The periodic solutions for general periodic impulsive population systems of functional differential equations and its applications

Abstract

a b s t r a c tIn this paper, the general periodic impulsive population systems of functional differential equations are investigated. By using the method of Poincare map and Horn's fixed point theorem, we prove that the ultimate boundedness of all solutions implies the existence of periodic solutions. As applications of this result, the existence of positive periodic solutions for the general periodic impulsive Kolmogorov-type population dynamical systems are discussed. We further prove that as long as the system is permanent, there must exist at least one positive periodic solution. In addition, the permanence and existence of positive periodic solutions are discussed for the periodic impulsive single-species logistic models and the periodic impulsive n-species Lotka-Volterra competitive models with delays.

The dynamical behavior of a predator–prey system with Gompertz growth function and impulsive dispersal of prey between two patches

“…As a consequence of Theorems 4 and 5, we have the following necessary and sufficient conditions on the global attractivity of periodic solution .u 10 .t/, u 20 .t/, 0/ and the permanence of species for model (3.1). Further, from the main results given Theorem 1 in [30] on the existence of periodic solutions for the general periodic impulsive differential equations, as consequence of Corollary 2, we have the following result.…”

“…Further, from the main results given Theorem 1 in on the existence of periodic solutions for the general periodic impulsive differential equations, as consequence of Corollary , we have the following result.Corollary Model has at least one positive τ ‐periodic solution if Proof From Corollary , we know that if holds, then model is permanent. Therefore, from Theorem 1 in , if an impulsive differential system is permanent, then there must exist at least one periodic solution. This completes the proof of Corollary .…”

Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations

“…And the human activities always happen in a short time or instantaneously. Thus, the corresponding models are subject to short-term perturbations, which are often assumed to be in the form of impulsive in the modeling process, one can see [21][22][23][24][25][26][27][28][29][30][31][32] and the references therein.…”

Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

“…Applying Theorem 1 given in [16], it is clear that when condition (33) holds, model (1) at least has one positive -periodic solution.…”

“…The discontinuity of human activities and the abrupt variation in the amount of the pest population, which occurs immediately after successful control measures (such as spraying pesticides, releasing natural enemies of the pest, and freeing infective pest individuals), may be described mathematically through making use of impulsive differential equations (see [5][6][7][8][9][10][11][12][13][14][15][16]). …”