Theory of the dependence of population levels on environmental history for semelparous species with short reproductive seasons.

A population that is strongly self-regulating through density-dependent effects is expected to be such that, if many generations have elapsed since its establishment, its present size should not be sensitive to its initial size but should instead be determined by the history of the variables that describe the influence of the environment on fecundity, mortality, and dispersal. Here we discuss the dependence of abundance on environmental history for a semelparous population in which reproduction is synchronous. It is assumed that at each instant t: (i) the rate of loss of members of age a by mortality and dispersal is given by a function rho of t, a, and the present number x = x(a,t) of such members; and (ii) the number x(0,t) of members born in the population is given by a function F of t and the number of x(a(f),t) at a specified age a(f) of fecundity. It is further assumed that the functions rho and F have the forms rho(x,a,t) = pi(1)(a,t)x + pi(2)(a,t)x(2) and F(x(a(f),t),t) = nu(t)x(a(f),t). For such a population, a change in the environment is significant only if it results in a change in nu(t) pi(1)(a,t), or pi(2)(a,t), and, hence, the history of the environment up to time t is described by giving nu(tau), pi(1)(a,tau), and pi(2)(a,tau) for each tau </= t and all a in [0, a(f)]. We show that the dependence of x on the history of the environment can be calculated explicitly and has certain properties of "fading memory"; i.e., environmental events that occurred in the remote past have less effect upon the present abundance than comparable events in the recent past.