### 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.

Original language | English |
---|---|

Pages (from-to) | 133-160 |

Number of pages | 28 |

Journal | Journal of Mathematical Biology |

Volume | 50 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2005 |

### Fingerprint

### Keywords

- Ecological niche
- Evolution of seed-size
- Limiting similarity
- Lotka-Volterra competition model
- Physiologically structured populations
- Regulated coexistence
- Structural stability

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*50*(2), 133-160. https://doi.org/10.1007/s00285-004-0283-5

**On the impossibility of coexistence of infinitely many strategies.** / Gyllenberg, Mats; Meszéna, G.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 50, no. 2, pp. 133-160. https://doi.org/10.1007/s00285-004-0283-5

}

TY - JOUR

T1 - On the impossibility of coexistence of infinitely many strategies

AU - Gyllenberg, Mats

AU - Meszéna, G.

PY - 2005/2

Y1 - 2005/2

N2 - 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.

AB - 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.

KW - Ecological niche

KW - Evolution of seed-size

KW - Limiting similarity

KW - Lotka-Volterra competition model

KW - Physiologically structured populations

KW - Regulated coexistence

KW - Structural stability

UR - http://www.scopus.com/inward/record.url?scp=13844271468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=13844271468&partnerID=8YFLogxK

U2 - 10.1007/s00285-004-0283-5

DO - 10.1007/s00285-004-0283-5

M3 - Article

C2 - 15614555

AN - SCOPUS:13844271468

VL - 50

SP - 133

EP - 160

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 2

ER -