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A discrete model of competing species sharing a parasite

Rafael Bravo de La Parra, Luis Sanz

TL;DR

The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number, and is a generalization of the Leslie-Gower competition model.

Abstract

In this work we develop a discrete model of competing species affected by a common parasite. We analyze the influence of the fast development of the shared disease on the community dynamics. The model is presented under the form of a two time scales discrete system with four variables. Thus, it becomes analytically tractable with the help of the appropriate reduction method. The 2-dimensional reduced system, that has the same the asymptotic behaviour of the full model, is a generalization of the Leslie-Gower competition model. It has the unfrequent property in this kind of models of including multiple equilibrium attractors of mixed type. The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number R0. In some cases, initial conditions decide among several exclusion or coexistence scenarios.

A discrete model of competing species sharing a parasite

TL;DR

The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number, and is a generalization of the Leslie-Gower competition model.

Abstract

In this work we develop a discrete model of competing species affected by a common parasite. We analyze the influence of the fast development of the shared disease on the community dynamics. The model is presented under the form of a two time scales discrete system with four variables. Thus, it becomes analytically tractable with the help of the appropriate reduction method. The 2-dimensional reduced system, that has the same the asymptotic behaviour of the full model, is a generalization of the Leslie-Gower competition model. It has the unfrequent property in this kind of models of including multiple equilibrium attractors of mixed type. The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number R0. In some cases, initial conditions decide among several exclusion or coexistence scenarios.
Paper Structure (6 sections, 6 theorems, 63 equations, 3 figures)

This paper contains 6 sections, 6 theorems, 63 equations, 3 figures.

Table of Contents

  1. The model.
  2. Analysis of the reduced model with homogeneous disease transmission and recovery.
  3. Parasite-mediated and parasite-modified competition.
  4. Conclusion.
  5. Disease dynamics in the case $\beta_{ij}=\beta$ and $\gamma _{i}=\gamma$ for $i,j=1,2$, and reduction results.
  6. Dynamics of the reduced system

Key Result

Lemma 2.1

Let $i\in\left\{ 1,2\right\}$ be fixed. The set $S_{i}:=\left\{ (x_{1},x_{2})\in\mathbb{R}^{2}:\phi_{i}(x_{1},x_{2})=1\right\}$ is a hyperbola that degenerates if and only if $c_{S1}^{i}c_{I2}^{i}=c_{S2}^{i}c_{I1}^{i}$, in which case it becomes two parallel lines. In addition, $S_{i}$ intersects $ where $\alpha_{j}^{i}:=r_{S}^{i}(c_{Ij}^{i}-1)+r_{I}^{i}(c_{Sj}^{i}-1),\ j=1,2$. Moreover, for each

Figures (3)

  • Figure 1: Different configurations of system (\ref{['sist']}) when $\phi _{i}(0,0)>1$ for $i=1,2$, in terms of the relative position of the intercepts of isoclines, $R_{ij}$\ref{['intercepts']} and the number of positive equilibria, as described in (\ref{['casos']})
  • Figure 2: Basins of attraction $B(E_{1}^{\ast})$, $B(E_{2}^{\ast})$ and $B(E_{4}^{\ast})$ of equilibria $E_{1}^{\ast}$, $E_{2}^{\ast}$ and $E_{4}^{\ast}$ and separatrix curves $\gamma_{3}$ and $\gamma_{5}$ for system \ref{['rm']} for parameters values: $\nu=0.5$, $b_{S}^{1}=13$, $b_{I}^{1}=3.6$, $b_{S}^{2}=3.4$, $b_{I}^{2}=8$, $c_{SS}^{11}=c_{SI}^{11}=0.9$, $c_{IS}^{11}=c_{II}^{11}=0.1$, $c_{SS}^{12}=c_{SI}^{12}=1.1$, $c_{IS}^{12}=c_{II}^{12}=5$, $c_{SS}^{21}=c_{SI}^{21}=6$, $c_{IS}^{21}=c_{II}^{21}=0.3$, $c_{SS}^{22}=c_{SI}^{22}=0.2$, $c_{IS}^{22}=c_{II}^{22}=0.8$.
  • Figure 3: Asymptotic behaviour cases of solutions of system (\ref{['rm']}) (Th. \ref{['prop:prop3']}) for parameters values: $\nu\in(0,1)$, $b_{S}^{1}\in[2,20]$, $b_{I}^{1}=2$, $b_{S}^{2}=4.4,b_{I}^{2}=9$, $c_{SS}^{11}=1.3$, $c_{SI}^{11}=0.5$, $c_{IS}^{11}=c_{II}^{11}=0.1$, $c_{SS}^{12}=1$, $c_{SI}^{12}=0.05$, $c_{IS}^{12}=8$, $c_{II}^{12}=3$, $c_{SS}^{21}=6$, $c_{SI}^{21}=c_{IS}^{21}=c_{II}^{21}=0.3$, $c_{SS}^{22}=c_{SI}^{22}=0.2$, $c_{IS}^{22}=c_{II}^{22}=0.8$.

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Proposition 1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2
  • proof
  • ...and 2 more