A simple geometrical condition for the existence of periodic solutions of planar periodic systems. Applications to some biological models (original) (raw)
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Advances in Difference Equations, 2016
We consider two-dimensional predator-prey systems with Beddington-DeAngelis-type functional response on periodic time scales. For this special case, we try to find the necessary and sufficient conditions for the considered system when it has at least one w-periodic solution. This study is mainly based on continuation theorem in coincidence degree theory and will also give beneficial results for continuous and discrete cases. Especially, for the continuous case, by using the study of Cui and Takeuchi (J. Math. Anal. Appl. 317:464-474, 2006), to obtain the globally attractive w-periodic solution of the given system, an inequality is given as a necessary and sufficient condition. Additionally, for the continuous case in this study, the open problem given in the discussion part of the study of Fan and Kuang (J. Math. Anal. Appl. 295:15-39, 2004) is solved.
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The criteria for an entirely bounded solution of a quasi-linear differential system are developed via asymptotic boundary value problems. The same principle allows us to deduce at the same time the existence of periodic orbits, when assuming additionally periodicity in time variables of the related right-hand sides. For almost periodicity, the situation is unfortu nately not so straightforward. Nevertheless, for the Lipschitzean uniformly almost periodic (in time variables) systems, we are able to show that every bounded solution becomes almost periodic as well.