Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

Abstract

• Spatio-temporal prey-predator model with cross-diffusion is studied. • Effects of cross-diffusion terms are studied–analytically and numerically. • Weakly nonlinear analysis is used to find the amplitude equations. • Numerical simulations are used to see how cross-diffusion influence Turing patterns. • Complete scenario of pattern formation in model with cross-diffusion is provided. Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.