Wave solutions of nonlocal delayed reaction -diffusion equations

Appearance of waveforms in population dynamics is a key element in studies of single and interacting species. The present work considers an age-structured model for single species that is in the form of a nonlocal delayed Reaction-Diffusion (RD) equation. The local and global stability of steady states are investigated as an intrinsic part of the wave studies. This is carried out through standard techniques such as linearization, Liapunov functionals and the method of characteristics. The present work differs from a large number of recent wave studies in two important respects. First, in contrast to several studies focused on the existence of the wave solutions, the emphasis is on the development and implementation of techniques for construction of the wave solutions. Secondly, it is not limited merely to the traveling wavefronts of the model but instead explores traveling and stationary wave solutions in the form of fronts and pulses. Considering wave solutions in such a broad context can reveal underlying physical and biological mechanisms that play crucial roles in dynamics of single species populations. Here, employing specific birth functions in the model, stationary wavefronts and wave pulses are obtained through an energy function method. By means of a number of techniques such as boundary layer and asymptotic expansion, the traveling wave solutions of the model are approximated. Although the age-structured model takes into account various key elements such as nonlocality and delay, it considers the unbounded one-dimensional domain. The present work also develops the model with respect to two-dimensional spatial domains. This enables further comparison between the outcomes of the model and those of laboratory experiments.