A Reaction-Diffusion Model with Spatially Inhomogeneous Delays

• Published in: Journal of dynamics and differential equations Vol. 36; no. 4; pp. 3743 - 3758
• Main Authors: Lou, Yijun, Wang, Feng-Bin
• Format: Journal Article
• Language: English
• Published: New York Springer US 28-03-2023; Springer Nature B.V
• Subjects: Applications of Mathematics; Diffusion rate; Heterogeneity; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Parameters; Partial Differential Equations; Population growth; Reaction-diffusion equations; Reaction-diffusion model; 92D25; 37N25; Stage-structured model; Population dynamics; 35K57; Spatially inhomogeneous delay
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-023-10254-6

Motivated by population growth in a heterogeneous environment, this manuscript builds a reaction-diffusion model with spatially dependent parameters. In particular, a term for spatially uneven maturation durations is included in the model, which puts the current investigation among the very few studies on reaction-diffusion systems with spatially dependent delays. Rigorous analysis is performed, including the well-posedness of the model, the basic reproduction ratio formulation and long-term behavior of solutions. Under mild assumptions on model parameters, extinction of the species is predicted when the basic reproduction ratio is less than one. When the birth rate is an increasing function and the basic reproduction ratio is greater than one, uniqueness and global attractivity of a positive equilibrium can be established with the help of a novel functional phase space. Permanence of the species is shown when the birth function is in a unimodal form and the basic reproduction ratio is greater than one. The synthesized approach proposed here is applicable to broader contexts of studies on the impact of spatial heterogeneity on population dynamics, in particular, when the delayed feedbacks are involved and the response time is spatially varying.