Doubly nonlocal reaction-diffusion equation and the emergence of species

TL;DR

This paper studies a complex reaction-diffusion model with nonlocal interactions, analyzing stability, wave solutions, and implications for species emergence, linking mathematical results to biological speciation concepts.

Contribution

It introduces a doubly nonlocal reaction-diffusion equation and proves existence of traveling waves and stationary pulses, connecting mathematical findings to biological speciation theories.

Findings

Existence of traveling wave solutions for narrow kernels.

Periodic traveling waves observed in simulations.

Species emergence linked to reproductive isolation constraints.

Abstract

The paper is devoted to a reaction-diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of travelling waves is proved in the case of narrow kernels of the integrals. Periodic travelling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin's definition of species as groups of morphologically similar individuals is related to Mayr's definition as groups of individuals that can breed only among themselves.