Demography

Ecology

Biology

Boundary value problem

Applied mathematics

Dirichlet boundary condition

Statistical physics

Type (biology)

Thermodynamics

Physics

Diffusion

Mathematics

Smoothness

Population

Sociology

Mathematical analysis

Semigroup

Reaction–diffusion system

Uniqueness

Dirichlet distribution

Compact space

Journal of Differential Equations

This paper is concerned with analysis of the global behavior for a type of reaction-diffusion system incorporating the pathogens' incubation period, which describes the infection of bacteria in a population. By using the operator semigroup theory and dynamical system approaches, the global dynamics with respect to the homogeneous Dirichlet problem are investigated. Firstly, the positivity, boundedness, compactness, and smoothness of the semiflow are obtained. Then, we study the topology on the ω-limit set and the dynamical behaviors, including permanence, existence, uniqueness and attractiveness of nontrivial steady-state solutions. Finally, our main results are employed to characterize threshold dynamics for two special models with Rick type and Mackey-Glass type infectivity nonlinearities respectively.

Global behavior of a reaction-diffusion model with time delay and Dirichlet condition