Effects of cyclic allele dominance rules and spatial structure on the dynamics of cyclic competition models

Kazunori Sato; Kazunori Sato

doi:10.3934/mbe.2020076

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1479-1494. doi: 10.3934/mbe.2020076

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Effects of cyclic allele dominance rules and spatial structure on the dynamics of cyclic competition models

Kazunori Sato ^,

Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu 432-8561, Japan

Received: 30 June 2019 Accepted: 03 November 2019 Published: 28 November 2019

Barreto et al. (2017) showed that the genotypic cyclic competition model with three phenotypes appearing as three morphs of male lizards' throats had the same equilibrium but a wider stability region as the corresponding phenotypic model. In this paper we re-examine stability conditions under the symmetric choice of parameters for the phenotypic model so we can set the same internal equilibrium densities for all three phenotypes. In this setting we compare the stability regions of cyclic allele dominance rule. Next we consider the dynamics on a two-dimensional square lattice space and then show the effect of this spatial structure on the stability of phenotypic model. We obtain the following results: (ⅰ) Cyclic allele dominance rule in a genotypic model gives a wider stable region of internal equilibrium than the allele dominance rule observed in lizards; and (ⅱ) spatial structure drastically changes dynamical behavior, especially when all three phenotypes coexist in almost all the parameter spaces when both competition and dispersal occur locally.
- rock-paper-scissors game,
- cyclic competition,
- phenotypic model,
- genotypic model,
- spatial structure
Citation: Kazunori Sato. Effects of cyclic allele dominance rules and spatial structure on the dynamics of cyclic competition models[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1479-1494. doi: 10.3934/mbe.2020076

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Abstract

Barreto et al. (2017) showed that the genotypic cyclic competition model with three phenotypes appearing as three morphs of male lizards' throats had the same equilibrium but a wider stability region as the corresponding phenotypic model. In this paper we re-examine stability conditions under the symmetric choice of parameters for the phenotypic model so we can set the same internal equilibrium densities for all three phenotypes. In this setting we compare the stability regions of cyclic allele dominance rule. Next we consider the dynamics on a two-dimensional square lattice space and then show the effect of this spatial structure on the stability of phenotypic model. We obtain the following results: (ⅰ) Cyclic allele dominance rule in a genotypic model gives a wider stable region of internal equilibrium than the allele dominance rule observed in lizards; and (ⅱ) spatial structure drastically changes dynamical behavior, especially when all three phenotypes coexist in almost all the parameter spaces when both competition and dispersal occur locally.

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