Local consideration of polymorphisms for populations coexisting in stable ecosystems

The stability problem for multiallelic genetic polymorphisms for n populations coexisting in stable ecosystems is considered. Taking into account only density-dependent interactions a generalization of Fisher's theorem is obtained. Specifically, the average fitness of a population must be locally maximized subject to the constraint that the equilibrium population sizes are fixed if the polymorphism is stable. Further, the quasi-equilibrium population sizes Ni corresponding to fixing the genetic structure of all populations in the ecosystem at various values have extrema at the equilibrium point. Such an equilibrium can be a maximum, minimum or saddle point depending upon the type of ecosystem under consideration. A simple test separating these cases on the basis of the so-called ecosystem matrix is suggested. The general equilibrium problem is reformulated as a maximization problem under some restrictions. Conditions under which the maximized function can be expressed as sigma ni=1 Ni are formulated.