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This paper deals with a diffusive logistic equation on one dimensional isotropically growing domain. The model equation on growing domains is first presented, and the comparison principle is then proved. The asymptotic behavior of temporal solutions to the reaction-diffusion problem is given by constructing upper and lower solutions. Our result shows that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast. Numerical simulations are also presented to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems. [ABSTRACT FROM AUTHOR]