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

• Published in: Advances in nonlinear analysis Vol. 11; no. 1; pp. 212 - 224
• Main Authors: Zheng, Bo, Yu, Jianshe
• Format: Journal Article
• Language: English
• Published: De Gruyter 01.01.2022
• Subjects: 37N25; 92B05; 92D30; Discrete model; Existence and uniqueness; infection frequency; Mosquito population; Periodic orbits; wolbachia infection frequency
• ISSN: 2191-9496; 2191-950X
• DOI: 10.1515/anona-2020-0194

In this paper, we study a discrete model on infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first generations with the release ratio , and the release is terminated from ( + 1)-th generation to -th generation. We find a release ratio threshold denoted by , ), and prove the existence of a -periodic solution for the model when (0, , )). For the special case when = 1 and = 2, we prove that the model has a unique -periodic solution which is unstable when (0, , )). While ≥ , ), no periodic phenomenon occurs and the 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.