A survey on constructing Lyapunov functions for reaction-diffusion systems with delay and their application in biology

Motivated by some biological and ecological problems given by reaction-diffusion systems with delays and boundary conditions of Neumann type and knowing their associated Lyapunov functions for delay ordinary differential equations, we consider a method for determining their Lyapunov functions to establish the local/global stability. The method is essentially based on adding integral terms to the corresponding Lyapunov function for ordinary differential equations. The new approach is not general but it is applicable in a wide variety of delays reaction-diffusion models with one discrete delay or more, distributed delay, and a combination of both of them. To illustrate our results, we present the method application to a reaction-diffusion epidemiological model with time delay (latency period) and indirect transmission effect.