Global dynamics and traveling wave solutions for a three-species model

In this work, we investigate the system of a three-species ecological model involving one predator-prey subsystem coupling with a generalist predator with negative effect on the prey. Without diffusive terms, all global dynamics of its corresponding reaction equations are proved analytically for all classified parameters. With diffusive terms, the transitions of different spatial homogeneous solutions, the existence of traveling wave solutions, are investigated by a higher dimensional shooting method, the Wazewski method. These mathematical results, under mild conditions, imply that a generalist predator can stabilize a predator-prey system even with negative effects of coupling. Finally, some biological implications are given and the interesting numerical simulations are performed.