Efficient numerical methods for spatially extended population and epidemic models with time delay

Reaction-diffusion models with time delay have been widely applied in population biology as well as epidemiology. This type of models can possibly exhibit complex dynamical behaviors such as traveling wave, self-organized spatial pattern, or chaos. Numerical methods play an essential role in the study of these dynamical behaviors. This paper concerns the finite element approximation for reaction-diffusion models with time delay. Two fully discrete schemes and corresponding a priori error estimates are derived. Generally, the research on evolution of population and epidemic needs to survey long-time dynamical behaviors of these models, so that it is important to improve the speed of numerical simulation. To this end, interpolation technique is used in our schemes to avoid numerical integration of reaction term. An outstanding advantage of using interpolation of reaction term is that it improves the operation speed greatly, meanwhile does not reduce convergence order. Applications are given to some model problems arising from population biology and epidemiology. From these simulations some interesting phenomena can be found and we try to explain them in biological significance.