(Non)local logistic equations with Neumann conditions

Abstract

We consider here a problem of population dynamics modeled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss situations that are more favorable for the survival of the species, in terms of the first positive eigenvalue. Quite surprisingly, the eigenvalue analysis for the one-dimensional case is structurally different from the higher-dimensional setting, and it sensibly depends on the nonlocal character of the dispersal.

The mathematical framework of this problem takes into consideration the equation $-\alpha \Delta u+\beta (-\Delta {)}^{s}u=(m-\mu u)u+\tau J\star u$ in $\Omega$, where $m$ can change sign. This equation is endowed with a set of Neumann conditions that combines the classical normal derivative prescription and the nonlocal condition introduced in Dipierro, Ros-Oton, and Valdinoci [Rev. Mat. Iberoam. 33 (2017), 377–416]. We will establish the existence of a minimal solution for this problem and provide a thorough discussion on whether it is possible to obtain nontrivial solutions (corresponding to the survival of the population).

The investigation will rely on a quantitative analysis of the first eigenvalue of the associated problem and on precise asymptotics for large lower and upper bounds of the resource. In this, we also analyze the role played by the optimization strategy in the distribution of the resources, showing concrete examples that are unfavorable for survival, in spite of the large resources that are available in the environment.