Delay-induced patterns in a predator-prey model on complex networks with diffusion

Reaction-diffusion (RD) systems with time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems. For predator-prey model with modified Leslie-Gower and Holling-type III schemes governed by RD equations, instability induced by time delay can generate spiral waves. Considering that populations are usually organized as networks instead of being continuously distributed in space, it is essential to study the predator-prey model on complex networks. In this paper, we investigate instability induced by time delay for the corresponding network organized system and explore pattern formations on several different networks including deterministic networks and random networks. We firstly obtain instability condition via linear stability analysis and then the condition is applied to study pattern formations for the model in question. The simulation results show that wave patterns can be generated on different networks. However, wave patterns on random networks differ significantly from patterns on deterministic networks. Finally, we discuss the influences of network topology on wave patterns from the aspects of amplitude and period, and reveal the ecology significance implied by these results.