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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 $x\sb i(t)$ and $x\sb m(t)$ 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 $$ \dot x\sb i(t)=\alpha x\sb m(t)-\gamma x\sb i(t)-e\sp{-\gamma \tau}\phi (t-\tau),\quad \dot x\sb m(t)=e\sp{-\gamma \tau}\phi (t-\tau)-\beta x\sp 2\sb m(t),\quad 0<t\le \tau; $$ $$ \dot x\sb i(t)=\alpha x\sb m(t)-\gamma x\sb i(t)-\alpha e\sp{-\gamma \tau}x\sb m(t-\tau),\quad \dot x\sb m(t)=\alpha e\sp{-\gamma \tau}x\sb m(t-\tau)-\beta x\sp 2\sb m(t),\quad t>\tau, $$ where $\phi$ (t) is the birth rate of $x\sb i(t)$ 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:
92D25Population dynamics (general)
34K20Stability theory of functional-differential equations
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References:
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