On the growth of populations with narrow spread in reproductive age: III. Periodic variations in the environment

Summary

The theory discussed in the first two papers, I and II, of this series is here generalized so that it is applicable to a population which obeys, at each instant t, the following two assumptions: (i) the rate- Dx(a, t) at which the population loses individuals of age a through death and dispersal is given by a function γ_t of a and the number x(a, t) of individuals which have age a, i.e.-Dx(a, t)=γ_t(x(a, t), a) and (ii) the number x(0, t) of newly born individuals is given by a function F_t of the number x(a_f, t) of individuals at a specified age a_f of fecundity, i.e. x(0, t)=F_t(x(a_f, t)). The ‘autonomous case’ in which the functions γ_t and F_t are independent of the subscript t corresponds to the theory developed in I and II. The present article contains a treatment of the case in which the population is in a ‘periodic environment’ in the sense that the mapping t ↦ (γ_t, F_t) is periodic with a period which is an integral multiple N of a_f. Under the assumption that for each pair (t, a) the function γ_t(·, a) is convex and the function F_t(·) is strictly increasing and concave, it is shown that when the environment is periodic, a given population can be expected to belong to one of three classes, regardless of initial conditions: (A) the class of ‘endangered populations’ for which the abundance function x eventually decays to zero, (B) the class of ‘asymptotically periodic populations’ for which as time increases x approaches a non-zero function x^* which is periodic in time with period Na_f, and (C) the class of populations which exhibit unbounded growth. The properties of the loss functions γ_t and fecundity functions F_t which determine the class to which a population belongs are found and discussed, and formulae are given for the stable periodic abundance function x^* of a population in class B. In a discussion of the domain of application of the theory, it is pointed out that when reproduction is seasonal and is followed by mortality, the assumption that an individual interacts only with others of the same age is a reasonable one.