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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Coexistence theorems of steady states for predator-prey interacting systems


Author: Lige Li
Journal: Trans. Amer. Math. Soc. 305 (1988), 143-166
MSC: Primary 35J60; Secondary 92A15
MathSciNet review: 920151
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Abstract: In this paper we give necessary and sufficient conditions for the existence of positive solutions of steady states for predator-prey systems under Dirichlet boundary conditions on $ \Omega \Subset {{\mathbf{R}}^n}$. We show that the positive coexistence of predatorprey densities is completely determined by the "marginal density," the unique density of prey or predator while the other one is absent, i.e. the $ ({u_0},\,0)$ or $ (0,\,{\nu _0})$. More specifically, the situation of coexistence is determined by the spectral behavior of certain operators related to these marginal densities and is also completely determined by the stability properties of these marginal densities. The main results are Theorems 1 and 4.2.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0920151-1
PII: S 0002-9947(1988)0920151-1
Article copyright: © Copyright 1988 American Mathematical Society