Biological models, monotonicity methods, and solving a discrete reaction-diffusion equation

where $n \geq 3$ , $A$ is an $(n - 1) \times (n - 1)$ matrix such that $- A$ is monotone, $ϕ : R^{n - 1} \to R^{n - 1}$ and $f : R^{n - 1} \to R^{n - 1}$ are smooth functions, and $λ$ is a positive real constant. Of particular interest is the case where $A$ is the discrete Laplacian and $f$ is the vector-valued logistic function. The function $ϕ (u)$ will encode boundary conditions. Our primary goal is to establish the existence of nonnegative solutions for several interesting choices of $ϕ$ . For each choice we use monotonicity methods to find nonnegative solutions for appropriate ranges of $λ$ .

Keywords

discrete nonlinear boundary value problem, reaction-diffusion equation, population model, sub- and supersolutions, density-dependent boundary conditions

Mathematical Subject Classification

Primary: 39A27

Secondary: 39A12, 39A60

Milestones

Received: 28 June 2022

Revised: 14 January 2023

Accepted: 16 January 2023

Published: 15 March 2024

Communicated by Suzanne Lenhart

Authors

Carson Rodriguez
Department of Mathematics Wake Forest University Winston-Salem, NC United States

Stephen B. Robinson
Department of Mathematics Wake Forest University Winston-Salem, NC United States

Vol. 17, No. 1, 2024

Carson Rodriguez and Stephen B. Robinson

Abstract

Keywords

Mathematical Subject Classification

Milestones

Authors