Turing Patterns in a Competition Lotka-Volterra Model with Nonlinear-Diffusion

Competition Lotka-Volterra models is studied, which is an extension of linear diffusion model in 25. In this paper, we study pattern formation solutions for this ecological system with a nonlinear diffusion part which represents both self-diffusion and cross-diffusion. The role of crucial cross diffusion β₂ in developing Patterns has been shown. Turing instability conditions have been derived and illustrated numerically. Furthermore, the Turing bifurcation Threshold region is verified. Labyrinthine patterns under cross-diffusion β₂ < 0 which may used for modeling the interactions in crystals has been found. Furthermore, spot patterns via self-diffusion α₁ < 0 have been shown. Numerical solutions through finite element method is applied to show those types of pattern formations for our model.