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

Renhao Tian, Jianfeng Huang, Yulin Zhao

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 Poincaré 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.

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

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 Poincaré 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.

Paper Structure

This paper contains 18 sections, 20 theorems, 86 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction and statements of main results
  2. Preliminaries
  3. "Normal" form of equation \ref{['equation0']}
  4. Rotated differential equations
  5. Derivative formulas in integral form of Poincaré map
  6. Proofs of Theorems \ref{["th1'"]}, \ref{['coro1']} and \ref{['th2']}
  7. Derivative formulas in discrete form of Poincaré map
  8. Characterization for the distribution of periodic solutions
  9. Applications to single-species models
  10. Application 1
  11. Application 2
  12. Application 3
  13. Strategy $T>\overline T$
  14. Strategy $T<\overline T$
  15. Appendix
  16. ...and 3 more sections

Key Result

Theorem 1.1

The Poincaré map $P(x_0)=x(1;0,x_0)$ of equation equation1, is $C^2$-differentiable on $V:=\{\rho\in \mathcal{D}|f_i(x(\frac{i-1}{n};0,\rho))\neq0,i=1,2,\cdots,n\}$. Furthermore, for $x_0\in V$ and $i=1,2,\ldots,n$, let $x_i:=x(\frac{i}{n};0,x_0)$. Then

Figures (4)

  • Figure 1: The relative positions of the zeros of $g_1$, $g_2$, and $g_1+g_2$ for different values of $h$.
  • Figure 2: Regional division on stability and multiplicity of limit cycles in the subcase $A\neq0$ with $f_1|_E>0$. The hyperbola and the regions ①, ② and ③ divided by the hyperbola and $x_1=x_0$ correspond to different stabilities of limit cycles.
  • Figure 3: The relative positions of the zeros of $h_1$, $h_2$, and $h_1+h_2$ for different values of $T$ and $c$.
  • Figure 4: The relative positions of the zeros of $h_1$, $h_2$, and $h_1+h_2$ for different values of $c$ and $T$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6: YULI5
  • Proposition 1.7: BOYU
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 31 more