Abstract.
We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the “interaction operator” which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of “limiting similarity” of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.
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Mathamatics Subject Classification (2000): 92D40, 92D15
Acknowledgement We thank Odo Diekmann, Peter Abrams and an anonymous referee for careful reading of the manuscript and many valuable suggestions that led to a considerable improvement of the paper. We have also benefited from discussions with Stefan Geritz, Patsy Haccou, Hans Metz, Yoh Iwasa, and Akira Sasaki. This work was financially supported by the Academy of Finland and by the grants OTKA T033097, FKFP 0187/1999 and NWO-OTKA N34028.
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Gyllenberg, M., Meszéna, G. On the impossibility of coexistence of infinitely many strategies. J. Math. Biol. 50, 133–160 (2005). https://doi.org/10.1007/s00285-004-0283-5
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DOI: https://doi.org/10.1007/s00285-004-0283-5