Patchy spread patterns in three-species bistable systems with facultative mutualism

A three-species population system under a facultative mutualistic relationship of one of the species is studied. The considered interactions are as follows: facultative between the first species and the second species, obligatory mutualism between the second species and the first one, and the third species is a predator of the first species. For this purpose, we extend the model proposed by Morozov et al., originally used to describe obligatory mutualism, to consider obligatory and facultative mutualism and prove that under adequately selected parameters this system produces a spatial patchy spread of populations or continuous wave fronts. Since the analytical treatment of a three-species model is often prohibitive, we first analyze the interaction between two mutualist species without diffusion and without the presence of the predator. Some parameters are fixed in the bistable regime of the mutualistic species to further consider the influence of the third species. The remaining parameters are then selected to produce patchy patterns under different mortality rates. Finally, the equations of the final three-species system are numerically solved to test the influence of different initial conditions in the formation of patchy populations. It is confirmed that the velocity and the profile of a traveling front are independent on the initial conditions. Our approach opens the way to study more general biological situations.