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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
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