Global stability and repulsion in autonomous Kolmogorov systems

Criteria are established for the global attraction, or global repulsion on a compact invariant set, of interior and boundary fixed points of Kolmogorov systems. In particular, the notions of diagonal stability and Split Lyapunov stability that have found wide success for Lotka-Volterra systems are extended for Kolmogorov systems. Several examples from theoretical ecology and evolutionary game theory are discussed to illustrate the results

Global attraction and repulsion of a heteroclinic limit cycle in three dimensional Kolmogorov maps

There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author's latest publications). In this paper, as one of the applications of the modified carrying simplex theory, a criterion is established for a three dimensional Kolmogorov map to have a globally repelling (attracting) heteroclinic limit cycle. As a concrete example, a discrete competitive model is investigated to illustrate the above criteria for global repulsion (attraction) of a hetericlinic limit cycle.Comment: 14 pages. arXiv admin note: text overlap with arXiv:2102.0927

Geometric method for global stability of discrete population models

A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary fixed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by finding the eigenvalues of the Jacobian matrix at the fixed point. An intriguing question is proposed: Can a globally attracting fixed point induce a homoclinic cycle

Geometric method for global stability and repulsion in Kolmogorov systems

A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established for global repulsion of an interior or a boundary equilibrium point on the carrying simplex. This method and the theorems can be viewed as a natural extension of those results for Lotka-Volterra systems in the literature

Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known

On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems

For a $C^1$ map $T$ from $C =[0, +\infty)^N$ to $C$ of the form $T_i(x) = x_if_i(x)$, the dynamical system $x(n) =T^n(x)$ as a population model is competitive if $\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$. A well know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008) 169--179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96--113) with various versions by others, states that, under certain conditions, the system has a compact invariant surface $\Sigma\subset C$ that is homeomorphic to $\Delta^{N-1} =\{x\in C: x_1+ \cdots + x_N=1\}$, attracting all the points of $C\setminus\{0\}$, and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the $N^2$ entries of the Jacobian matrix $Df = (\frac{\partial f_i}{\partial x_j})$ to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of $Df$ to be nonpositive and each $f_i$ is strictly decreasing in $x_i$. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others

Global attraction and repulsion of a heteroclinic limit cycle in three dimensional Kolmogorov maps

There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author's latest publications). In this paper, as one of the applications of the modified carrying simplex theory, a criterion is established for a three dimensional Kolmogorov map to have a globally repelling (attracting) heteroclinic limit cycle. As a concrete example, a discrete competitive model is investigated to illustrate the above criteria for global repulsion (attraction) of a hetericlinic limit cycle

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

This paper deals with global asymptotic behaviour of the dynamics for $N$-dimensional competitive Kolmogorov differential systems of equations $\frac{dx_i}{dt} =x_if_i(x), 1\leq i\leq N, x\in \R^N_+$. A theory based on monotone dynamical systems was well established by Morris W Hirsch (Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51--71). One of his theorems is outstanding and states the existence of a co-dimension 1 compact invariant submanifold $\Sigma$ that attracts all the nontrivial orbits under certain assumptions and, in practice, under the condition that the system is totally competitive (all $N^2$ entries of the Jacobian matrix $Df$ are negative). The submanifold $\Sigma$ has been called carrying simplex since then and the theorem has been well accepted with many hundreds of citations. In this paper, we point out that the requirement of total competition is too restrictive and too strong; we prove the existence and uniqueness of a carry simplex under the assumption of strong internal competition only (i.e. $N$ diagonal entries of $Df$ are negative), a much weaker condition than total competition. Thus, we improve the theorem significantly by dramatic cost reduction from requiring $N^2$ to $N$ negative entries of $Df$. As an example of applications of the main result, the existence and global attraction (repulsion) of a heteroclinic limit cycle for three-dimensional systems is discussed and two concrete examples are given to demonstrate the existence of such heteroclinic cycles