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Positive periodic solutions for an integrodifferential model of mutualism. (English) Zbl 0981.45002
The paper deals with a system of two nonlinear integro-differential equations of first order which presents a model of mutualism. The ecological meaning of such type systems may be found in the book by K. Gopalsamy [Stability and oscillations in delay differential equations of population dynamics (1992; Zbl 0752.34039)]. Sufficient conditions are given for the existence of at least one positive periodic solution of the system under consideration. The proof is based on the continuation theorem on the existence of at least one solution of the operator equation with Fredholm operator of index zero; in this connection see the book by R. E. Gaines and J. L. Mawhin [Coincidence degree and nonlinear differential equations (1977; Zbl 0339.47031)].

MSC:
45J05 Integro-ordinary differential equations
45M15 Periodic solutions of integral equations
45M20 Positive solutions of integral equations
92D25 Population dynamics (general)
45G15 Systems of nonlinear integral equations
92D40 Ecology
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[1] Vandermeer, J.H.; Boucher, D.H., Varieties of mutualistic interaction models, J. theor. biol., 74, 549-558, (1978)
[2] Boucher, D.H.; James, S.; Keeler, K.H., The ecology of mutualism, Ann. rev. syst., 13, 315-347, (1982)
[3] Dean, A.M., A simple model of mutualism, Amer. natural, 121, 409-417, (1983)
[4] Wolin, C.L.; Lawlor, L.R., Models of facultative mutualism: density effects, Amer. natural, 144, 843-862, (1984)
[5] Boucher, D.H., The biology of mutualism: ecology and evolution, (1985), Croom Helm London
[6] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Boston · Zbl 0752.34039
[7] Gaines, R.E.; Mawlin, J.L., Coincidence degree and nonlinear differential equations, Lecture notes in math., 568, (1977), Springer-Verlag Berlin
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