Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency

Abstract

In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio α, and the release is terminated from (N + 1)-th generation to T-th generation. We find a release ratio threshold denoted by α∗(N, T), and prove the existence of a T-periodic solution for the model when α ∈ (0, α∗(N, T)). For the special case when N = 1 and T = 2, we prove that the model has a unique T-periodic solution which is unstable when α ∈ (0, α∗(N, T)). While α ≥ α∗(N, T), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.