Coexistence States and Global Attractivity for Some Convective Diffusive Competing Species Models

López-Gómez, Julián; Lis, José C. Sabina de

doi:10.2307/2155205

Cited by 24 publications

(36 citation statements)

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“…Hence, in the autonomous case our results agree with all the classical results of uniqueness and stability of the non-semitrivial steady states of (1.3) for the three cases of competition, symbiosis and prey-predator (see for instance Theorem 4.4 in Furter and López-Gómez [13] and Corollary 4.3 in López-Gómez and Sabina de Lis [29] in the competition case, and Corollary 9.5 in Delgado et al [12] in the symbiosis case).…”

Section: (λ(T X) − A(t X)u − B(t X)v) X ∈ ω T > S V T − D 2 ∆V = supporting

confidence: 88%

“…However, in the competition case it is well-known that if λ ≤ Λ 0 or µ ≤ Λ 0 , then one of the two species (or both of them) will be driven to extinction (see López-Gómez and Sabina [29] for an improvement of this result). Similar results can be obtained in the other cases, see [6] and [30].…”

Section: (λ(T X) − A(t X)u − B(t X)v) X ∈ ω T > S V T − D 2 ∆V = mentioning

confidence: 99%

See 1 more Smart Citation

Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

Langa¹,

Robinson²,

Rodriguez-Bernal³

et al. 2009

SIAM J. Math. Anal.

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Abstract. Lotka-Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis or prey-predator behaviour involving different interacting species in a fixed habitat. Much of the work on these models has been within the framework of infinite-dimensional dynamical systems, but this has frequently been extended to allow explicit time dependence, generally in a periodic, quasiperiodic or almost periodic fashion. The presence of more general non-autonomous terms in the equations leads to non-trivial difficulties which have stalled the development of the theory in this direction. However, the theory of non-autonomous dynamical systems has received much attention in the last decade, and this has opened new possibilities in the analysis of classical models with general non-autonomous terms. In this paper we use the recent theory of attractors for non-autonomous PDEs to obtain new results on the permanence and the existence of forwards and pullback asymptotically stable global solutions associated to non-autonomous Lotka-Volterra systems describing competition, symbiosis or prey-predator phenomena. We note in particular that our results are valid for prey-predator models, which are not order-preserving: even in the 'simple' autonomous case the uniqueness and global attractivity of the positive equilibrium (which follows from the more general results here) is new.

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Section: (λ(T X) − A(t X)u − B(t X)v) X ∈ ω T > S V T − D 2 ∆V = supporting

confidence: 88%

Section: (λ(T X) − A(t X)u − B(t X)v) X ∈ ω T > S V T − D 2 ∆V = mentioning

confidence: 99%

Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

Langa¹,

Robinson²,

Rodriguez-Bernal³

et al. 2009

SIAM J. Math. Anal.

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“…Observe that conditions (4.8), (4.9) and (4.10) define regions in the plane (λ, µ) which could eventually be empty. For the semilinear case, that is g 1 ≡ g 2 ≡ 0, it can be shown, see for example [18] and [17], that these regions are not empty, imposing some conditions (b or c small). The study of these regions are out of the scope of this paper, but let us remark some aspects.…”

Section: 2mentioning

confidence: 99%

A sub-supersolution method for nonlinear elliptic singular systems with natural growth and some applications

2016

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Abstract. In this paper we give a sub-supersolution method for nonlinear elliptic singular systems with quadratic gradient whose model system is the followingwhere Ω is a smooth bounded domain of R N (N ≥ 3), β, µ ≥ 0, 0 < α, γ < 1 and regular f 1 , f 2 functions. Moreover, we apply it to prove existence of solution for some systems, including the classical Lotka-Volterra models with gradient terms. Specifically, we study the competition and the symbiotic LoktaVolterra systems.

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“…This eigenvalue plays a predominant rôle in the qualitative study of nonlinear boundary value problems via bifurcation theory and in the method of sub-and supersolutions (cf. [37], [51], [53], [54], [57], [58], and the references therein). Consequently, we investigate in some detail questions of existence and continuous dependence on the data of the principal eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

Maximum Principles and Principal Eigenvalues

Amann

2005

10 Mathematical Essays on Approximation in Analysis and Topology

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Coexistence States and Global Attractivity for Some Convective Diffusive Competing Species Models

Cited by 24 publications

References 0 publications

Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka–Volterra Models with Diffusion

A sub-supersolution method for nonlinear elliptic singular systems with natural growth and some applications

Maximum Principles and Principal Eigenvalues

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