Phase transition and time emergence in a statistical model of species competition

Abstract

In this paper, we expand the application of Gibb’s canonical ensemble to ecological modeling. Hitherto, the ensemble has been used to model species that are competing for resources in an environment with a fixed carrying capacity. Within that context, a statistical time intrinsic to the ecology emerges, as an analogue of the inverse temperature in the usual thermodynamical applications. Now, we extend the model to include the population growth before the carrying capacity is reached. In the extended model, there is a first-order phase transition to the resource-limited case which governs the emergence of the statistical time. The new model is presented as an elementary demonstration of a rigorous method for non-perturbative modeling of a phase transition in a finite system, using entropy maximization that is subject to a weak inequality constraint. In this approach, the phase transition occurs as the constraint switches between slack and binding.