Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system

Aïnseba, Bedr’Eddine; Aniţa, Sebastian

doi:10.1016/j.na.2004.09.055

Cited by 30 publications

(39 citation statements)

References 11 publications

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“…The nonnegativity of u 1 and u 2 is a natural requirement due to the biological significance of u 1 and u 2 . For some stabilizability results of the population dynamics with diffusion we refer to [13], for predator-prey systems, see [14], [15], whereas for general stabilization results for distributed systems we refer to [16].…”

Section: Remarkmentioning

confidence: 99%

On the stabilization of reaction–diffusion systems modeling a class of man‐environment epidemics: A review

Aniţa¹,

Capasso

2010

Math Methods in App Sciences

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A two-component reaction-diffusion system modeling a class of spatially structured epidemic systems is considered. The system describes the spatial spread of infectious diseases mediated by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so-called zero-stabilization. In a series of papers, necessary conditions and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. In order to model the possible seasonal variability of the environmental conditions, the relevant parameters need to be assumed to be periodic, all with the same period. Corresponding results for the time homogeneous case are presented too.

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Section: Remarkmentioning

confidence: 99%

On the stabilization of reaction–diffusion systems modeling a class of man‐environment epidemics: A review

Aniţa¹,

Capasso

2010

Math Methods in App Sciences

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“…For a model taking into consideration the diffusion process but with the very same interaction functions f and g, see Ainseba-Aniţa [1]. One limiting case, motivated by practical examples, corresponds to K = +∞ and to the multi-valued choice of g as…”

Section: An Examplementioning

confidence: 99%

Viability for nonlinear multi-valued reaction–diffusion systems

Roșu

2010

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper we consider a nonlinear evolution reactiondiffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K be a nonempty and locally closed subsetY a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (τ, ξ, η) ∈ K, the next system Mathematics Subject Classification (2000). Primary 34G20, 47J35, 35K57; Secondary 35K45, 35K65.

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“…In the past two decades, the reaction-diffusion models, especially in population dynamics, have been studied extensively. For example, Ainseba and Aniţa in [1] considered a 2 × 2 system of semilinear partial differential equations of parabolic-type to describe the interactions between a prey population and a predator population and obtained some necessary and sufficient conditions for stabilizability. Xu and Ma in [21] studied a reaction-diffusion predator-prey system with non-local delay and Neumann boundary conditions and established some sufficient conditions on the global stability of the positive steady state and the semitrivial steady state.…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

Liu

Zhang

et al. 2018

Journal of Biological Dynamics

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In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end. ARTICLE HISTORY

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Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system

Cited by 30 publications

References 11 publications

On the stabilization of reaction–diffusion systems modeling a class of man‐environment epidemics: A review

On the stabilization of reaction–diffusion systems modeling a class of man‐environment epidemics: A review

Viability for nonlinear multi-valued reaction–diffusion systems

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

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