Interaction of maturation delay and nonlinear birth in population and epidemic models.

Abstract

A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N'(t)=B(N(t-T)) N(t-T) e(-)d(1)T- dN(t) governs the adult population, with the death rate in previous life stages d(1)>==0. Standard assumptions are made on B(N) so that a unique equilibrium N(e) exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d(1)>0, as T increases the equilibrium N(e) can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R(0) is identified. When R(0)<1, the disease dies out; when R(0)>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay.