Some Simple Models for Nonlinear Age-Dependent Population Dynamics

This paper presents two simple models for nonlinear age-dependent population dynamics. In these models the basic equations of the theory reduce to systems of ordinary differential equations. We discuss certain qualitative aspects of these systems; in particular, we show that for many cases of interest periodic solutions are not possible. 1. BASIC EQUATIONS Recently, Gurtin and MacCamy [l] introduced a nonlinear’ theory of population dynamics with age dependence. This theory is based on the equations’ p,(a,t)+p,(a,t)+CL(a,P(t))p(a,t)=O, B(t)=p(O,r)=lmp(a,P(r))p(a,r)da, (1.1) 0 where p(a,t) is the age distribution, that is, the number of individuals of age a at time t (a > 0, t > 0); P(t) is the total population; B(t) is the birth rate; ~(a, P) is the survival function; P(a, P) is the materniv function. ‘The linear theory is discussed in detail by Hoppensteadt [2]. 2Subscripts indicate partial differentiation. MATHEMATICAL BIOSCIENCES 43: 199-211 (1979) 199 ~Elsevier North Holland, Inc., 1979 0025-55&I/79/020199+ 13502.25 200 MORTON E. GURTIN AND RICHARD C. MAcCAMY The functions &a,P) and j?(a,P) (a > 0, P > 0) are assumed prescribed, as is the initial age distribution cp. The corresponding initial-value problem then consists in finding a solution p(a, t) of (1.1) which satisfies the initial condition P(G 0) = cp(a). (l-2) The system (1.1) (1.2) can be converted to a pair of nonlinear integral equations, which can be used, under reasonable hypotheses, to establish existence and uniqueness [l]. Here we will show that when ~1 and j3 have certain simple forms, (1.1) and (1.2) reduce to an initial-value problem for a nonlinear system of ordinary differential equations. Before discussing these systems, however, we note that the partial differential equation (1. l), [with the conditions ~(0, t) = B(t), p(u, 0) = q(u)] can be integrated along characteristics to give3 p(a,t)=rp(a-t)~(a-t,a,O), a at, p(u,t)=B(t-u)?r(O,u,t-a), t >a, (1.3) where ~(u,,u,t)=exp i J Op(a,P(a-u~+t))da . a0 I Thus a knowledge of the quantities P and B will completely determine the age distribution of the population for all time. 2. MODEL WITH TWO DIFFERENTIAL EQUATIONS Before discussing this model we prove the following LEMMA Assume that the survival function y is independent of age. Let p be a solution of (l.l), and let g be a (sufficiently nice) function of age with4 g(u)p(u, t)+O us u+oo. (2.1) Define G(t)=~mg(u)~(~,f)d~, 0 ~(t)=/~g’(a)p(u,t)du. 0 (2.2) 3Cf. [I], pp. 284285. %is assumption is reasonable. Indeed, one can show, as a consequence of (1. 1)1, that p(a,l) as a function of a will have compact support at each t provided ‘p has compact support. NONLINEAR ALE-DEPENDENT ~PULATION D~A~~CS Then 201 d+p(P)G-g(O)B=H. (2.3) Proo$ We multiply (1. I)] by g and integrate from a =0 to a = co; the result is d(t) + / mg(~)p,(~,t)~~+~(~)G(t)=O, 0 and if we integrate the second term by parts and use (2.1), then (2.3) follows. We are now ready to state the basic assumptions of our model; they are P(% P) = Y(P)> P(a, P) = &(P)e-““. (24 Assumption (2.4), asserts that the expected number of births is a monotone decreasing function of age (and hence is m~imum at age a =O). Such an assumption is, of course, ludicrous if taken literally, but probably Ieads to a decent approximation if the population reproduces at a fairly young age. As we shall see, the importance of this model is that it leads to a pair of ordinary differential equations and hence is amenable to analysis. In Sec. 3 we will discuss a model with a more realistic birth law. We now turn to a derivation of the differential equations. By (2.3) with g=l 1 i;+p(P)PB=O. On the other hand, (2.3) with g(u) = e-ao yields c&(P)GB+~G=O, (2.5)