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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 τ>0. The model takes the form

x ˙ i (t)=αx m (t)-γx i (t)-e -γτ φ(t-τ),x ˙ m (t)=e -γτ φ(t-τ)-βx m 2 (t),0<tτ;
x ˙ i (t)=αx m (t)-γx i (t)-αe -γτ x m (t-τ),x ˙ m (t)=αe -γτ x m (t-τ)-βx m 2 (t),t>τ,

where φ (t) is the birth rate of x i (t) at time t, -τt0, and α,β,γ>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