Some simple models for nonlinear age-dependent population dynamics

Abstract

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. BASIC EQUATIONSRecently, 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);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.

Diffusive and nondiffusive population models

“…This equation is sometimes referred to as the renewal equation since it describes how a population is renewed [59]. When we assume that the life span is finite, a ∈ [0, a + ] with a + < ∞, this problem can be formulated as a Volterra equation of second kind (for the variable B(t) := u(0, t)), which is called the renewal or Lotka equation [64].…”

“…For i = 1 and c 1 (a) = 1, we obtain the Gurtin-MacCamy model introduced in [59]. The existence and uniqueness of solutions to this model is proved in [146] using a semigroup approach.…”

Self-thinning and community persistence in a simple size-structured dynamical model of plant growth

“…The original Sharpe-Lotka-McKendrick-von Foerster equation is linear. Nonlinear age-and size-structured population models have been first proposed by Gurtin and MacCamy (1974) and Hoppensteadt (1974) (see also Webb, 1985;Metz and Diekmann, 1986;Cushing, 1998;Diekmann et al, 2001). Traditionally, the source of nonlinearity is the density dependence of some of the characteristic population rates (birth, death, migration) on the total population size…”

Hierarchical models of intra-specific competition: scramble versus contest

“…One approach taken by many authors has been to study classes of models that can mathematically be reduced, by means of some trick or other, to more tractable equations, such as ordinary differential equations or integral equations. Examples include socalled ''separable equations'' [28,2], ''linear chain trickery'' [17,26], and ''hierarchically structured'' models [9,10]. The latter type of models are particularly useful for studying contest and scramble competition because the definitions of these modes of competition involve a hierarchical ranking within the population [22].…”

“…For density regulated populations they are also dependent on the distribution function , usually by means of a linear functional of . In the simplest density-dependent models and depend on total population size [16] …”