# Dynamics of Partial Differential Equations

## Volume 3 (2006)

### Perturbation of a nonlinear elliptic biological interacting model

Pages: 281 – 293

DOI: http://dx.doi.org/10.4310/DPDE.2006.v3.n4.a2

#### Authors

Joon Hyuk Kang (Department of Mathematics, Andrews University, Berrien Springs, Michigan, U.S.A.)

Jungho Lee (Department of Mathematics, Michigan State University, East Lansing, Michigan, U.S.A.)

Kami Lizarraga (Department of Mathematics, Andrews University, Berrien Springs, Michigan, U.S.A.)

#### Abstract

In this paper, we investigate the effects of perturbations on thecoexistence state of the general competition model for two species.Previous work by Kang, Lee, and Oh establishedsufficient conditions for the uniqueness of the positive solution tothe following general elliptic system for two competing species ofanimals:$$\left\{ \begin{array}{l}\left.\begin{array}{l}\Delta u + ug(u,v) = 0\\\Delta v + vh(u,v) = 0\end{array} \right.\;\;\mbox{in}\;\;\Omega,\\u|_{\partial\Omega} = v|_{\partial\Omega} = 0.\end{array} \right.$$That is, they proved that under certain conditions, two species cancoexist and that the coexistence state is unique at fixed rates. Inthis paper, we extend their uniqueness results by perturbingfunctions $g$ and $h$ of the above model, and applying super-subsolutions, maximum principles and spectrum estimates. Our argumentsalso rely on some detailed properties for the solution of logisticequations. By applying these techniques, we obtain sufficientconditions for the existence and uniqueness of a time independentcoexistence state for the perturbed general competition model.

#### Keywords

Lotka Volterra competition model, coexistence state

#### 2010 Mathematics Subject Classification

35Axx

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