The discrete dynamics of symmetric competition in the plane (Q1118550)

This paper is a detailed study of the map \(F:R^ 2_+\mapsto R^ 2_+\) defined by \(F(x,y)=(f(x,y),\quad f(y,x))\) where \(f(x,y)=\exp (r- rx-sy),\) (r,s non-negative parameters). The iterates of this map represent the dynamics of competition between two species with discrete generation time. The map is symmetric: the intrinsic growth ``exponents'' r and the exponents representing the interspecific competition s are the same at the two species. Therefore the model is non-generic; nevertheless, its interesting and rich dynamics may help to understand what happens in such a situation. Fixed points, their stability, invariant attractive sets, period-two cycles, their stability, Hopf bifurcations from stable period-two cycles treated as fixed points of \(F^ 2\), and a possible cascade of period doubling bifurcations occur and are studied in an elegant exposition.