Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model.

*(English)*Zbl 0954.92027Summary: We investigate a generalized \(n\)-species Gilpin-Ayala competition system [M.E. Gilpin and F.J. Ayala, Proc. Natl. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)] with several deviating arguments in periodic environment, which is more general and more realistic than the classical Lotka-Volterra competition systems. By using the method of coincidence degree, a set of easily verifiable sufficient conditions is derived for the existence of at least one strictly positive (componentwise) periodic solution.

Some new results are obtained. As applications, we also apply our main results to some special cases of the system we consider here, including the classical \(n\)-species Lotka-Volterra competition systems and \(n\)-species Gilpin-Ayala competition model, which have been studied extensively in the literature. Some known results are improved and generalized. The examples show that our criteria are new, general, and easily verifiable.

Some new results are obtained. As applications, we also apply our main results to some special cases of the system we consider here, including the classical \(n\)-species Lotka-Volterra competition systems and \(n\)-species Gilpin-Ayala competition model, which have been studied extensively in the literature. Some known results are improved and generalized. The examples show that our criteria are new, general, and easily verifiable.

##### MSC:

92D40 | Ecology |

34C25 | Periodic solutions to ordinary differential equations |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

##### Keywords:

positive periodic solutions; generalized Gilpin-Ayala competition system; deviating arguments; Gilpin-Ayala competition system; coincidence degree
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\textit{M. Fan} and \textit{K. Wang}, Comput. Math. Appl. 40, No. 10--11, 1141--1151 (2000; Zbl 0954.92027)

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