Doubly nonlocal reaction–diffusion equations and the emergence of species

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.

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

“…Among the most recent developments of the Bellomo theories we cite: (i) the theory of thermostatted active particles, which allows to impose physically backgrounded constraints to the activity of individuals, developed by Bianca and Menale [21][22][23]; (ii) the stochastic evolutionary theory of tumor adaptation developed by Clairambault, Delitala, Lorenzi and coworkers [24][25][26][27][28][29]. Finally, it is worth mentioning the mathematical modeling of Darwinian species emergence by Volpert and colleagues that represent the evolution of active particles uniquely subject to a Brownian force and logistic non-local growth [30][31][32].…”

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

“…Instead he assumed that consumption of resources at a spatial location does not solely depend on the local population density but also on the population's weighted average at that point. For some variants of nonlocal Fisher equation, it was shown that the spatially homogenous solution becomes unstable when nonlocal competition is taken into account (Genieys et al 2006;Aydogmus 2015;Fuentes et al 2003Fuentes et al , 2004Banerjee et al 2016). The effect of nonlocal interactions for integro-difference equations was also studied by Aydogmus et al (2017).…”

Prey-predator model with nonlocal and global consumption in the prey dynamics

“…Movement of the individuals to the nearby location occurs in a faster time scale compared to the movement from one location to the other one. This modifies the modeling approach and gives rise to an integro-differential equation describing the nonlocal consumption of resources [8,10,27]. This type of models manifest stationary patchy distribution, and periodic traveling waves in addition to the steady traveling wave solutions.…”

“…The models with nonlocal consumption of resources present complex dynamics for the single species models [4,3,18,19,26]. A limiting case of nonlocal consumption of resources, where the individual can consume resource not only from the neighboring region, but from any point of the entire considered domain, is termed to be global consumption [10,27]. Models with global consumption of resources, under proper parametric conditions and appropriate choice of initial conditions, give rise to pulse solutions.…”