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

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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 01, 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.

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