Global existence for Lotka-Volterra models of predation and competition with diffusion and predator/competitor taxis in the spatial dimension n = 2,3

We study a prey-predator system and the competition system, both with Lotka-Volterra reaction terms, assuming, in addition to the diffusive movement of both species, an avoidance strategy of one of them modeled as repulsive taxis. For the predator-prey case, also called predator-taxis, the global existence of classical solutions is proven for the 2D case, assuming prey density dependent velocity suppression whose strength depends on some parameter σ - an assumption unnecessary in the prey-taxis case, whose role is confirmed by numerical simulations presented in the paper. This assumption is also useful in proving global classical solutions to the 3D competition model with competitor-taxis and the velocity suppression; it also serves as a regularization of the original model, which allows us to prove in the limit, $\sigma \to 0$, the existence of weak distributional solutions to the 3D competition model. The velocity suppression turns out to be unnecessary assumption to prevent blow-up of solutions to the competition model with competitor-taxis in 2D case. Finally, we emphasize that our numerical simulations starting from appropriately selected initial conditions, in addition to their illustrative function, indicate directions for further research.