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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)

Citations:

Zbl 0339.47031
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References:

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