Existence and Uniqueness of Nontrivial Periodic Solutions to a Discrete Switching Model

To control the spread of mosquito-borne diseases, one goal of the World Mosquito Program’s Wolbachia release method is to replace wild vector mosquitoes with Wolbachia-infected ones, whose capability of transmitting diseases has been greatly reduced owing to the Wolbachia infection. In this paper, we propose a discrete switching model which characterizes a release strategy including an impulsive and periodic release, where Wolbachia-infected males are released with the release ratio α1 during the first N generations, and the release ratio is α2 from the (N+1)-th generation to the T-th generation. Sufficient conditions on the release ratios α1 and α2 are obtained to guarantee the existence and uniqueness of nontrivial periodic solutions to the discrete switching model. We aim to provide new methods to count the exact numbers of periodic solutions to discrete switching models.

Periodic Solutions with Prescribed Minimal Period for 2 n th-Order Nonlinear Discrete Systems

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.

A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings.

Multiple periodic solutions for damped vibration systems with superquadratic terms at infinity

By using variational methods, we obtain infinitely many nontrivial periodic solutions for a class of damped vibration systems with superquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature. Besides, some examples are given to illustrate our results.

Bifurcation of Nontrivial Periodic Solutions for a Food Chain Beddington–DeAngelis Interference Model with Impulsive Effect

In this paper, a food chain Beddington–DeAngelis interference model with impulsive effect is studied. The trivial periodic solution is locally asymptotically stable if the release rate or the release period is suitable. Conditions for permanence of the model are obtained. The existence of nontrivial periodic solutions and semi-trivial periodic solutions are established when the trivial periodic solution loses its stability under different conditions.

Nontrivial periodic solutions to delay difference equations via Morse theory

In this paper some sufficient conditions are obtained to guarantee the existence of nontrivial 4T + 2 periodic solutions of asymptotically linear delay difference equations. The approach used is based on Morse theory.

Chemotherapy models

A chemotherapeutic treatment model for cell population with resistant tumor is considered. We consider the case of two drugs one with pulsed effect and the other one with continuous effect. We investigate stability of the trivial periodic solutions and the onset of nontrivial periodic solutions by the mean of Lyapunov-Schmidt bifurcation. Nous considérons un modèle de chimiothérapie pour une population de cellules avec ré-sistance. Nous considérons le cas de deux médicaments le premier avec effet impulsif et le deuxième avec effet continu. Nous étudions la stabilité des solutions périodiques triviales et l'apparition des solutions périodiques nontriviales en utilisant la bifurcation de Lyapunov-Schmidt