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

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 3333 stable nullcline classes, and show that in 2727 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 55 of the remaining 66 classes, and that only the classes 2626 and 2727 admit Hopf bifurcations and the other 4 do not. The class 2727, 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 2828 with two limit cycles.