On the study of the limit cycles for a class of population models with time-varying factors

TL;DR

This paper develops new mathematical tools to analyze limit cycles in population models with time-varying factors, providing bounds on their number and applying these to ecological models with seasonal effects.

Contribution

It introduces derivative formulas and a criterion for limit cycle existence in piecewise smooth population models, unifying analysis methods and improving bounds on limit cycles.

Findings

A maximum of two limit cycles in seasonal harvesting models.

At most three limit cycles in Abel equation-based models.

Application to mosquito population suppression models.

Abstract

In this paper, we study a class of population models with time-varying factors, represented by one-dimensional piecewise smooth autonomous differential equations. We provide several derivative formulas in "discrete" form for the Poincar\'{e} map of such equations, and establish a criterion for the existence of limit cycles. These two tools, together with the known ones, are then combined in a preliminary procedure that can provide a simple and unified way to analyze the equations. As an application, we prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao in 2016. We also apply our results to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles, is three. Finally, a population suppression model for mosquitos considered by Yu and Li in 2020 and Zheng et al. in 2021 is studied using our approach.