Global stability and repulsion in autonomous Kolmogorov systems

Hou, Zhenting; Baigent, Stephen

doi:10.3934/cpaa.2015.14.1205

Cited by 12 publications

(14 citation statements)

References 26 publications

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“…The virtue of working with the May-Leonard model (2) is that it has identical linear growth rates which means that it can be reduced to (6), (7). The May-Leonard system (2) provides a simple model where the balance simplex can be easily shown to exist, and can be computed via (14). While the notion of a carrying simplex is traditionally confined to models where species interactions are all of a competitive nature (so that the invariant manifold identified with the carrying simplex is an unordered manifold), the balance simplex of (2) allows us to study the effect of a mixture of competitive, cooperative, and predatorprey interactions and allows for invariant manifolds without the restrictive requirement that they are unordered.…”

Section: Discussionmentioning

confidence: 99%

“…It is known that the presence of a carrying simplex, and the fact that it is the boundary of repulsion basins, can help to understand global stability or repulsion (in the carrying simplex) in Kolmogorov systems through the Split Kolmogorov method [2,14,23] or the index theorem approach of [16] (and [15] for discrete dynamics), so it would be interesting to ask what the presence of a balance simplex says in the context of global stability.…”

Section: Discussionmentioning

confidence: 99%

“…In portraits 7,8,9,10,11,12,14,15,17 there is a unique interior equilibrium. Of these, in portraits 7, 9, 15, 17 the interior equiilbrium p has B(p) = int( ).…”

Section: Appendix Using the Classification Of Bomzementioning

confidence: 99%

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Balance simplices of 3-species May-Leonard systems

Baigent¹,

Ching²

2020

Journal of Biological Dynamics

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We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is 'growth when rare' and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex. ARTICLE HISTORY

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Balance simplices of 3-species May-Leonard systems

Baigent¹,

Ching²

2020

Journal of Biological Dynamics

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“…Now we turn to the discretetime version of the Split Lyapunov method introduced for competitive Lotka-Volterra differential equations in [32] and developed further for general Lotka-Volterra systems in [1,14] and for general Kolmogorov differential equations in [15].…”

Section: Split-lyapunov Stability Of Interior Fixed Pointsmentioning

confidence: 99%

Global stability of discrete-time competitive population models

Baigent¹,

Hou

2017

Journal of Difference Equations and Applications

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Abstract. 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.

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“…It has been proved as a powerful tool to investigate global dynamics of competitive systems, especially for the lower dimensional systems. The reader can consult, for instance, [44,80,83,85,86,61,35,7,4,51,45,11], for more results on continuous-time competitive systems via the carrying simplex.…”

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confidence: 99%

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

Gyllenberg¹,

Jiang

Niu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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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.

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Global stability and repulsion in autonomous Kolmogorov systems

Cited by 12 publications

References 26 publications

Balance simplices of 3-species May-Leonard systems

Balance simplices of 3-species May-Leonard systems

Global stability of discrete-time competitive population models

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

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