Stationary, Oscillatory, Spatio-Temporal Patterns and Existence of Global Solutions in Reaction-Diffusion Models of Three Species

The goal of my Ph.D. research is to analyze three species models in order to describe the behavior of an ecological community. In particular, two reaction-diffusion systems describing different local interactions between three species have been considered to obtain species coexistence, diversity, and distribution patterns. The first analyzed model describes intraguild predation: there are an IG-predator species, an IG-prey species, and a common resource species, which is shared by both of them. The IGP interaction is of Lotka-Volterra type, coupled with nonlinear diffusion, since we assume that the IG-prey moves towards lower density areas of the IG-predator. In this model, the extinction of species has been surveyed. Performing the linear stability analysis in the neighborhood of the coexistence point, the conditions for the occurrence of Hopf instability have been established. Cross-diffusion is able to induce Turing instability for this system, which would not admit this bifurcation in presence of only classical diffusion terms. Moreover, the effect of each parameter on Turing and Turing-Hopf instability has been detected. Numerical solutions of the system have been computed using spectral method, showing the rich dynamics of the model, including the Turing pattern, time oscillation pattern, Turing-Hopf pattern, and chaotic behavior. The weakly nonlinear analysis also has been employed to predict the amplitude of patterned solutions have been compared with numerical spectral solutions of the reaction-diffusion system. Furthermore, we have used multiscale methods to determine normal form of the model around Turing-Hopf codimension-2 points. Finally, by utilizing the fixed point argument and energy estimate, the existence of the global solution to the system has been established, assuming some conditions on initial data. The second three species model describes the dynamics of two predators competing with each other to feed on the same prey. The functional response of predators is the Holling type. This local dynamics has been coupled with linear cross-diffusion terms taking into account the movement of each species towards lower-density areas of the other species. We have applied linear analysis of the system with and without diffusion to obtain the necessary conditions of stability and the occurrence of Hopf and Turing instability. In particular, weakly nonlinear analysis, Turing regions, and maximum growth rate have been investigated. Using a numerical finite elements method, Turing patterns have also been displayed and compared with WNL solutions. Finally, to prove the existence of global in time of the solutions, a rectangular invariant method has been presented for a particular case.