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A time-delay model of single-species growth with stage structure. (English) Zbl 0719.92017
The authors consider the global asymptotic stability of the positive equilibrium of a single-species growth model with stage structure consisting of immature and mature stages. Let $$x_ i(t)$$ and $$x_ m(t)$$ denote the concentration of immature and mature populations, respectively. We assume that the populations entering the environment over a time interval equal to the length of time from birth to maturity is $$\tau >0$$. The model takes the form $\dot x_ i(t)=\alpha x_ m(t)-\gamma x_ i(t)-e^{-\gamma \tau}\phi (t-\tau),\quad \dot x_ m(t)=e^{-\gamma \tau}\phi (t-\tau)-\beta x^ 2_ m(t),\quad 0<t\leq \tau;$
$\dot x_ i(t)=\alpha x_ m(t)-\gamma x_ i(t)-\alpha e^{-\gamma \tau}x_ m(t-\tau),\quad \dot x_ m(t)=\alpha e^{-\gamma \tau}x_ m(t-\tau)-\beta x^ 2_ m(t),\quad t>\tau,$ where $$\phi$$ (t) is the birth rate of $$x_ i(t)$$ at time t, -$$\tau\leq t\leq 0$$, and $$\alpha,\beta,\gamma >0$$ are constants. Oscillation and nonoscillation of solutions are addressed analytically and numerically. The effect of the delay on the population at equilibrium is also considered.

##### MSC:
 92D25 Population dynamics (general) 34K20 Stability theory of functional-differential equations
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##### References:
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