Diffusion and discrete temporal models of the population growth of domestic cats in urban areas

The survival of the domestic cat (Felis catus) in various ecosystems has become increasingly relevant due to its impact on wildlife, public health, and society. In countries like Mexico, social factors such as abandonment have led to the feralization of the species and an unexpected increase in its population in urban areas. To design and implement effective population control methods, a thorough analysis of the species' population dynamics, along with the social factors influencing it, is necessary, this being the aim of the present paper. We propose a reaction-diffusion model to simulate the natural dispersal of the population within a bounded domain. After exploring the species' spreading ability, we construct a discrete dynamical system based on the biological characteristics of cats and their intraspecific and interspecific interactions, which we explain and study in detail. The model includes both fixed parameters, i.e. determined by the demographic data available, and stochastic parameters, that are adjusted according to some random distribution, to enhance the realism of the simulations. Our results indicate that the population reaches a non-trivial equilibrium in the case of the reaction-diffusion model and a periodic equilibrium in the discrete dynamical system, highlighting the need for control methods combined with social regulations to achieve sustainability in the system.