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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 ${x}_{i}\left(t\right)$ and ${x}_{m}\left(t\right)$ 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

${\stackrel{˙}{x}}_{i}\left(t\right)=\alpha {x}_{m}\left(t\right)-\gamma {x}_{i}\left(t\right)-{e}^{-\gamma \tau }\phi \left(t-\tau \right),\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{m}\left(t\right)={e}^{-\gamma \tau }\phi \left(t-\tau \right)-\beta {x}_{m}^{2}\left(t\right),\phantom{\rule{1.em}{0ex}}0
${\stackrel{˙}{x}}_{i}\left(t\right)=\alpha {x}_{m}\left(t\right)-\gamma {x}_{i}\left(t\right)-\alpha {e}^{-\gamma \tau }{x}_{m}\left(t-\tau \right),\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{m}\left(t\right)=\alpha {e}^{-\gamma \tau }{x}_{m}\left(t-\tau \right)-\beta {x}_{m}^{2}\left(t\right),\phantom{\rule{1.em}{0ex}}t>\tau ,$

where $\phi$ (t) is the birth rate of ${x}_{i}\left(t\right)$ at time t, -$\tau \le t\le 0$, 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