On the impossibility of  coexistence of infinitely many strategies

Mats Gyllenberg 
Department of Mathematics, University of Turku
FIN-20014 Turku, Finland 
matsgyl@utu.fi 

Geza Meszena 
Department of Biological Physics, Eotvos University 
Pazmany Peter setany 1A, H-1117 Budapest, Hungary
Collegium Budapest, Institute for Advanced Studies
Szentharomsag ter 2, H-1014 Budapest, Hungary 
geza.meszena@elte.hu



Journal of Mathematical Biology, in press


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.