Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Abstract

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of 33 stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes 1 − 25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26 − 31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29, 31, 33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.Received xxxx 20xx; revised xxxx 20xx.

Heteroclinic Cycles of A Symmetric May-Leonard Competition Model with Seasonal Succession

“…To be more explicit, when ϕ varies in [0, 1], system (4.1) enjoys different dynamic features: the orbit emanating from initial value x 0 = (0.5, 0.6, 0.4) for the Poincaré map S tends to the heteroclinic cycle Γ(see Figure 4) → invariant closed curves (see Figure 6 and Figure 7) → the positive fixed point (see Figure 8-12) → the interior fixed point of coordinate plane {x 2 = 0} (see Figure 13) → the interior fixed point of x 1 -axis (see Figure 14) → the interior fixed point of the coordinate plane {x 3 = 0} (see Figure 15) → the interior fixed point of x 2 -axis (see Figure 16) → the trivial fixed point O (see Figure 17). From these numerical examples, one can see that the dynamic features and patterns generated by system (4.1) are much richer , which are different from concrete discrete-time competitive maps analyzed in [6,7,13,14].…”

“…Hence, it suffices to investigate the dynamics of the discrete-time system {S n } n≥0 . However, compared to the concrete discrete-time competitive maps discussed in [4,6,7,13,14,16], there is no explicit expression of the Poincaré map S for system (1.2). This makes the research much more difficult and complicated on the dynamics of system (1.2).…”

Chaotic attractors in the four-dimensional Leslie–Gower competition model

“…n = 1) competitive maps admitting a carrying simplex, there is a globally attracting positive fixed point as the carrying simplex, so the dynamics is trivial. For 2D competitive maps admitting a carrying simplex, every trajectory also converges to a fixed point (see [11]), because the map restricted to the one-dimensional carrying simplex is a homeomorphism. For 3D competitive maps admitting a carrying simplex, nontrivial dynamics such as Neimark-Sacker bifurcations and heteroclinic cycles can occur (see, for example, [8,9,10,11,12]).…”

“…For 2D competitive maps admitting a carrying simplex, every trajectory also converges to a fixed point (see [11]), because the map restricted to the one-dimensional carrying simplex is a homeomorphism. For 3D competitive maps admitting a carrying simplex, nontrivial dynamics such as Neimark-Sacker bifurcations and heteroclinic cycles can occur (see, for example, [8,9,10,11,12]). Neimark-Sacker bifurcation is the birth of an invariant cycle from a fixed point in discrete-time systems, and either all orbits are periodic, or all orbits are dense on the invariant cycle (in this case, it is called a quasiperiodic curve).…”

A note on global stability of three-dimensional Ricker models

“…However, for most competitive discrete-time models on R n + , one can usually find a forward invariant and globally attracting hyperrectangle [0, u] with u 0 (see [9]), such as the Ricker map (2) (see [8]). Thus, in this note, we try to study the global stability of the Ricker map (2) in Gyllenberg et al's class 33 restricted to a forward invariant hyperrectangle directly without constructing a monotone region by the method in [1].…”