The solution and the stability of a nonlinear age-structured population model

Norhayati, G. C. Wake
2003 ANZIAM journal (Print)  
We consider an age-structured population model achieved by modifying the classical Sharpe-Lotka-McKendrick model, incorporating an overcrowding effect or competition for resources term. This term depends on the whole population rather than on any specific age group, in the case of overcrowding or limitation of resources. We investigate the solutions for arbitrary initial conditions. We consider the existence of a steady age distribution and its stability and are able to determine this for a
more » ... le illustrative case. If the non-trivial steady age distribution is unstable, there is a critical initial population size beyond which the population explodes. This watershed is independent of the shape of the initial age distribution. It is only relatively recently that the effect of age of the individuals in a population on the growth of that population has been considered. Among the first 'continuous' population models incorporating age effects were those of Sharpe and Lotka [10] and McKendrick [7]. Basically, the Sharpe-Lotka-McKendrick model assumes that birth and mortality processes are linear functions of population density. Leslie [4] used discrete age compartments (giving rise to Leslie matrices) and in [3] used both discrete time and age compartments. The methods used here generalise these for a completely non-discrete model introduced below. Gurtin and MacCamy [1] and Hoppensteadt [2] introduced the first models of nonlinear continuous age-dependent population dynamics. Wake et al. [11] considered age-structured disease models but with nonlinear terms in the disease-transmission formulation. Here we consider the effect of overall population density on the age distribution as suggested by Pollak [9]. Consequently, the equations of that model, as in the case of the Verhulstian models, contained nonlinear terms involving the total population. By analogy with the Verhulstian models of age-independent population dynamics, these nonlinearities provided a mechanism by which the population might stabilise to a non-trivial equilibrium state as time evolves. The formulation of the age-structured model The classical model of linear age-dependent population dynamics represented by the Sharpe-Lotka equation is formulated as follows. Let n(a, t) be the population density (or age distribution) with respect to age a of a population at time t. The units of n(a, t) are given in units of population divided by units of time. Accordingly, the total number of individuals between ages a\ and a 2 is given by / Ja 2 n(a,t)da, where n(a, t) is taken to be a smooth function of (a, t). The total population at time / of all members of the population is given by N(t) = / n(a,t)da. Jo The upper limit of the chronological age of the species considered is, of course, some finite number, but it will be assumed for convenience to be infinite with n(oo, t) = 0 and with N(t) being finite. This density function satisfies the so-called balance law (or ageing process of the population). The equation describing the growth of the population is formulated as follows. available at Downloaded from IP address:, on 27 Jul 2018 at 17:41:56, subject to the Cambridge Core terms of use, Case A. For the trivial solution, first of all we have that (5.2) takes the form R(0) = 1 = p/n and (6.4) takes the form /?(0) = 0/(/x + X). However, R(X) is a decreasing function of X, so that if R(0) < 1 we have X < 0 and the trivial solution is asymptotically stable. For fi > ii, in contrast we have A. > 0 and the trivial solution is unstable. Case B. (i) If ft > ii and k > 0, then the term in the denominator of (4.3) which dominates at large t is /xkNo so that N(r) -*• (/} -ix)j\ik. This means that on available at https://doi.
doi:10.1017/s1446181100013237 fatcat:v4like67s5hczp27zjh7zpasqi