Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Highlights • It is proposed a model based on non-local operators capable of reproduce several of the equations traditionally used in research on population dynamics, as well as having additional features, useful for describing the phenomenon of pattern formation. • The combination of two of the parameters present in the model, namely, α and β, determines the existence of a phase space, characterized by regions with and without patterns. • The patterns formed will have M peaks and M valleys, with M ≥ 2. • There are (α, β) combinations which lead to the formation of degenerate states, where the system seems not to forget its initial conditions. • The model allows the study of real systems, such is the case of bacterial populations subjected to inhomogeneous growth conditions. • The parameter β represents a macroscopic measure of changes that occur at an individual level in the concentrations of proteins in bacteria, as a product of significant variations in the environment.
Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.