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

Carson Rodriguez and Stephen B. Robinson

Vol. 17 (2024), No. 1, 65–84

The problem of interest is a discrete reaction-diffusion equation motivated by models in population biology. We consider

Au + ϕ(u) + λf(u) = 0 for u Rn1,

where n 3, A is an (n1) × (n1) matrix such that A is monotone, ϕ : Rn1 Rn1 and f : Rn1 Rn1 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.

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
Received: 28 June 2022
Revised: 14 January 2023
Accepted: 16 January 2023
Published: 15 March 2024

Communicated by Suzanne Lenhart
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