Positive Steady-State Solutions of a Competing Reaction-Diffusion System

Ruan, Weihua; Pao, C. V.

doi:10.1006/jdeq.1995.1059

Journal of Differential Equations

1995

DOI: 10.1006/jdeq.1995.1059

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Positive Steady-State Solutions of a Competing Reaction-Diffusion System

Weihua Ruan¹,

C. V. Pao²

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Cited by 27 publications

(22 citation statements)

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“…Let S y be the maximal linear subspace of X contained in W y . Assume that T is a compact and Fréchet differentiable operator on X such that y ∈ W is a fixed point of T and T (W ) ⊆ W. Then the Fréchet derivative T (y) of T at y leaves W y and S y invariant (see [4,26]). If there exists a closed linear subspace X y of X such that X = S y ⊕ X y and W y is generating, then the index of T at y can be found by analyzing certain eigenvalue problems in X y and S y as follows.…”

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confidence: 99%

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The Effect of Inhibitor on the Plasmid‐Bearing and Plasmid‐Free Model in the Unstirred Chemostat

Wu¹,

Nie²,

Wolkowicz³

2007

SIAM J. Math. Anal.

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Abstract. This paper deals with a chemostat model with an inhibitor in the context of competition between plasmid-bearing and plasmid-free organisms. First, sufficient conditions for coexistence of the steady-state are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which represents the effect of the inhibitor, plays a very important role in deciding the number of the coexistence solutions. The results show that if μ is sufficiently large this model has at least two coexistence solutions provided that the maximal growth rate a of u lies in a certain range and has only one unique asymptotically stable coexistence solution when a belongs to another range. Finally, extensive simulations are done to complement the analytic results. The main tools used here include degree theory in cones, bifurcation theory, and perturbation technique. 1. Introduction. The chemostat is a common model in microbial ecology. It is used as an ecological model of a simple lake, as a model of waste treatment, and as a model for commercial production of fermentation processes. It is important in ecology because the parameters are readily measurable and, thus, the mathematical results are readily testable. For a general discussion of competitive systems see [29], while a detailed mathematical description of competition in the chemostat can be found in [30].Our study focuses on a chemostat model in the context of competition between plasmid-bearing and plasmid-free organisms. This issue has recently received considerable attention. Levin, and Waldstätter [23]. In industry, genetically altered organisms are frequently used to manufacture a desired product, for instance, a pharmaceutical. The alteration is accomplished by introducing a piece of DNA into the cell in the form of a plasmid. The burden imposed on the cell by the task of production can result in the genetically altered (the plasmid-bearing) organism being a less able competitor than the plasmid-free organism. Unfortunately, the plasmid can be lost in the reproductive process. Thus, it is possible for the plasmid-free organism to take over the culture. To avoid "capture" of the process by the plasmid-free organism, the obvious choice is to alter the medium in such a way as to favor the plasmid-bearing organism. An example of this would be to introduce an antibiotic into the feed bottle. See [10,15,16]

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“…Let Q : X → X y be the projection operator of X y along S y . In view of Theorems 2.1 and 2.2 of [26], index W (T, y) exists if the Fréchet derivative T (y) of T at y has no nonzero fixed point in W y . Furthermore, Next, we derive some a priori estimates for positive solutions of (EP).…”

mentioning

confidence: 99%

The Effect of Inhibitor on the Plasmid‐Bearing and Plasmid‐Free Model in the Unstirred Chemostat

Wu¹,

Nie²,

Wolkowicz³

2007

SIAM J. Math. Anal.

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“…For each of these three types of (ps, Ns) we determine its stability and give a condition under which nonuniform steady-state solutions exist. The technique adopted in this section is the so-called "stability-existence" method developed in [14,15], which uses the stability of certain steady-state solutions of a parabolic system in a convex invariant set to determine the existence of other steadystate solutions. Suppose E is a bounded closed convex subset of the Banach space X = C(Q) ® C(£2), and suppose that E is forward invariant for the semiflow U generated by the parabolic system…”

mentioning

confidence: 99%

“…Here "cl" denotes the topological closure of the set in X . It is shown in [15,Theorem 3.3] that whenever the equation…”

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confidence: 99%

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A reaction-diffusion system arising in population genetics

Ruan¹

1996

Quart. Appl. Math.

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Abstract. This paper is concerned with a nonweakly coupled system of parabolic equations that models the reaction-diffusion process of a population of continuously reproducing diploids with two alleles at one locus of genes. We assume that the fitnesses of genotypes are density dependent and the boundary condition is Dirichlet type with the boundary data given in the form such that the system has a unique uniform steady-state solution. The existence of nonuniform steady-state solutions is examined in relation with the stability of uniform steady-state solutions. In several cases we show that a nonuniform steady-state solution exists if the uniform one is unstable. We also investigate the stability and attractivity of the uniform steadystate solution in various cases. Some results of global convergence of solutions to a uniform steady-state solution are also given.

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Effects of a Degeneracy in the Competition Model

2002

Journal of Differential Equations

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We study the competition modelwhere the coefficient functions are strictly positive over the underlying spatial region W except b(x), which vanishes in a nontrivial subdomain of W, and is positive in the rest of W. We show that there exists a critical number l g such that if l < l g , then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when l > l g , new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v)=(., 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part II, where these two kinds of steady-state solutions will be used to determine the dynamics of the model. © 2002 Elsevier Science (USA)

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Positive Steady-State Solutions of a Competing Reaction-Diffusion System

Cited by 27 publications

References 0 publications

The Effect of Inhibitor on the Plasmid‐Bearing and Plasmid‐Free Model in the Unstirred Chemostat

The Effect of Inhibitor on the Plasmid‐Bearing and Plasmid‐Free Model in the Unstirred Chemostat

A reaction-diffusion system arising in population genetics

Effects of a Degeneracy in the Competition Model

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