A general monotone scheme for elliptic systems with applications to ecological models

Korman, Philip; Leung, Anthony W.

doi:10.1017/s0308210500026391

Cited by 49 publications

(33 citation statements)

References 15 publications

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“…See, e.g., [1], [5], [10], [11], [20], [22] and [25]. In particular, the existence of positive solutions for this case was completely understood, see Dancer [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Some uniqueness and exact multiplicity results for a predator-prey model

Lou

1997

Trans. Amer. Math. Soc.

117

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Abstract. In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter m in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when m is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large m reveal interesting contrast with that for the well-studied case m = 0, i.e., the classical Lotka-Volterra predator-prey model.

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“…See, e.g., [1], [5], [10], [11], [20], [22] and [25]. In particular, the existence of positive solutions for this case was completely understood, see Dancer [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Some uniqueness and exact multiplicity results for a predator-prey model

Lou

1997

Trans. Amer. Math. Soc.

117

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“…Observe that this method applies for cooperative elliptic systems, which is the case here, since the functions f are quasi-monotone. For the reader's convenience we state the following result, which will be sufficient for our purposes (see for example [23] for more general statement).…”

Section: Lemma 22 Under the Hypotheses Of Either Theorem 2 Or Theorementioning

confidence: 99%

On the Ambrosetti–Prodi Problem for Non-Variational Elliptic Systems

Figueiredo

Sirakov

2007

Djairo G. De Figueiredo - Selected Papers

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“…The dependence on the space variable x of these coefficient functions represents the fact that the two species u and v are competing in a spatially heterogeneous environment. In a spatially homogeneous environment, these coefficient functions should be replaced by positive constants; in such a case, the above system is known as the classical Lotka-Volterra competition model with diffusion, which has attracted considerable amounts of study, see, for example, [BB1,BB2,CL,DD,DB,EFL,GL,HL,KL,RP] and the references therein.…”

Section: =La 1 (X) U − B(x) U 2 − C(x) Uv V T (X T) − D 2 (X) Dv(xmentioning

confidence: 99%

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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A general monotone scheme for elliptic systems with applications to ecological models

Cited by 49 publications

References 15 publications

Some uniqueness and exact multiplicity results for a predator-prey model

Some uniqueness and exact multiplicity results for a predator-prey model

On the Ambrosetti–Prodi Problem for Non-Variational Elliptic Systems

Effects of a Degeneracy in the Competition Model

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