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

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 eq uation N'(t) = B(N(t - T))N(t - T)e (-d1T) - dN(t) governs the adult popula tion, with the death rate in previous life stages d(1) greater than or equa l to 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 chan ge 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 (w ith numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a thre shold 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 val ue or oscillating about this value. Numerical simulations indicate that osc illations can also be induced by disease related death in a model with matu ration delay.