Some remarks on a singular reaction-diffusion system arising in predator prey modeling

This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.