tiring instability and travelling waves in diffusive plankton models with delayed nutrient recycling

In this paper we propose a reaction-diffusion system with two distributed delays to stimulate the growth of plankton communities in the lakes/oceans in which the plankton feeds on a limiting nutrient supplied at a constant rate. The limiting nutrient is partially recycled after the death of the organisms and a distributed delay is used to model nutrient recycling. The second delay is involved in the growth response of the plankton to nutrient uptake. We first show that there are oscillations (Hopf bifurcations) in the delay model induced by the second delay. Then we study Turing (diffusion-driven) instability of the reaction-diffusion system with delay. Finally, it is shown that if the delay model has a stable periodic solution, then the corresponding reaction-diffusion model with delay has a family of travelling waves.