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Existence and global attractivity of positive periodic solutions of competitor-competitor-mutualist Lotka-Volterra systems with deviating arguments. (English) Zbl 1200.34102

Summary: We study the existence and global attractivity of periodic solutions of competitor-competitor-mutualist Lotka-Volterra systems with deviating arguments
\[ \begin{cases} &x_1'(t)=x_1(t)(r_1(t)-a_{11}(t)x_1(t-\tau_{11}(t))-a_{12}(t)x_2(t-\tau_{12}(t))+a_{13}(t)x_3(t-\tau_{13}(t)))\\ &x_2'(t)=x_2(t)(r_2(t)-a_{21}(t)x_1(t-\tau_{21}(t))-a_{22}(t)x_2(t-\tau_{22}(t))+a_{23}(t)x_3(t-\tau_{23}(t)))\\ &x_3'(t)=x_3(t)(r_3(t)-a_{31}(t)x_1(t-\tau_{31}(t))-a_{32}(t)x_2(t-\tau_{32}(t))+a_{33}(t)x_3(t-\tau_{33}(t)))\end{cases}\tag{*} \]
where \(x_1(t)\) and \(x_2(t)\) denote the densities of competing species at time \(t\), \(x_3(t)\) denotes the density of cooperating species at time \(t\), \(r_i,a_{ij}\in C(R,[0,\infty))\) and \(\tau_{ij}\in C(R,R)\) are \(w\)-periodic functions \((\omega>0)\) with
\[ \bar r_i=\frac1w\int^w_0r_i(s)\,ds>0; \quad \bar a_{ij}=\frac1w \int^w_0 a_{ij}(s)\geq 0,\quad i,j=1,2,3. \]
We obtain sufficient conditions for the existence and global attractivity of positive periodic solutions of (*) by Krasnoselskii’s fixed point theorem and the construction of Lyapunov functions.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
47N20 Applications of operator theory to differential and integral equations
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References:

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