## Periodic solutions of delayed predator-prey model with the Beddington-DeAngelis functional response.(English)Zbl 1155.34361

The authors study the following delayed predator-prey model with Beddington-DeAngelis type functional response
\begin{aligned} \dot x(t)&= x(t)\biggl[a(t)-b(t)x(t-\tau(t,x(t),y(t)))- \frac{c(t)y(t)}{1+nx(t)+my(t)}\biggr],\\ \dot y(t)&= y(t) \biggl[\frac{f(t)x(t-\sigma(t,x(t),y(t)))}{1+nx(t-\sigma(t,x(t),y(t)))+ my(t-\sigma(t,x(t),y(t)))}-d(t)\biggr], \end{aligned}\tag{1}
where $$x(t)$$ and $$y(t)$$ represent the densities of the prey and the predator population at time $$t$$, respectively. $$a(t), b(t), c(t), d(t), f(t)\in C(\mathbb R,\mathbb R^+)$$, $$\mathbb R^+=(0,+\infty)$$, are $$\omega$$-periodic functions, $$\tau$$ and $$\sigma\in C(\mathbb R,\mathbb R)$$ are nonnegative $$\omega$$-periodic functions with respective the first argument, $$m$$ and $$n$$ are positive constants. Sufficient conditions are derived for the existence of periodic solutions to system (1) by using the continuation theorem developed by [R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Berlin etc.: Springer-Verlag (1977; Zbl 0339.47031)].

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general)

### Keywords:

periodic solution; delay; predator-prey model

Zbl 0339.47031
Full Text:

### References:

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