Invasions in a four-species fractured cyclic ecological system

We consider the invasion problem for a four-species cyclic ecological community. When the cyclic interspecies competition is stronger than the intraspecies competition (crowding), the system is dominated by two two-species alliances which are the competing entities. We assume that one of the alliances is fractured due to internal competition and predation. The invasion problem can then be reduced to a traveling wave problem and the two alliances will be equally matched under standstill conditions, i.e., when the speed of the traveling wave is zero. We determine the standstill condition and the role of fracturing on standstill in two regimes: (i) balanced competition, when the interspecies competition is comparable to the intraspecies competition, so that there is a significant region where the four species can live together and (ii) strong competition, where species from the two alliances cannot coexist except in a very narrow band. We employ a suitable coordinate transformation for the regime of balanced competition and a suitable linearization for the case of strong competition. In both cases we determine the role of fracturing on standstill conditions. We validate our results with numerical computations.