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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.

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