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## 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, doi: 10.3934/dcdsb.2019204
##### References:
 [1] L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Mathematical Biosciences, 124 (1994), 83-105. doi: 10.1016/0025-5564(94)90025-6. Google Scholar [2] B. Bolker, M. Holyoak, V. Krivan, L. Rowe and O. Schmitz, Connecting theoretical and empirical studies of trait-mediated interactions, Ecology, 84 (2003), 1101-1114. doi: 10.1890/0012-9658(2003)084[1101:CTAESO]2.0.CO;2. Google Scholar [3] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, Reduction of discrete dynamical systems with applications to dynamics population models, Mathematical Modelling of Natural Phenomena, 8 (2013), 107-129. doi: 10.1051/mmnp/20138608. Google Scholar [4] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, Discrete models of disease and competition, Discrete Dynamics in Nature and Society, (2017), Art. ID 5310837, 13 pp. doi: 10.1155/2017/5310837. Google Scholar [5] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, A discrete predator-prey ecoepidemic model, Mathematical Modelling of Natural Phenomena, 12 (2017), 116-132. doi: 10.1051/mmnp/201712207. Google Scholar [6] Y. Chow and S. R.-J. Jang, Multiple attractors in a Leslie-Gower competition system with Allee effects, Journal of Difference Equations and Applications, 20 (2014), 169-187. doi: 10.1080/10236198.2013.815166. Google Scholar [7] J. M. Cushing, R. F. Costantino, B. Dennis, R. Desharnais and S. M. Henson, Chaos in Ecology: Experimental Nonlinear Dynamics, Theoretical Ecology Series, Vol. 1. Academic Press (Elsevier Science), New York, 2003.Google Scholar [8] J. M. Cushing, S. M. Henson and C. C. Blackburn, Multiple mixed-type attractors in a competition model, Journal of Biological Dynamics, 1 (2007), 347-362. doi: 10.1080/17513750701610010. Google Scholar [9] J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, Journal of Difference Equations and Applications, 10 (2004), 1139-1151. doi: 10.1080/10236190410001652739. Google Scholar [10] J. L. Edmunds, A Study of a Stage-Structured Model of Two Competing Species, Thesis (Ph.D.)-The University of Arizona, 2001, 64 pp. Available from: https://repository.arizona.edu/handle/10150/289978 Google Scholar [11] J. L. Edmunds, Multiple attractors in a discrete competition model, Theoretical Population Biology, 72 (2007), 379-388. doi: 10.1016/j.tpb.2007.07.004. Google Scholar [12] J. Edmunds, J. M. Cushing, R. F. Costantino, S. M. Henson, B. Dennis and R. A. Desharnais, Park's tribolium competition experiments: A non-equilibrium species coexistence hypothesis, Journal of Animal Ecology, 72 (2003), 703-712. Google Scholar [13] M. J. Hatcher, J. T. A. Dick and A. M. Dunn, How parasites affect interactions between competitors and predators, Ecology Letters, 9 (2006), 1253-1271. doi: 10.1111/j.1461-0248.2006.00964.x. Google Scholar [14] M. J. Hatcher and A. M. Dunn, Parasites in Ecological Communities: From Interactions to Ecosystems, Cambridge University Press, Cambridge, 2011. Google Scholar [15] P. Klepac and H. Caswell, The stage-structured epidemic: Linking disease and demography with a multi-state matrix approach model, Theoretical Ecology, 4 (2011), 301-319. doi: 10.1007/s12080-010-0079-8. Google Scholar [16] P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika, 45 (1958), 316-330. doi: 10.1093/biomet/45.3-4.316. Google Scholar [17] P. H. Leslie, T. Park and D. B. Mertz, The effect of varying the initial numbers on the outcome of competition between two tribolium species, Journal of Animal Ecology, 37 (1968), 9-23. doi: 10.2307/2708. Google Scholar [18] M. Marvá and R. Bravo de la Parra, Coexistence and superior competitor exclusion in the Leslie-Gower competition model with fast dispersal, Ecological Modelling, 306 (2015), 247-256. doi: 10.1016/j.ecolmodel.2014.10.039. Google Scholar [19] T. Park and M. Burton Frank, The fecundity and development of the flour beetles, tribolium confusum and tribolium castaneum, at three constant temperatures, Ecology, 29 (1948), 368-374. doi: 10.2307/1930996. Google Scholar [20] L. Sanz, R. Bravo de la Parra and E. Sánchez, Approximate reduction of non-linear discrete models with two time scales, Journal of Difference Equations and Applications, 14 (2008), 607-627. doi: 10.1080/10236190701709036. Google Scholar [21] H. L. Smith, Planar competitive and cooperative difference equations, Journal of Difference Equations and Applications, 3 (1998), 335-357. doi: 10.1080/10236199708808108. Google Scholar [22] E. E. Werner and S. D. Peacor, A review of trait-mediated indirect interactions in ecological communities, Ecology, 84 (2003), 1083-1100. doi: 10.1890/0012-9658(2003)084[1083:AROTII. Google Scholar

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##### References:
 [1] L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Mathematical Biosciences, 124 (1994), 83-105. doi: 10.1016/0025-5564(94)90025-6. Google Scholar [2] B. Bolker, M. Holyoak, V. Krivan, L. Rowe and O. Schmitz, Connecting theoretical and empirical studies of trait-mediated interactions, Ecology, 84 (2003), 1101-1114. doi: 10.1890/0012-9658(2003)084[1101:CTAESO]2.0.CO;2. Google Scholar [3] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, Reduction of discrete dynamical systems with applications to dynamics population models, Mathematical Modelling of Natural Phenomena, 8 (2013), 107-129. doi: 10.1051/mmnp/20138608. Google Scholar [4] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, Discrete models of disease and competition, Discrete Dynamics in Nature and Society, (2017), Art. ID 5310837, 13 pp. doi: 10.1155/2017/5310837. Google Scholar [5] R. Bravo de la Parra, M. Marvá, E. Sánchez and L. Sanz, A discrete predator-prey ecoepidemic model, Mathematical Modelling of Natural Phenomena, 12 (2017), 116-132. doi: 10.1051/mmnp/201712207. Google Scholar [6] Y. Chow and S. R.-J. Jang, Multiple attractors in a Leslie-Gower competition system with Allee effects, Journal of Difference Equations and Applications, 20 (2014), 169-187. doi: 10.1080/10236198.2013.815166. Google Scholar [7] J. M. Cushing, R. F. Costantino, B. Dennis, R. Desharnais and S. M. Henson, Chaos in Ecology: Experimental Nonlinear Dynamics, Theoretical Ecology Series, Vol. 1. Academic Press (Elsevier Science), New York, 2003.Google Scholar [8] J. M. Cushing, S. M. Henson and C. C. Blackburn, Multiple mixed-type attractors in a competition model, Journal of Biological Dynamics, 1 (2007), 347-362. doi: 10.1080/17513750701610010. Google Scholar [9] J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, Journal of Difference Equations and Applications, 10 (2004), 1139-1151. doi: 10.1080/10236190410001652739. Google Scholar [10] J. L. Edmunds, A Study of a Stage-Structured Model of Two Competing Species, Thesis (Ph.D.)-The University of Arizona, 2001, 64 pp. Available from: https://repository.arizona.edu/handle/10150/289978 Google Scholar [11] J. L. Edmunds, Multiple attractors in a discrete competition model, Theoretical Population Biology, 72 (2007), 379-388. doi: 10.1016/j.tpb.2007.07.004. Google Scholar [12] J. Edmunds, J. M. Cushing, R. F. Costantino, S. M. Henson, B. Dennis and R. A. Desharnais, Park's tribolium competition experiments: A non-equilibrium species coexistence hypothesis, Journal of Animal Ecology, 72 (2003), 703-712. Google Scholar [13] M. J. Hatcher, J. T. A. Dick and A. M. Dunn, How parasites affect interactions between competitors and predators, Ecology Letters, 9 (2006), 1253-1271. doi: 10.1111/j.1461-0248.2006.00964.x. Google Scholar [14] M. J. Hatcher and A. M. Dunn, Parasites in Ecological Communities: From Interactions to Ecosystems, Cambridge University Press, Cambridge, 2011. Google Scholar [15] P. Klepac and H. Caswell, The stage-structured epidemic: Linking disease and demography with a multi-state matrix approach model, Theoretical Ecology, 4 (2011), 301-319. doi: 10.1007/s12080-010-0079-8. Google Scholar [16] P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika, 45 (1958), 316-330. doi: 10.1093/biomet/45.3-4.316. Google Scholar [17] P. H. Leslie, T. Park and D. B. Mertz, The effect of varying the initial numbers on the outcome of competition between two tribolium species, Journal of Animal Ecology, 37 (1968), 9-23. doi: 10.2307/2708. Google Scholar [18] M. Marvá and R. Bravo de la Parra, Coexistence and superior competitor exclusion in the Leslie-Gower competition model with fast dispersal, Ecological Modelling, 306 (2015), 247-256. doi: 10.1016/j.ecolmodel.2014.10.039. Google Scholar [19] T. Park and M. Burton Frank, The fecundity and development of the flour beetles, tribolium confusum and tribolium castaneum, at three constant temperatures, Ecology, 29 (1948), 368-374. doi: 10.2307/1930996. Google Scholar [20] L. Sanz, R. Bravo de la Parra and E. Sánchez, Approximate reduction of non-linear discrete models with two time scales, Journal of Difference Equations and Applications, 14 (2008), 607-627. doi: 10.1080/10236190701709036. Google Scholar [21] H. L. Smith, Planar competitive and cooperative difference equations, Journal of Difference Equations and Applications, 3 (1998), 335-357. doi: 10.1080/10236199708808108. Google Scholar [22] E. E. Werner and S. D. Peacor, A review of trait-mediated indirect interactions in ecological communities, Ecology, 84 (2003), 1083-1100. doi: 10.1890/0012-9658(2003)084[1083:AROTII. Google Scholar
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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