On the complete classification of nullcline stable competitive three-dimensional Gompertz models

Abstract

We investigate a competitive model of three species, each of which, in isolation, admits Gompertz growth. A well-known theorem by M.W. Hirsch guarantees the existence of carrying simplex. Based on this, we compare three dimensional competitive Gompertz models with three dimensional competitive Lotka–Volterra models, and we find that each Gompertz model has a corresponding Lotka–Volterra model with identical nullclines. We then present the complete classification of nullcline stable models and arrive at a total of 33 stable nullcline classes, and show that in 27 of these classes all the compact limit sets are equilibria. Despite the common results, we go on to show that the behavior on the carrying simplex of Gompertz systems is subtly different from that on Lotka–Volterra systems. The number of limit cycles is finite in 5 of the remaining 6 classes, and that only the classes 26 and 27 admit Hopf bifurcations and the other 4 do not. The class 27, which has a heteroclinic cycle, contains a system having May–Leonard phenomenon: the existence of nonperiodic oscillation, and still admitting at least two limit cycles. The numerical simulation reveals that there are some systems in class 28 with two limit cycles.