The evolutionary dynamics of a population model with a strong Allee effect

An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).