495 693 106 33 744 inertia 7861 50 315 736 101 532 506 476 33 266 282 488 514 381 592 622 740 635 78 155 557 661 42 154168 609 645 413 469 721 848 774 21 141 25 480 839584 824 32 17 312 352 402 423 709 605 24 9 380520 565 3 597 633 96 25 58 401 359 392 907 117 128 169 422 49 43 429 539532861 584 49 52 772 197 inertia 41 43 624 528 16 456 545 372 572 557 675 691 607 960 444467 8 41 3 10 480 772 597 504 706744 8 10 978870 2 106 184 333 446 813 651 711 301 339 506 870 582 742 70 176 26 52 320 625 44 519 742 633653 555 26 502 681 452 811 35 832 90 169 16 493 580 dampingRatio 141 44155 907 824 530 683 713 571 117 711 dampingRatio 544 128 35 476 4 11 651 605 27 422444 493 661 943 624 32 4 467 832 11 681 519 528 24 571 2 19 502 Fig. 5 Biplots of principal component analysis on non-normality metrics (prefix ‘N’) and ecological measures of population dynamics. a Using unscaled metrics; b scaled. Frob Frobenius, Hen Henrici, lambda1 λ i.e., dominant eigenvalue, reac reactivity Henrici metric, and year (positively). The second loaded Discussion onto the Frobenius metric (positively), damping ratio (nega- tively), and year again (positively). Note that the Ruhe met- This is, to the best of our knowledge, the first comprehensive ric is not represented by either of the significant principal continental-scale comparative assessment of the susceptibil- components. Using the scaled non-normality metrics, the ity of human populations to transient dynamics. We quan- two significant principal components explained more of the tified this transient potential using non-normality metrics: variance (86%) than the unscaled case. Loadings differed, overall, these increased for European populations between but directions did not: the first principal component loaded 1960 and 2014 (Fig. 2). The patterns of non-normality onto the Henrici metric, λ , the Ruhe metric positively, the metrics were correlated with transient indices (Figs. 2, 3): Frobenius metric, and transient indices; the second compo- relationships were strong and positive, with the peaks in the nent loaded onto damping ratio, year, and inertia again, but scaled Henrici and Ruhe metrics echoed in those for reactiv- this time negatively. ity and inertia—implying increasing influence of transient Scaling moved all non-normality metrics into the same dynamics on these populations. Although we caution against part of the plot (in Fig. 5b), whereas when unscaled, the the potential loss of information in restricting analyses to a Frobenius and Henrici metrics were almost orthogonal to single measure of non-normality, where a streamlined evalua- each other (in Fig. 5a). The Frobenius and Ruhe metrics tion is desired we particularly recommend the Henrici metric, appeared to be most susceptible to asymptotic growth rate, since in our study it proved to be least affected by the scaling moving more than the other variables when the effect of λ issue and most strongly correlated with transient indices. was removed; this reiterates the relatively low sensitivity Focusing on these transient indices, we found a very strong of the Henrici metric to scaling. In both plots the damping and significant correlation between reactivity (transient change ratio was orthogonal to the axis with λ and transient indi- in population size after one timestep) and inertia (asymptotic ces (unscaled plot) or non-normality (scaled), suggesting change in population size due to transience), as did Stott et al. that it describes something fundamentally different to both (2011). Our transient indices rarely yielded attenuation, i.e., asymptotic and transient dynamics—which should perhaps values smaller than one, which reflect decreases relative to be unsurprising since it is supposedly a measure of duration the asymptotic trajectory. In contrast, using the same metrics rather than amplitude. Two groups of points (labelled as: 36, on orchids, Tremblay et al. (2015) showed transient decline 53, 70; and 553, 580) are notable outliers on both biplots: to be much more common than amplification; this suggests the former represent Iceland in the 1960s; the latter Bulgaria that the western human populations that are most common in the late 1990s. in our database tend towards transient increases, while plants 1 3 Population Ecology (2018) 60:185–196 193 may more often decrease. While we found a greater likelihood varying non-normality across datasets. Both overall trends of transient increase when populations were declining overall and turning points illustrate that non-normality cannot be (and vice versa), since both transient indices were opposed to considered static for a given country, rather as changing tem- λ (Fig. 5), the opposite was found in a study of over 100 plant porally—perhaps similarly to momentum which is a process species, where faster-growing populations tended towards that plays out over time (Blue and Espenshade 2011). While greater reactivity (along with other measures of transience; a non-normality value for a single matrix reveals little about Stott et al. 2010). Stott et al. (2010) argued from their results the impact of the transient at that snapshot in time, its rela- that vital rates impacted short- and long-term dynamics simi- tion to others in the dataset integrate multiple sources and larly, but pointed out that animal populations including humans forms of stochasticity with respect to the impact of varying appear to be more sensitive to initial conditions. transient dynamics on population trajectories. Historically, The opposition of short- and long-term dynamics is fur- the damping ratio has been used to quantify transient impact, ther drawn out in the contrast of decreasing λ through time, but it exhibits orthogonal behaviour to inertia and reactivity whereas reactivity and inertia peak around the millennium. (Fig. 5). Over and above the methodological limitations of The first observation is increasingly recognised: for many the damping ratio already discussed, a key consideration is countries worldwide, and especially in Europe, a ‘second the fact that the damping ratio is a proxy for the duration of demographic transition’ is underway, with total fertility rate transient fluctuations, while reactivity and inertia provide dropping below replacement, driving population decline in immediate and eventual measures of the transient ampli- the absence of immigration (Harper 2013; van Daalen and fication in population size. It remains to be seen how the Caswell 2015). Any reason for a peak in transience is less three non-normality metrics perform across other systems obvious. Lutz et al. (2003) found that “for the [then] 15 mem- and stage structures, and whether their interrelationships ber countries of the EU, low fertility brought the population with population dynamic indices remain consistent. Com- to the turning point from positive to negative momentum parative studies using the COMPADRE and COMADRE around the year 2000” (p. 1991). However, inspection of demographic databases (Salguero-Gómez et al. 2015, 2016) country-stratified data suggests that the humps are a combi- could prove particularly insightful here. nation of different types of trajectory, rather than all countries peaking simultaneously. Perhaps some are related to preced- Caveats ing and ongoing disturbances such as the dismantling of the socialist economic model in Central and Eastern Europe Matrix outputs are affected by matrix dimension (Tenhum- (Sobotka 2002), the reunification of Germany (1989), and berg et al. 2009), with potential implications for non-normal- the armed conflict in the former Yugoslavia (1991–1999). As ity. A study on cacti found larger matrices to generate lower a specific example, Bulgaria’s economic instability during the asymptotic growth rates (Rojas-Sandoval and Meléndez- 1990s could have driven the transient effects suggested by the Ackerman 2013). With our data, single-year matrices (of PPMs for 1997 and 1998, which were outliers on the biplots dimension 85 × 85) generated λ values up to 9% larger or and had the highest values for the scaled Henrici metric. smaller than those from the 18 × 18 matrices used here, with Returning to the non-normality metrics, we found all a mean difference of + 3% (unpublished data); we employed three measures to have similar temporal trends once the the smaller matrices in this study for consistency with stand- effect of declining asymptotic growth had been factored out. ard approaches in human demography and because they cap- This follows Stott et al.’s (2011) recommendation that tran- ture the vast majority of variation whilst enabling expansion sient analyses are more usefully performed on standardised to other regions and time periods for which annual data are matrices. When studying the whole system, using raw matri- not available. Influence of matrix dimension on transients is ces, the different non-normality metrics told varying stories: more contested: while a study of six bird and mammal spe- Frobenius suggests a negative quadratic relationship, Henrici cies with varied life histories found no effect (Koons et al. increases to a plateau, and Ruhe shows very little change. 2005), a piece of research on pea aphids and another on a This impact of λ is especially notable given the relatively wide range of plants found positive correlations (Tenhum- small range of values seen across human populations as berg et al. 2009; Williams et al. 2011). Furthermore, the opposed to other animals or plants: this study saw 0.89–1.11, potential for transients has been found to affect the magni- compared to 0.80–1.12 within one metapopulation of mar- tude of changes in λ with matrix dimensionality (Ramula mots (Ozgul et al. 2009), and approximately 0.7–2.1 across and Lehtilä 2005). Although Stott et al. (2010) are concerned 20 plant species (Crone et al. 2013). The effect of λ should that such effects could “perhaps [be] signifying a potentially therefore be acknowledged in all comparative studies of tran- worrying artefact of basic model parameterisation” (p. 302), sients (Stott et al. 2011). Ellis (2013) reassures that these relationships are likely to Furthermore, we suggest that longitudinal (as well as be weaker when considering case-specific transient indices comparative) studies should consider the potential for 1 3 194 Population Ecology (2018) 60:185–196 ab Fig. 6 Pseudospectra for Bulgaria in 2014, as a contour plot (a) and lower-valued contours (e.g., contour 4 around λ ) would shift only a perspective plot (b). Compare to the spectrum in Fig. 1. Contours under large perturbations, while those encapsulated by higher eigen- correspond to perturbations of the original matrix, with an inverse values (e.g., contour 12 around eigenvalues 6–8) are more easily per- relationship: small-valued contours correspond to large perturba- turbed. Human PPMs have multiple zero eigenvalues, which explains tions, and vice versa. The original, unperturbed, eigenvalues have a the ‘volcano’ pattern in the perspective plot, as these eigenvalues are ‘height’ of infinity (= 1/0): they are seen as dots in the contour plot sensitive to even small perturbations and sharp peaks in the perspective plot. Eigenvalues encapsulated by (‘realistic’ scenarios, as here), compared to bounds (extreme Trefethen et al. 1993; Trefethen and Embree 2005) for appli- hypothetical cases; see Stott et al. 2011). cations in fluid dynamics, but with the recognition that the A further fundamental caveat is the lack of migration techniques also apply to related problems across the mathe- among populations, which is increasingly considered essen- matical sciences. Trefethen (1997) believes that visual repre- tial when modelling human populations (Azose et al. 2016; sentations aid interpretation by “supplementing the abstract Willekens 2016). Ozgul et al. (2009) shows how transients notion of a matrix [with] a picture in the complex plane” (p. unfold differently when incorporating migration between 383). He suggested that pseudospectra give matrices ‘per- patches in metapopulations. Inclusion of such complexity sonality’, and that they may allow us “to notice things that reveals highly variable transient responses (Espenshade and went unnoticed before” (p. 404). Pseudospectra can now be Tannen 2015, and the unpublished EU study therein), with interrogated via perturbation analysis and transient bound eminent policy implications. calculation (Townley et al. 2007). A more significant limitation to our study is the obser - Figure 6 shows two different types of plot for pseudospec- vation that differing behaviours of non-normality metrics tra corresponding to the spectrum shown in Fig. 1 (for Bul- with respect to matrix standardisation remind us that these garia in 2014). Pseudospectra ‘look beyond’ eigenvalues to measures may be well-defined mathematically but less so express how they change under perturbation (Trefethen 1992; with relevance to demography. Even in their original formu- Trefethen and Embree 2005). Here it can be helpful to bear lations, “scalar measures of nonnormality suffer from a basic in mind that errors in parameter estimation mean that the limitation: Non-normality is too complex to be summarised ‘true’ model may actually lie within the pseudospectral set in a single number” (Trefethen and Embree 2005, p. 446). of slightly perturbed matrices. Pseudospectra can capture There is therefore still a need to develop more reliable meas- transient dynamics more holistically than eigenvalues— ures. One response (Gheorghiu 2003) to Elsner and Paarde- “although pseudospectra rarely give an exact answer, they kooper’s (1987) review of non-normality metrics considered detect and quantify transients that eigenvalues miss” (Trefe- scalar instruments to be just one of two ‘major concepts’ in then and Embree 2005, p. 135). Another reason we restricted their measurement—the other being pseudospectra analysis. analyses to the Frobenius norm is that it defines a special case where pseudospectra exactly determine matrix norm A future direction: pseudospectra analysis behaviour (Greenbaum and Trefethen 1993). Inferences about for population ecology non-normality can be made by studying eigenvalue encap- sulation by the pseudospectra contours: the lower the value Pseudospectra are visual representations of non-normality of contours encapsulating the eigenvalues, the less stable the developed by Trefethen and colleagues (Trefethen 1992; matrix and the greater its proneness to transient behaviour. 1 3 Population Ecology (2018) 60:185–196 195 credit to the original author(s) and the source, provide a link to the Concluding remarks Creative Commons license, and indicate if changes were made. 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Population Ecology – Springer Journals
Published: Jun 5, 2018
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