Global Bifurcation of Positive Solutions in Some Systems of Elliptic Equations

Blat, Josep; Brown, K. J.

doi:10.1137/0517094

Cited by 162 publications

(128 citation statements)

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“…We note that the global bifurcation arguments in [2] do not seem easily applicable to our problem here. First, in the abstract global bifurcation result in [2] the fixed point index in the entire space is required, which was calculated there based on a good understanding of the eigenvalues of (4.3) with ξ = 1.…”

Section: Yihong Du and Sze-bi Hsumentioning

confidence: 94%

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On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

Du¹,

Hsu²

2010

SIAM J. Math. Anal.

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Abstract.In this paper we analyze a nonlocal reaction-diffusion model which arises from the modeling of competition of phytoplankton species with incomplete mixing in a water column. The nonlocal nonlinearity in the model describes the light limitation for the growth of the phytoplankton species. We first consider the single-species case and obtain a complete description of the longtime dynamical behavior of the model. Then we study the two-species competition model and obtain sufficient conditions for the existence of positive steady states and uniform persistence of the dynamical system. Our approach is based on a new modified comparison principle, fixed point index theory, global bifurcation arguments, elliptic and parabolic estimates, and various analytical techniques.Key words. phytoplankton, competition for light, uniform persistence, reaction-diffusion equation, steady state AMS subject classifications. 35J55, 35J65, 92D25DOI. 10.1137/090775105 Introduction.In this paper we analyze a reaction-diffusion model which describes the growth of phytoplankton species in a eutrophic environment. In such environments there are ample nutrients and the phytoplankton species typically compete for light. In [17,18,25] Huisman and Weissing developed a theory of interspecific competition for light that assumes complete mixing of phytoplankton species. This theory is based on a system of ordinary differential equations (ODEs) and predicts that complete mixing leads to competitive exclusion similar to that in [13,12,1,23]; namely, the species with the lowest "critical light intensity" wins the competition.However, in many aquatic environments, phytoplankton species are not thoroughly mixed. To understand the effect of incomplete mixing on the growth of phytoplankton species in a eutrophic environment, Huisman, van Oostveen, and Weissing [16] introduced a reaction-diffusion model and analyzed the model through numerical simulations. But a thorough mathematical treatment of the model has been lacking. The purpose of this paper is to prove some basic mathematical facts for this model which provide a basis for further rigorous mathematical analysis of the involved reaction-diffusion system. Our uniform persistence result indicates that with incomplete mixing of the phytoplankton species, competitive exclusion does not always happen, and coexistence can occur in some parameter ranges.

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Section: Yihong Du and Sze-bi Hsumentioning

confidence: 94%

“…Global bifurcation analysis. Let us now fix δ > 0 small enough such that (4.1) has no positive solution for d 2 …”

Section: Yihong Du and Sze-bi Hsumentioning

confidence: 99%

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On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

Du¹,

Hsu²

2010

SIAM J. Math. Anal.

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“…These last nonlinearities arise in population dynamics. Indeed, when K = 0, f 1 is the classical logistic reaction term and for K = 0 the predation one Ku/(1 + u) is called the Holling-Tanner term, see for example [7] for an ecological interpretation. In order to state our main results we need some notations.…”

Section: Introductionmentioning

confidence: 99%

Some indefinite nonlinear eigenvalue problems

Fernández

2004

The First 60 Years of Nonlinear Analysis of Jean Mawhin

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“…Bifurcation methods have been used in many texts concerning interacting species (competition models, predator-prey systems), see [20,21,23,22] and more recently in the study of some age structured models, see [8,9]. In that respect, we wish to stress that the chemostat involves a fairly specific mathematical structure, a fact that plays a crucial role below: the nonlinear coupling in (1.2), say, only involves terms of the form f i (x, R) U or f i (x, R) V ; in other words the two species U and V in (1.2) are only coupled through the resource R. This observation holds in any chemostat model and allows, in some situations, to reduce the original model to a standard competition system by eliminating the equation on the resource, see [13,12,11,3,18,19,10].…”

Section: Introductionmentioning

confidence: 99%

Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

Castella

Madec

2012

J. Math. Biol.

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To cite this version:François Castella, Sten Madec. Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates. Journal of Mathematical Biology, Springer Verlag (Germany), 2014, 68 (1-2), pp.377-415. <10.1007/s00285-012-0633-7>. Biological Mathematics manuscript No. (will be inserted by the editor) Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion ratesFrançois Castella · Sten Madec the date of receipt and acceptance should be inserted later Abstract We study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit coexistence phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a diffusion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a distinct diffusion constant. This is a key difficulty in our analysis: the specific (and classical) case where all diffusion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simplifies the analysis and the qualitative phenomena.Using the global bifurcation theory, we prove that the underlying 2-species, stationary, diffusive, chemostat-like model, does possess coexistence solutions, where both species survive. On top of that, we identify the domain, in the space of the identified bifurcation parameters, for which the system does have coexistence solutions.

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Global Bifurcation of Positive Solutions in Some Systems of Elliptic Equations

Cited by 162 publications

References 10 publications

On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

Some indefinite nonlinear eigenvalue problems

Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

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