A singular reaction–diffusion system modelling prey–predator interactions: Invasion and co-extinction waves

We consider a singular reaction–diffusion system arising in modelling prey–predator interactions in a fragile environment. Since the underlying ODEs system exhibits a complex dynamics including possible finite time quenching, one first provides a suitable notion of global travelling wave weak solution. Then our study focusses on the existence of travelling waves solutions for predator invasion in such environments. We devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co-extinction tail for both species. Under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere, such a property arising for every admissible wave speeds.