# American Institute of Mathematical Sciences

June  2020, 25(6): 2121-2142. doi: 10.3934/dcdsb.2019204

## A discrete model of competing species sharing a parasite

 1 U.D. Matemáticas, Ed. Ciencias, Universidad de Alcalá, 28871 Alcalá de Henares, Spain 2 Dpto. Matemática Aplicada a la Ingeniería, ETSI Industriales, Univ. Politécnica de Madrid, 28006 Madrid, Spain

Received  February 2019 Published  September 2019

Fund Project: Authors are supported by Ministerio de Economía y Competitividad (Spain), project MTM2014-56022-C2-1-P.

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 asymptotic behaviour as 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 $R_0$. In some cases, initial conditions decide among several exclusion or coexistence scenarios.

Citation: Rafael Bravo De La Parra, Luis Sanz. A discrete model of competing species sharing a parasite. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2121-2142. doi: 10.3934/dcdsb.2019204
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Different configurations of system (10) when $\phi _{i}(0,0)>1$ for $i = 1,2$, in terms of the relative position of the intercepts of isoclines, $R_{ij}$ (11) and the number of positive equilibria, as described in (13)
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 (9) for parameters values: $\nu = 0.5$, $b_{S}^{1} = 13$, $b_{I}^{1} = 3.6$, $b_{S}^{2} = 3.4$, $b_{I}^{2} = 8$, $c_{SS}^{11} = c_{SI}^{11} = 0.9$, $c_{IS}^{11} = c_{II}^{11} = 0.1$, $c_{SS}^{12} = c_{SI}^{12} = 1.1$, $c_{IS}^{12} = c_{II}^{12} = 5$, $c_{SS} ^{21} = c_{SI}^{21} = 6$, $c_{IS}^{21} = c_{II}^{21} = 0.3$, $c_{SS}^{22} = c_{SI} ^{22} = 0.2$, $c_{IS}^{22} = c_{II}^{22} = 0.8$
Asymptotic behaviour cases of solutions of system (9) (Th. (3.3)) for parameters values: $\nu\in(0,1)$, $b_{S}^{1}\in[2,20]$, $b_{I}^{1} = 2$, $b_{S}^{2} = 4.4,b_{I}^{2} = 9$, $c_{SS}^{11} = 1.3$, $c_{SI}^{11} = 0.5$, $c_{IS}^{11} = c_{II}^{11} = 0.1$, $c_{SS}^{12} = 1$, $c_{SI}^{12} = 0.05$, $c_{IS}^{12} = 8$, $c_{II}^{12} = 3$, $c_{SS}^{21} = 6$, $c_{SI}^{21} = c_{IS} ^{21} = c_{II}^{21} = 0.3$, $c_{SS}^{22} = c_{SI}^{22} = 0.2$, $c_{IS}^{22} = c_{II}^{22} = 0.8$
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