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Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent. (English) Zbl 1147.34056

The authors study two species time-delayed predator-prey Lotka-Voltera type dispersal systems with periodic coefficients in which the prey species can disperse among \(n\) patches, while the density-independent predator species is confined to one of patches cannot disperse
\[ \begin{aligned} & {dx_1(t)\over dt}= x_1(t)\Biggl[a_1(t)- b_1(t) x_1(t)- c(t)\int^0_{-\infty} k_{1,2}(s)y(t+ s)\,ds\Biggr]+\\ &\hskip 4cm \sum^n_{j=1} [\alpha_{1,j}(t) d_{1,j}(t) x_j(t- \tau_{1,j})- d_{j,1}(t) x_1(t)],\\ & {dx_i(t)\over dt}= x_i(t)[a_1(t)- b_i(t) x_i(t)]+ \sum^n_{j=1} [\alpha_{i,j}(t) d_{i,j}(t) x_j(t-\tau_{i,j})- d_{j,i}(t)x_i(t)],\;i= 2,\dots, n,\\ & {dy(t)\over dt}= y(t)\Biggl[- e(t)- f(t) \int^0_{-\infty} k_{2,1}(s) x_1(t+ s)\,ds\Biggr],\;t\in [0,\infty),\end{aligned}\tag{\(*\)} \]
\(x_i\) denote the population density of prey species in the \(i\) patch and \(y\) is the population density of predator species. Sufficient conditions on boundedness, permanence and existence of positive periodic solution for \((*)\) are established. The theoretical results are confirmed by a special example and numerical simulation.

MSC:

34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics (general)
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K13 Periodic solutions to functional-differential equations
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