Analysis of patterns in an additional food-provided predator-prey reaction diffusion model using amplitude equations

A two-species predator-prey reaction diffusion model where additional food was provided to the predator has been considered in the presence of both self- and cross-diffusion terms in a two-dimensional space. Importance of self- and cross-diffusion terms are discussed ecologically. Linear stability analysis has been done to determine the Turing instability region of the parameter space. Impacts of the cross-diffusion term on the stability behaviour are discussed. Using weak nonlinear analysis, the amplitude equations has been formulated to predict the stability of hexagonal, stripes and their mixture patterns near the Turing thresholds analytically. Finite difference method has been used for numerical simulations of the model under no-flux boundary conditions. Numerical simulation results are in good agreement with the theoretical predictions of patterns near the Turing threshold using amplitude equations. Additional food can play a significant role in pattern forming instabilities because near the Turing thresholds small fluctuation of additional food parameter can produce a wide variety of Turing patterns.