Separable models in age-dependent population dynamics
This paper treats nonlinear models of age structured populations. The models have a certain separable form and include equations such as \[ u_ a+u_ t+\mu (a)u=F(t,P(t)),\quad u(a,0)=u_ 0(a),\quad u(0,t)=\int^{\infty}_{0}\beta (a)u(a,t)da, \] where \(P(t)=\int^{\infty}_{0}u(a,t)da\). The problem is recast in terms of a partial differential equation for the age profile \(w(a,t)=u(a,t)/P(t)\) and an ordinary nonlinear differential equation for P(t). The equation for w(a,t) is analyzed using classical linear renewal theory. Results concerning asymptotic behavior of solutions are obtained for a variety of models. In particular the effects of maturation periods are studied for interacting species and predator-prey models. Hopf bifurcation theory is used to establish that increases in maturation period can have stabilizing or destabilizing consequences, and in certain cases successive exchanges in stability. This paper develops a very useful approach to a general class of nonlinear models of age dependent population dynamics.
nonlinear models of age structured populations
scientific article; zbMATH DE number 3918170