Analysis of spatiotemporal patterns in a single species reaction–diffusion model with spatiotemporal delay

• The conditions of both spot and stripe patterns are given.
• The multiple scale method is used to obtain the amplitude equations.
• The patterns are plotted to show the physical implication.

Employing the theories of Turing bifurcation in the partial differential equations, we investigate the dynamical behavior of a single species reaction–diffusion model with spatiotemporal delay. The linear stability and the conditions for the occurrence of Turing bifurcation in this model are obtained. Moreover, the amplitude equations which represent different spatiotemporal patterns are also obtained near the Turing bifurcation point by using multiple scale method. In Turing space, it is found that the spatiotemporal distributions of the density of this researched species have spots pattern and stripes pattern. Finally, some numerical simulations corresponding to the different spatiotemporal patterns are given to verify our theoretical analysis.