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Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. (English) Zbl 1168.34052
The authors study the problem $$\align \dot{y}_1(t)&= y_1(t)\left(r-ay_1(t-\tau)-\frac{\alpha_1 y_2(t)}{1+\beta_1 y_1(t)}\right),\\ \dot{y}_2(t)&= y_2(t)\left(\frac{\alpha_2 y_1(t)}{1+\beta_2 y_1(t)}-d\right), \quad t\not=nT,\ n=1,\dots,\\ \Delta y_1(nT)&= -py_1(nT^-), \quad y_i(t)=\varphi_i (t)>0,\ t<0,\ i=1,2. \endalign$$ They obtain the following results. If $(1-p)e^{rT}<1$, then the system is extinct. If $(1-p)e^{rT}>1$ then the zero solution becomes unstable. Moreover if $\alpha_2>d\beta_2$, and $(\alpha_2-d\beta_2)(rT+\ln (1-p))> \operatorname{ad} Te^{2rT+\ln (1-p)}$ then the system has at least one $T$ periodic solution.

34K45Functional-differential equations with impulses
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
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