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Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments. (English) Zbl 1186.34118

Summary: The model discussed in this paper is described by the following periodic 3-species Lotka-Volterra predator-prey system with several deviating arguments:
\[ \begin{cases} x'_1(t)=x_1(t)(r_1(t)-a_{11}(t)x_1(t-\tau_{11}(t))-a_{12}(t)x_2(t-\tau_{12}(t))-a_{13}(t) x_3 (t-\tau_{13}(t))) \\ x'_2(t)=x_2(t)(r_2(t)-a_{21}(t)x_1(t-\tau_{21}(t))-a_{22}(t)x_2(t-\tau_{22}(t))-a_{23}(t) x_3 (t-\tau_{23}(t))) \\ x'_3(t)=x_3(t)(r_2(t)-a_{31}(t)x_1(t-\tau_{31}(t))-a_{32}(t)x_2(t-\tau_{32}(t))-a_{33}(t) x_3 (t-\tau_{33}(t))) \end{cases} \tag{*} \]
where \(x_1(t)\) denotes the density of prey species at time \(t\), \(x_2(t)\) and \(x_3(t)\) denote the density of predator species at time \(t\), \(r_i,a_{ij}\in C(\mathbb{R},[0, \infty))\) and \(\tau_{ij}\in C(\mathbb{R},\mathbb{R})\) are \(w\)-periodic functions with
\[ \overline r_i=\frac1w\int^w_0 r_i(s) \,ds>0; \quad \overline a_{ij}=\frac1w\int^w_0 a_{ij}(s)>0, \quad I,j=1,2,3. \]
By using Krasnoselskii’s fixed point theorem and the construction of Lyapunov function, a set of easily verifiable sufficient conditions are derived for the existence and global attractivity of positive periodic solutions of (*).

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
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References:

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