Competitive exclusion and coexistence in ann-species Ricker model

Abstract

We analyse a discrete-time Ricker competition model with n competing species and give sufficient conditions, which depend on the competition coefficients only, for one species to survive (not necessarily at an equilibrium) and to drive all the other species to extinction. Our results complement and extend similar existing results from the literature. For the model reduced to three species (n = 3), we also investigate various scenarios under which all species coexist, in the sense that each species is robustly uniformly persistent. We provide a few numerical simulations to illustrate that coexistence does not necessarily mean convergence to the interior equilibrium, and that the interior dynamics can be quite complex.

On the dynamics of multi-species Ricker models admitting a carrying simplex

“…In [49] Ruiz-Herrera provided an exclusion criterion for discrete-time competitive models of two or three species via carrying simplices. Jiang and Niu [32] deduced an index formula on the sum of the indices of all fixed points on Σ for the three-dimensional map T of type (1):…”

“…Such an invariant closed curve corresponds to either subharmonic or quasiperiodic solutions in continuous-time systems. In [34], Jiang et al studied the occurrence of heteroclinic cycles via carrying simplices for competitive maps (1) and provided their stability criteria.…”

“…Suppose that T ∈ CRC(2) is stable relative to ∂Σ and possesses a positive fixed point p. Then det U = 0, and the positive fixed point p is unique. Moreover, p attracts on Σ if and only if det U > 0 (Lemma 4.1(c)) and repels on Σ if and only if det U < 0 (Lemma 4.1(d)).Biologically, det U > 0 means that both species can invade, while det U < 0 means that none of them can (Remark 4.1(i)).For other related results on the dynamics of the 2D Ricker map(17), see, for example,[1,6,7,31,40,50,51].…”

“…The trajectory emanating from x0 = (0.8, 0.8, 1) for the map T ∈ CRC(3) with the parameters U 26 given in Example 4.1 and ν1 = 0.04, ν2 =1 9 , ν3 = 1 3 tends to an attracting closed invariant curve, i = 1, 2, 3. p. 4. For i = 29, we let ν 1 = 7/17, ν 2 = s, ν 3 = 1/43, and ν = (ν 1 , ν 2 , ν 3 ).…”

Dynamics of the degenerate 2D Ricker equation

“…15 The case where the carrying capacities are such that the map is monotone was proved in Smith. 16 In spite of the fact that the two-dimensional case has not been completely solved, there are results for higher dimensions for some parameter values; see Ackleh and Salceanu, 17 Balreira, Elaydi, and Luis, 18 and Baigent and Hou. 19 It is natural to adopt the view that the carrying capacities p, q are fixed and examine how the dynamical behavior changes as the coupling parameters are tuned.…”