Two predators feeding on two prey species: a result on permanence

For biological populations the precise asymptotic behavior of the corresponding dynamic system is probably less important than the question of extinction and survival of species. An ecological differential equation is called permanent if there exists some level k greater than 0 such that if the number xi(0) of species i at time 0 is positive for i = 1,2, ..., n then xi(t) greater than k for all sufficiently large times t Characterizations for permanence in a four-species prey-predator system modeled by the Lotka-Volterra equation are presented. The method used is based on a combination of two well-known approaches to dealing with permanence. An interesting feature is the occurrence of heteroclinic cycles.