Permanence criteria for Kolmogorov systems with delays

Hou, Zhenting

doi:10.1017/s0308210512000297

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Some research projects were inspired by Hirsch's theorem although they could not use theorem 1.1 because some of its conditions were not met. For example, Hou [11][12][13][15][16][17] investigated permanence and stability without assuming the existence of a carrying simplex; Liang and Jiang [29] studied the dynamical behaviour of type-K competitive systems; Mierczyński and Schreiber [34] established permanence conditions; Tu and Jiang [36] found coexistence conditions for systems with limited competition; Zeeman [43] picked up a condition for extinction for some species; Yu et al [39] gave a criterion for global stability of three-dimensional system. 4.…”

Section: Introductionmentioning

confidence: 99%

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

Hou

2020

Nonlinearity

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For a C 1 map T from C = [0, +∞) N to C of the form Ti(x) = xifi(x), the dynamical system x(n) = T n (x) as a population model is competitive if ∂f i ∂x j ≤ 0 (i = 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 Σ ⊂ C that is homeomorphic to ∆ N−1 = {x ∈ C : x1 + • • • + xN = 1}, attracting all the points of C \ {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 = (∂f i ∂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 fi is strictly decreasing in xi. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

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Section: Introductionmentioning

confidence: 99%

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

Hou

2020

Nonlinearity

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“…For a particular system with N ≥ 4, since J-permanence is another specialised active research area, we need to search the literature for available Jpermanence results (e.g. [10]) or analyse the location of the global attractor of the system restricted to j∈I + π j and j∈I − π j . Remark 6.…”

Section: P G Attractionmentioning

confidence: 99%

Global stability and repulsion in autonomous Kolmogorov systems

Hou¹,

Baigent²

2015

CPAA

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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.

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“…Some research projects were inspired by Hirsch's theorem although they could not use theorem 1.1 because some of its conditions were not met. For example, Hou [16][17][18] investigated permanence and stability without assuming the existence of a carrying simplex; Liang and Jiang [33] studied the dynamical behaviour of type-K competitive systems; Mierczyński and Schreiber [38] established permanence conditions; Tu and Jiang [43] found coexistence conditions for systems with limited competition; Yu et al [47] gave a criterion for global stability of three-dimensional system. The concept of carrying simplex has been extended to discrete competitive dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

On global dynamics of type-K competitive Kolmogorov differential systems

Hou

2023

Nonlinearity

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This paper deals with global asymptotic behaviour of the dynamics for N-dimensional type-K competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-K monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set Σ which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets S H and S V , their restriction to the interior of the first orthant are ( N − 1 ) -dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either S H or S V : each trajectory in the interior of the first orthant is asymptotic to one in S H and one in S V . Geometric and asymptotic features of the global attractor Σ are investigated. It is proved that the partition Σ = Σ H ∪ Σ 0 ∪ Σ V holds such that Σ H ∪ Σ 0 ⊂ S H and Σ V ∪ Σ 0 ⊂ S V . Thus, Σ0 contains all the ω-limit sets for all interior trajectories of any type-K subsystems and the closure Σ H ∪ Σ V ‾ as a subset of Σ is invariant and the upper boundary of the basin of repulsion of the origin. This Σ has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below Σ is asymptotic to one in Σ and the ω-limit set is in Σ for every other nontrivial trajectory.

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Permanence criteria for Kolmogorov systems with delays

Cited by 4 publications

References 21 publications

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

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

Global stability and repulsion in autonomous Kolmogorov systems

On global dynamics of type-K competitive Kolmogorov differential systems

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