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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 (·),b i (·)C(R,[0,+)); τ i (·),ρ i (·)C 1 (R,[0,+)); c i (·)C(R,(0,+));r i (·)C(R,R)(i=1,2) be ω-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-τ 1 (t))+c 1 (t)y 2 (t-ρ 1 (t)))],
y 2 ' (t)=y 2 (t)[r 2 (t)-a 2 (t)y 2 (t)-b 2 (t)y 2 (t-τ 2 (t))+c 2 (t)y 1 (t-ρ 2 (t)))],

where 0 ω 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.

MSC:
34K60Qualitative investigation and simulation of models
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)