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On the stable periodic solutions of a delayed two-species model of facultative mutualism. (English) Zbl 1143.34054
Let \(a_i(\cdot), b_i(\cdot)\in C(R,[0,+\infty));\) \(\tau_i(\cdot),\rho_i(\cdot)\in C^1(R,[0,+\infty));\) \(c_i(\cdot)\in C(R,(0,+\infty)); r_i(\cdot)\in C(R,R)\, (i=1,2)\) be \(\omega\)-periodic functions. The authors establish sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions for the following two-species system modelling “facultative mutualism”: \[ y'_1(t)=y_1(t)[r_1(t)-a_1(t)y_1(t)-b_1(t)y_1(t-\tau_1(t))+c_1(t)y_2(t-\rho_1(t)))], \] \[ y'_2(t)=y_2(t)[r_2(t)-a_2(t)y_2(t)-b_2(t)y_2(t-\tau_2(t))+c_2(t)y_1(t-\rho_2(t)))], \] where \(\int_0^\omega r_i(t)dt>0\,(i=1,2).\) These results are extended to a more general two-species facultative mutualism system involving multiple delays. Some applications and biological interpretations are discussed.

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)
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