Lotka–Volterra models with fractional diffusion

Abstract

In this paper we study the Lotka-Volterra models with fractional Laplacian. For that, we study in detail the logistic problem and show that the sub-supersolution method works for the scalar problem and in case of systems as well. We apply this method to show existence and non-existence of positive solutions in terms of the system parameters.

Extension of the Lotka-Volterra competition model

“…In [2] by the method of upper and lower solutions and its associated monotone iterations, existence and non-existence results was derived for both the scalar problem and for systems with fractional diffusion. Also, in [7], the authors gave simple extensions of the Lotka-Volterra prey-Predator model.…”

On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

“…In recent years, nonlocal operators, and notably fractional ones, are a classical topic in harmonic analysis and operator theory, and they are becoming impressively popular because of their connection with many real‐world phenomena, such as, the thin obstacle problem, 13 ecology, 14–17 finance, 18 and anomalous diffusion 19 ; for more details, see Di Nezza et al 20 and references therein. Stinga and Volzone 21 considered that the diffusion of the concentration of the chemical is nonlocal and studied the steady states of the local–nonlocal type for the Keller–Segel system: $$$$ \left\{\begin{array}{cc}{D}_1\Delta u-\chi \nabla \cdotp \left(u\nabla \log v\right)=0\kern0.60em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{D}_2{\left(-\Delta \right)}^{1/2}v+ av- bu=0\kern0.30em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{\partial}_{\nu u}={\partial}_{\nu v}=0\kern0.30em & \mathrm{on}\kern0.5em \mathrm{\partial \Omega }.\end{array}\right.$$…”

“…In recent years, nonlocal operators, and notably fractional ones, are a classical topic in harmonic analysis and operator theory, and they are becoming impressively popular because of their connection with many real-world phenomena, such as, the thin obstacle problem, 13 ecology, [14][15][16][17] finance, 18 and anomalous diffusion 19 ; for more details, see Di Nezza et al 20 and references therein. Stinga and Volzone 21 considered that the diffusion of the concentration of the chemical is nonlocal and studied the steady states of the local-nonlocal type for the Keller-Segel system:…”