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A note on the periodic orbits of Wolbachia spread dynamics in mosquito populations in periodic environments

J. S. Canovas

Abstract

We consider the periodic model introduced in [20] and disprove the conjectures on the number of periodic orbits the model can have. We rebuild the conjecture to prove that for periodic sequences of maps of any period, the number of non-zero periodic trajectories is bounded by two.

A note on the periodic orbits of Wolbachia spread dynamics in mosquito populations in periodic environments

Abstract

We consider the periodic model introduced in [20] and disprove the conjectures on the number of periodic orbits the model can have. We rebuild the conjecture to prove that for periodic sequences of maps of any period, the number of non-zero periodic trajectories is bounded by two.

Paper Structure

This paper contains 4 sections, 9 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notation, known results and technical lemmas
  3. Main results
  4. Discussion

Key Result

Proposition 1

Let $[\varphi _{1},...,\varphi _{T}]$ be a periodic sequence of maps and let $x$ be a fixed point of $\varphi _{T}\circ ...\circ \varphi _{1}$. Then, $x$ is a fixed point for all the maps $\varphi _{1},...,\varphi _{T}$, if and only if $x$ is a fixed point for $[\varphi _{1},...,\varphi _{T}]$.

Figures (3)

  • Figure 1: For $T=2$, $s_{f_{1}}=0.2$, $s_{h_{1}}=0.45$, $s_{f_{2}}=0.4$, $s_{h_{2}}=0.9$ and $\mu_1=\mu_2=0$, graph of the maps $f_1$ (thick), $f_2$ (dashed) and $f_2\circ f_1$ (thin). The common fixed point is $4/9$.
  • Figure 2: (a) For $T=2$, $s_{f_{1}}=0.1$, $s_{h_{1}}=0.9$, $s_{f_{2}}=0.3$, $s_{h_{2}}=0.9$, $\mu_1=\mu_1^*$ and $\mu_2=0$, graph of the maps $f_2\circ f_1$ (thick) and the identity (thin). (b) For $T=2$, $s_{f_{1}}=0.1$, $s_{h_{1}}=0.9$, $s_{f_{2}}=0.3$, $s_{h_{2}}=0.9$, $\mu_1=\mu_1^*$ and $\mu_2=\mu_2^*$, graph of the maps $f_2\circ f_1$ (thick) and the identity (thin). There are no fixed points except for $0$.
  • Figure 3: For $T=2$, $s_{f_1} = 0.1$, $s_{h_1} = 0.9$, $s_{f_2} = 0.8$, $s_{h_2} = 0.9$, $\mu_1 =0.0975309$ and $\mu_2=0.00863972$, graph of the map $f_2\circ f_1$ (thick) and the identity (thin). The fixed point is approximately $0.7949203$.

Theorems & Definitions (17)

  • Conjecture 1
  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1: Singer's Theorem
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Example 6
  • ...and 7 more