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Interaction of maturation delay and nonlinear birth in population and epidemic models. (English) Zbl 0945.92016
Summary: 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 $N'(t)=B \bigl(N(t-T) \bigr)N(t-T)e^{-d_1T}-dN(t)$ governs the adult population, with the death rate in previous life stages $$d_1\geq 0$$. 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.

##### MSC:
 92D30 Epidemiology 34D23 Global stability of solutions to ordinary differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations
maturation delay
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