Demographic heterogeneity impacts density-dependent population dynamics

Abstract

Among-individual variation in vital parameters such as birth and death rates that is unrelated to age, stage, sex, or environmental fluctuations is referred to as demographic heterogeneity. This kind of heterogeneity is prevalent in ecological populations, but is almost always left out of models. Demographic heterogeneity has been shown to affect demographic stochasticity in small populations and to increase growth rates for density-independent populations. The latter is due to “cohort selection,” where the most frail individuals die out first, lowering the cohort’s average mortality as it ages. The importance of cohort selection to population dynamics has only recently been recognized. We use a continuous-time model with density dependence, based on the logistic equation, to study the effects of demographic heterogeneity in mortality and reproduction. Reproductive heterogeneity is introduced in three ways: parent fertility, offspring viability, and parent–offspring correlation. We find that both the low-density growth rate and the equilibrium population size increase as the magnitude of mortality heterogeneity increases or as parent–offspring phenotypic correlation increases. Population dynamics are affected by complex interactions among the different types of heterogeneity, and trade-off scenarios are examined which can sometimes reverse the effect of increased heterogeneity. We show that there are a number of different homogeneous approximations to heterogeneous models, but all fail to capture important parts of the dynamics of the full model.

Appendix: Population structure

We solve $\frac{\rm d}{{\rm d}t}n(t)=0$ and $\frac{\rm d}{{\rm d}t}q(t)=0$ for $u=\frac{q}{n}$ and arrive at two quadratic equations, one for n(t) = n ^* and the other for n(t) = 0. The equilibrium structure, u ^*, is the solution to the equation au ² + bu + c = 0 whose coefficients are

$$ a=(vh+f)+\sigma_\delta(vf+h), $$

$$ b=(1-h)\Big((1-vf)+\sigma_\delta(v-f)\Big), $$

and

$$ c=-\Big((v+hf)+\sigma_\delta(1+vhf)\Big). $$

If a ≠ 0, then the equation is quadratic, and there are two candidate solutions. When $f>-h\frac{v+\sigma_\delta}{1+\sigma_\delta v}$, u ^* is given by the ‘+’ root, and alternatively it is given by the ‘−’ root. When a = 0 the equation is linear, and u ^* = − c/b.

If $v>-\frac{hf+\sigma_\delta}{1+hf\sigma_\delta}$, then phenotype two dominates the equilibrium population (u ^* > 0), and if $v<-\frac{hf+\sigma_\delta}{1+hf\sigma_\delta}$, then phenotype one dominates the equilibrium population (u ^* > 0).

The structure along the eigenvector out of the origin, u ^o, is the solution to the equation au ² + bu + c = 0 whose coefficients are

$$ a=\beta(vh+f)+\sigma_\delta, $$

$$ b=\beta(1-h)(1-vf), $$

and

$$ c=-\beta(v+hf)-\sigma_\delta. $$

Similar to the equilibrium structure, when the coefficient a > 0, then we take the “+” root, and when a < 0, then we take the “−” root.