Evolution of stochastic demography with life history tradeoffs in density-dependent age-structured populations

Abstract

We analyze the stochastic demography and evolution of a density-dependent age- (or stage-) structured population in a fluctuating environment. A positive linear combination of age classes (e.g., weighted by body mass) is assumed to act as the single variable of population size, [Formula: see text], exerting density dependence on age-specific vital rates through an increasing function of population size. The environment fluctuates in a stationary distribution with no autocorrelation. We show by analysis and simulation of age structure, under assumptions often met by vertebrate populations, that the stochastic dynamics of population size can be accurately approximated by a univariate model governed by three key demographic parameters: the intrinsic rate of increase and carrying capacity in the average environment, [Formula: see text] and [Formula: see text], and the environmental variance in population growth rate, [Formula: see text] Allowing these parameters to be genetically variable and to evolve, but assuming that a fourth parameter, [Formula: see text], measuring the nonlinearity of density dependence, remains constant, the expected evolution maximizes [Formula: see text] This shows that the magnitude of environmental stochasticity governs the classical trade-off between selection for higher [Formula: see text] versus higher [Formula: see text] However, selection also acts to decrease [Formula: see text], so the simple life-history trade-off between [Formula: see text]- and [Formula: see text]-selection may be obscured by additional trade-offs between them and [Formula: see text] Under the classical logistic model of population growth with linear density dependence ([Formula: see text]), life-history evolution in a fluctuating environment tends to maximize the average population size.

Keywords: K-selection; age distribution; environmental variance; evolutionary maximization; r-selection.

Fig. 1.

(A) Mean, $\text{E}[N/K]$, and (B) SD, ${\sigma }_{N/K}$, as functions of the composite parameter $a={\sigma }_e^2/(2r_0)$, in the stochastic $\theta$-logistic model of density dependence for $\theta =0.5$ (red), $1$ (black), and $2$ (blue). (C) Stationary probability distribution $\varphi (N/K)$ in the stochastic logistic model ($\theta =1$) for two values of $a$. Solid lines calculated from diffusion approximation (Appendix). Each point in A or B and each distribution in C gives statistics for a sample of ${10}^4$ observations from a stage-structured model of an avian raptor with age at first breeding of 4 y and $r_0=0.04$, simulated for ${10}^6$ y and recorded once every 100 y (Supporting Information).

Fig. 2.

The same as Fig. 1 but with simulations for a passerine bird life history with age at first breeding of 1 y and $r_0=0.1$.