Malay Banerjee, Vitali Vougalter, Vitaly Volpert
Doubly nonlocal reaction-diffusion equations and the emergence of species
(1381K, pdf)

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 indivuals that can breed only among themselves.