Digital Object Identifier (DOI): 10.1007/s00285-004-0283-5 J. Math. Biol. 50, 133–160 (2005) Mathematical Biology Mats Gyllenberg · G´ eza Mesz´ ena On the impossibility of coexistence of infinitely many strategies Received: 15 April 2002 / Revised version: 19 March 2004 / Published online: 20 December 2004 – c  Springer-Verlag 2004 Abstract. We investigate the possibility of coexistence of pure, inherited strategies belong- ing 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 convolu- tion 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. 1. Introduction Competitive exclusion and limiting similarity are classical, but, still, controver- sial concepts of ecology. In the formulation of MacArthur and Levins [26], the principle of competitive exclusion states that the number of coexisting strategies cannot be larger than the number of limiting resources. Later Levin [24] extended the idea beyond the case of resource competition by introducing the notion of lim- iting factors. In the terminology of Diekmann et al. [10,11] limiting factors are environmental interaction variables through which the self-regulating feedback of the ecosystem operates. Using the concept of environmental interaction variable Diekmann et al. [11] reformulated the principle of competitive exclusion for a large class of structured population models as follows: The dimension of the M. Gyllenberg: Department of Mathematics and Statistics, University of Helsinki, 20014 Helsinki, Finland. e-mail: mats.gyllenberg@helsinki.fi G. Mesz´ ena: Department of Biological Physics, E ¨ otv¨ os University, P´ azm´ any P´ eter s´ et´ any 1A, 1117 Budapest, Hungary and Collegium Budapest, Institute for Advanced Studies, Szenth´ ar- oms´ ag t´ er 2, 1014 Budapest, Hungary. e-mail: geza.meszena@elte.hu Mathamatics Subject Classification (2000): 92D40, 92D15 Key words or phrases: Limiting similarity – Ecological niche – Regulated coexistence – Lotka-Volterra competition model – Physiologically structured populations – Evolution of seed-size – Structural stability.