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28.08.2019

Evolutionary Substitution and Replacement in N-Species Lotka–Volterra Systems

Zeitschrift:
Dynamic Games and Applications
Autoren:
Ross Cressman, Miklós Koller, M. Barnabás Garay, József Garay
Wichtige Hinweise
Financial assistance from the Natural Sciences and Engineering Research Council (NSERC) of Canada (to R.C.) as well as from grants GINOP 2.3.2-15-2016-00057 (to J.G.) and NKFIH No. 115926 (to M.B.G.) of Hungary is gratefully acknowledged. This project has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 690817. We also thank the referee for helpful comments on the original submission.

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Abstract

The successful invasion of a multi-species resident system by mutants has received a great deal of attention in theoretical ecology but less is known about what happens after the successful invasion. Here, in the framework of Lotka–Volterra (LV) systems, we consider the general question where there is one resident phenotype in each species and the evolutionary outcome after invasion remains one phenotype in each species, but these include all the mutant phenotypes. In the first case, called evolutionary substitution, a mutant appears in only one species, the resident phenotype in this species dies out, and the mutant coexists with the original phenotypes of the other species. In the second case, called evolutionary replacement, a mutant appears in each species, all resident phenotypes die out, and the evolutionary outcome is coexistence among all the mutant phenotypes. For general LV systems, we show that dominance of the resident phenotype by the mutant (i.e. the mutant is always more fit) in each species where the mutant appears leads to evolutionary substitution/replacement. However, it is shown by example that, when dominance is weakened to only assuming the average fitness of the mutants is greater than the average for the resident phenotype, the residents may not die out. We also show evolutionary substitution occurs in two-species competitive LV systems when the initial invasion of the resident system (respectively, of the new coexistence system) is successful (respectively, unsuccessful). Moreover, if sequential evolutionary substitution occurs for either order that the two mutant phenotypes appear (called historically independent replacement), then it is shown evolutionary replacement occurs using a generalization of the dominance argument.

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