Existence of periodic solutions in predator–prey and competition dynamic systems |
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Authors: | Martin Bohner Meng Fan Jimin Zhang |
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Affiliation: | aDepartment of Mathematics and Statistics, University of Missouri–Rolla, Rolla, MO 65401, USA;bSchool of Mathematics and Statistics, and Key Laboratory for Vegetation Ecology, Northeast Normal University, Changchun, Jilin 130024, PR China |
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Abstract: | In this paper, we systematically explore the periodicity of some dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator–prey systems and competition systems) in mathematical biology governed by differential equations and difference equations. Easily verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models when the time scale is chosen as or , respectively. The main approach is based on a continuation theorem in coincidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on general time scales. |
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Keywords: | Time scales Periodic solution Coincidence degree Predator– prey system Beddington– DeAngelis response Holling-type response Competition system Gilpin– Ayala system |
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