Seasonal spatial distributions of amphibians and Gibbs measures: a new approach inspired by Hamiltonian mechanics

Abstract We introduce a model that describes the population dynamics of species in ecology. Our model employs a Hamiltonian framework from physics, where the balance between species dispersal and environmental attraction is governed by two competing energy terms: a kinetic-like term driving homogeneous spatial spread, and a potential-like term generating attraction to key landscape features such as rivers, with a coupling constant modulating their seasonal balance. An innovative aspect that we address in this paper is a dynamical picture in ecology in which we model the different distributions of individuals throughout the year (four seasons). Our main contribution is focused on applied mathematics and statistics. Our model presents a new perspective on how Markov random fields (Gibbs measures) can be used to describe spatial statistics in the context of ecology. We present a case study of our model using Plectrohyla sagorum , a vulnerable species of amphibians. Our model indicates that Plectrohyla sagorum concentrates near the river in the dry season and disperses as precipitation increases. We show our results using heat maps describing this seasonal variation, showing peak density on the river in winter.