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Dynamic behaviors of a single-species population model with birth pulses in a polluted environment. (English) Zbl 1178.34051

The authors consider the following system with impulses:
\[ \begin{aligned} \frac{dx(t)}{dt} &=-rc(t)x(t)-dx(t), \\ \frac{dc(t)}{dt}&=kf(t)-gc(t), \\ \frac{df(t)}{dt}&=-hf(t). \end{aligned} \]
This system describes the dynamics of a single-species model with birth pulses, pulse harvesting and pulse toxicant input in a polluted environment.
Using a discrete dynamical system determined by the stroboscopic map, they obtain an exact 1-periodic solution of system whose birth function is the Ricker function or Beverton-Holt function, and obtain the threshold conditions for their stability.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34A37 Ordinary differential equations with impulses
34C25 Periodic solutions to ordinary differential equations
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