Another look at the eigenvalues of a population matrix model

Introduction

Population matrix models (PMMs) are used to assess strategies to manage populations for resource purposes. These models are regularly found in quantitative ecology texts (e.g., Fryxell, Sinclair & Caughley, 2014), and have been used in many applied settings (Salguero-Gómez et al., 2014; Salguero-Gómez et al., 2016). But how familiar are broad audiences with the mathematics behind the PMM, particularly those that generate the most prominent model output, the finite rate of growth? We use a visual approach to provide an alternative way to understand how targeted managerial alterations to a life cycle can alter the finite rate of growth.

The life history traits of a species (“vital rates”) govern the mathematical structure of the PMM (De Kroon, Van Groenendael & Ehrlen, 2000). The vital rates include stage-wise fertilities, survival, and transition probabilities (Caswell, 2001). The population-scale demographic properties obtained from the PMM include stage abundances, the finite rate of growth (hereafter “ λ”), the stable stage distribution, net reproductive values, sensitivities and elasticities (Caswell, 2001), plus a host of transient quantities (Stott, Hodgson & Townley, 2012). For example, let an arbitrary 3-stage PMM be: (1)${\displaystyle \mathbf{A}=\left[\begin{array}{ccc}\hfill A_{11}\hfill & \hfill A_{12}\hfill & \hfill A_{13}\hfill \\ \hfill A_{21}\hfill & \hfill A_{22}\hfill & \hfill A_{23}\hfill \\ \hfill A_{31}\hfill & \hfill A_{32}\hfill & \hfill A_{33}\hfill \end{array}\right],}$ which contains matrix elements A_ij (i = 1, 2, 3, and j = 1, 2, 3) and where A_1j is the average fertility in the jth stage, A_ii is average survival of the ith stage without transition out of the ith stage, and A_ij is the average survival of the jth stage with transition from the jth stage to the ith stage. Stage-wise abundances are projected through time using (Caswell, 2001): (2)${\displaystyle \left[\begin{array}{c}\hfill J_1\hfill \\ \hfill S_1\hfill \\ \hfill B_1\hfill \end{array}\right]=\mathbf{A}\left[\begin{array}{c}\hfill J_0\hfill \\ \hfill S_0\hfill \\ \hfill B_0\hfill \end{array}\right],}$ where J₀, S₀, and B₀ are the number of stage one, stage two, and stage three individuals at time 0, and J₁, S₁, and B₁ are the number of stage one, stage two, and stage three individuals at time 1. Recursive calculation of Eq. (2) from time 0 through time n yields: (3)${\displaystyle \left[\begin{array}{c}\hfill J_n\hfill \\ \hfill S_n\hfill \\ \hfill B_n\hfill \end{array}\right]={\mathbf{A}}^n\left[\begin{array}{c}\hfill J_0\hfill \\ \hfill S_0\hfill \\ \hfill B_0\hfill \end{array}\right].}$ As n gets large, Aⁿ behaves like a single number, denoted λ (Caswell, 2001). Depending on the citation, λ is called the “dominant eigenvalue”, the “intrinsic rate of growth”, the “asymptotic growth rate”, or simply “the growth rate”. Regardless of which name is chosen, the λ is the geometric rate of population change once stage oscillations damp out and the stages approach stable proportions. In other words (Caswell, 2001): (4)${\displaystyle Aw=\lambda w,}$ where w gives the stable stage proportions.

But to broader audiences, Eqs. (1)–(4) do not necessarily clarify the link between cause and effect in population dynamics, especially as they pertain to decision making towards a resource goal. Herein, we attempt to clarify the nature of λ as it pertains to strategic planning of targeted managerial interventions in resource management settings.

We introduce the interactive software (“IsoPOPd”) that represents λ in a visual framework. The graphics provide an alternative way to think about how managerial alterations to matrix elements (hereafter “A_ij”) produce changes in λ. We illustrate the use of the software with white-tailed deer (Odocoileus virginianus) but remark that the software may be used to understand cause and effect in any 2- or 3-stage PMM. The software is at: https://cwhl.vet.cornell.edu/tools/isopopd.

Methods

A common life history contains three stages that we define as the juvenile (non-reproducing individual), early adult (with reproduction rates consistent with mid-life fertility), and late adult (with reproduction rates consistent with late-life fertility) stages. We use A to represent the PMM for such a species. The characteristic equation for A is found in linear algebra textbooks (e.g., Strang, 2016): ${\displaystyle {\lambda }^3+\left(-A_{11}-A_{22}-A_{33}\right){\lambda }^2+\left(A_{11}{A}_{22}+A_{22}{A}_{33}+A_{11}{A}_{33}-A_{32}{A}_{23}-A_{21}{A}_{12}-A_{13}{A}_{31}\right)\lambda }$ (5)${\displaystyle +\left(A_{11}{A}_{32}{A}_{23}+A_{12}{A}_{21}{A}_{33}+A_{31}{A}_{13}{A}_{22}-A_{11}{A}_{22}{A}_{33}-A_{21}{A}_{13}{A}_{32}-A_{31}{A}_{12}{A}_{23}\right)=0.}$

Using the notation for the superparameters (“coefficients of the characteristic equation”) in Hanley & Dennis (2019), the superparameters are: (6)${\displaystyle p=\left(-A_{11}-A_{22}-A_{33}\right),}$ (7)${\displaystyle q=\left(A_{11}{A}_{22}+A_{22}{A}_{33}+A_{11}{A}_{33}-A_{32}{A}_{23}-A_{21}{A}_{12}-A_{13}{A}_{31}\right),}$ and (8)${\displaystyle r=\left(A_{11}{A}_{32}{A}_{23}+A_{12}{A}_{21}{A}_{33}+A_{31}{A}_{13}{A}_{22}-A_{11}{A}_{22}{A}_{33}-A_{21}{A}_{13}{A}_{32}-A_{31}{A}_{12}{A}_{23}\right),}$ yielding an abbreviated form of the characteristic equation: (9)${\displaystyle {\lambda }^3+p{\lambda }^2+q\lambda +r=0.}$ The software focuses on the link between p, q, and r and λ. In particlar, the characteristic equation (Eq. (9)) contains four 3-dimensional volumes (“orthotopes”) that must always balance positive and negative contributions of p, q, and r to zero. In the context of resource management, targeted intervention activities modify the values of p, q, or r, leaving λ to respond in a manner that maintains the overall equality. Indeed, Eq. (9) is the set of rules by which λ must respond to any managerial strategy. Positive and negative influences on λ appear in Table 1.

IsoPOPd was developed to provide an alternative way to understand how λ responds to managerial alterations to p, q, and r. The IsoPOPd software converts any PMM into a visual representation of their p, q, and r values, from which λ is obtained. The user defines the structure of a 3-stage PMM, then designates the values of each of the nine A_ijs. The Characteristic equation tab displays the inputs (the PMM itself), the output (the corresponding λ), and the mathematical linkage between the two (Eq. (9)). The remaining tabs use a visual framework to illustrate the nature of the linkage. The Breakdown of the λ³ term tab shows the first orthotope, λ³, a 3-D cube scaled to dimensions (λ ×λ × λ). The Breakdown of the pλ² term tab shows the second orthotope, pλ², the 3-D volume scaled on one side by the vital rates in p (Eq. (6)), and on the remaining two sides by λ. The negative sign in front of pλ² (and the light blue shading in the software graphics) represents a negative contribution to Eq. (9), but an opposite (positive) contribution to λ. The Breakdown of the qλ term tab shows qλ, the trio of sub-orthotopes whose dimensions that may be better understood by rearranging Eq. (7): (10)${\displaystyle q=\left(A_{11}{A}_{22}-A_{21}{A}_{12}\right)+\left(A_{22}{A}_{33}-A_{32}{A}_{23}\right)+\left(A_{11}{A}_{33}-A_{13}{A}_{31}\right).}$ The trio of sub-orthotopes in qλ are scaled on two sides by the vital rates in Eq. (10) and scaled on the third side by λ. A positive sign (and the black shading in the software graphics) represents a positive contribution to Eq. (9), but an opposite (negative) contribution to the final value of λ, while a negative sign (and the light blue shading in the software graphics) represents a negative contribution to Eq. (9), but an opposite (positive) contribution to λ. The sub-orthotopes in Eq. (10) therefore make positive and negative contributions to q, but only the net change is reflected in the final value of qλ. Finally, the Breakdown of the rterm tab shows the last orthotope, r, the 3-D volume that is scaled on all three sides by vital rates. The Geometric interpretation of the characteristic equation tab reveals the entire set of graphical orthotopes (λ³, pλ², qλ, and r) that comprise Eq. (9). Black and light blue shadings in these volumes must always balance.

We use the IsoPOPd software to demonstrate the effects on λ from hypothetical managerial interventions in white-tailed deer (Odocoileus virginianus). We assume A tracks a single sex (female), the A_ij are static, the probability of transition in each stage is one, and each A_ij constitutes an average for members of the jth stage (Caswell, 2001). Let the PMM be (Chitwood et al., 2015), reduced to two digits): (11)${\displaystyle A=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.58\hfill & \hfill 0.70\hfill \\ \hfill 0.14\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.77\hfill & \hfill 0.80\hfill \end{array}\right].}$ We investigated how a 10% increase in survival in the 2nd stage (A₃₂) alters the configuration of the orthotopes in Eq. (9), and ultimately, λ. We then investigated how a 10% increase in survival in the 3rd stage (A₃₃) alters the orthotopes and λ. Finally, we investigated a combination scenario where survival is increased by 10% in both 2nd and 3rd stages.

Results

The characteristic equation for the unperturbed deer scenario (Eq. (11)) is: (12)${\displaystyle {\lambda }^3+\left(-A_{33}\right){\lambda }^2+\left(-A_{21}{A}_{12}\right)\lambda +\left(A_{21}{A}_{12}{A}_{33}-A_{21}{A}_{13}{A}_{32}\right)=0,}$ or equivalently, (13)${\displaystyle {\lambda }^3+\left(-0.8\right){\lambda }^2+\left(-0.081\right)\lambda +\left(-0.0105\right)=0.}$

Figures 1–4 show the λ³, pλ², qλ, and r orthotopes (respectively) in Eq. (13). Each dimension in the orthotopes are scaled to the appropriate A_ij values (as specified in Eq. (11)) and to the λ that balances them.

A 10% increase in the survival of early-breeding adults (A₃₂) gives: (14)${\displaystyle A=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.58\hfill & \hfill 0.70\hfill \\ \hfill 0.14\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.84\hfill & \hfill 0.80\hfill \end{array}\right],}$ which yields (15)${\displaystyle {\lambda }^3+\left(-0.8\right){\lambda }^2+\left(-0.081\right)\lambda +\left(-0.0173\right)=0,}$ and a corresponding 1% increase in λ. This management strategy alters the characteristic equation by (a) increasing the volume of the λ³ orthotope (Fig. 5), (b) rescaling the two λ dimensions in the pλ² orthotope to decrease its overall contribution, (c) rescaling the λ dimensions in the trio of qλ sub-orthotopes to decrease their overall net contribution, and by (d) decreasing the overall contribution of the r orthotope (Fig. 6).

Alternatively, a 10% increase in survival of late-breeding adults (A₃₃) gives: (16)${\displaystyle A=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.58\hfill & \hfill 0.70\hfill \\ \hfill 0.14\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.77\hfill & \hfill 0.88\hfill \end{array}\right],}$ which yields (17)${\displaystyle {\lambda }^3+\left(-0.88\right){\lambda }^2+\left(-0.081\right)\lambda +\left(0.004\right)=0,}$ which equates to a 7% increase in λ. This change alters the characteristic equation by (a) increasing the volume of the λ³ orthotope (Fig. 7), (b) decreasing the contribution of the pλ² orthotope (Fig. 8), (c) rescaling the λ dimensions in the trio of qλ sub-orthotopes to decrease their overall net contribution, and by (d) increasing the overall contribution of the r orthotope (Fig. 9).

Finally, a 10% simultaneous increase in survival of early- and late-breeding adults (A₃₂, A₃₃) gives: (18)${\displaystyle A=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.58\hfill & \hfill 0.70\hfill \\ \hfill 0.14\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.84\hfill & \hfill 0.88\hfill \end{array}\right],}$ (19)${\displaystyle {\lambda }^3+\left(-0.88\right){\lambda }^2+\left(-0.081\right)\lambda +\left(-0.0108\right)=0,}$ which also equates to a 7% increase in λ. The changes alter the characteristic equation by (a) increasing the volume of the λ³ orthotope (Fig. 10), (b) decreasing the contribution of the pλ² orthotope, (c) rescaling the λ dimensions in the trio of qλ sub-orthotopes to decrease their overall net contribution, and by (d) slightly decreasing the overall contribution of the r orthotope (Fig. 11).

Discussion

The behavior of λ regularly defies intuition. But the linkage between a managerial action and its effect on λ can become more transparent when cause and effect is thought of as a two-step process. A targeted managerial change to an A_ij first alters the configurations of the p, q, and r orthotopes, then the value of λ (and the configuration of the λ³ orthotope) adjusts to rebalance the overall equation to zero. In this two-step process, changes to the A_ijs always alter the configuration of the p, q, or r orthotopes, but only the net changes to these orthotopes alter λ. In actuality, the response of λ happens instantaneously, but we use this procedural metaphor to help readers of think more deeply about the linkage.

The mathematical configurations of A_ijs in Eqs. (6)–(9) are governed by definitions in linear algebra (Strang, 2016), yet the configurations of sums, differences, and products of A_ijs suggest that not all A_ijs contribute equally to λ. Such asymmetries in the configurations of A_ijs suggest that targeted alterations to some A_ij are more likely to produce net effects than alterations to others. Indeed, traditional sensitivity analyses (in equation form) have long exposed these differentials (Caswell, 2001). We suggest, however, that IsoPOPd may be used by broader audiences to garner a deeper understanding of why this is so. For example, the deer example illustrated that alterations to an A_ii (that contributes to the p orthotope) had higher propensity to influence net changes in the characteristic equation (and therefore λ) than the A_ijs that contributes only to r. Calculations using traditional sensitivity analysis for white-tailed deer corroborate this result; A₃₃ has a higher influence on λ than A₃₂.

It might seem intuitive that a management strategy designed to improve survival of all individuals would proportionally increase the growth rate. But we just showed that in some life cycles, this intuition may be flawed. The deer demonstration revealed that a management strategy designed to simultaneously increase survival in the 2nd and 3rd stages is no better off than a management strategy designed to increased survival in the 3rd stage only. Due to the asymmetries of the vital rates in Eq. (9), simultaneous modification of A₃₂ and A₃₃ differentially altered the characteristic equation at the normal scale, the quadratic scale, and produced two (opposing!) influences at the cubic scale, lending an overall unintuitive (and non-linear) response in λ. Indeed, it was the counterbalancing of the positive and negative expansions from A₃₂ and A₃₃ in the r orthotope that neutralized any additive benefit of increasing A₃₂ (Fig. 11). This seemingly peculiar result arose from the life history of the deer, and may or may not extend to management in other species. Afterall, sensitivity analysis is situational (De Kroon, Van Groenendael & Ehrlen, 2000).

It is natural to ponder the biological interpretations of p, q, and r, but such a discussion is outside of the scope of this work. Rather, the scope of this demonstration is to provide a general tool for non-specialized audiences to heuristically discover why the behavior of λ can defy intuition, and how to leverage the mathematics to their benefit in their strategic planning. Simply stated, management strategies that influence vital rates in the r orthotope are not as influential on λ as management strategies that influence the p or q orthotopes. As well, any alteration that counteracts another in magnitude and directionality will result in unchanged net values, which in turn will fail to alter λ. Consideration of these mathematical relationships may aid managers in identifying the most efficient intervention strategies for attaining their population goals.

In our demonstration and discussion, we have used the concept of vital rates and matrix elements interchangeably. In the deer example, stage one individuals can die or advance to stage two, stage two individuals can die or advance to stage three, and stage three individuals can die or remain in stage three, so the probability of any transition (given survival) is equal to one. However, in more complicated life histories, transitional matrix elements (A_ij where i = 2, 3, and j = 1, 2, 3) are defined as the product between survival and transition (Caswell, 2001). This definition becomes important when individuals can transition through their life history in more ways than one. For example, in PMMs where A₂₂ is non-zero, animals in stage two could (1) die, (2) survive and remain in stage two, or (3) survive and mature into stage three. In this case, the probability of transition among stages two and three must be incorporated into the values of both A₂₂ and A₃₂ (and not just in A₃₂ as we did in our example). Although the compound nature of transition and survival are undoubtedly important in PMMs, we leave it to the user to properly calculate each A_ij for input in the IsoPOPd software.

Conclusion

Population matrix models are foundational in the study of ecology, population dynamics, wildlife management, and conservation biology (Caswell, 2001), and sensitivity investigations of the relationships between A_ij and λ are not novel (see Salguero-Gómez et al., 2014; Salguero-Gómez et al., 2016). However, it is our hope that the IsoPOPd software illuminates to broader audiences the behavior of λ in the context of decision making in population management, and how such knowledge can be used as leverage to achieve conservation goals.

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