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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 869147, 11 pages
http://dx.doi.org/10.5402/2012/869147
Research Article

Existence of Alternate Steady States in a Phosphorous Cycling Model

1Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
2Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA

Received 20 January 2012; Accepted 8 February 2012

Academic Editors: J.-F. Colombeau, G. L. Karakostas, and P. Omari

Copyright © 2012 Dagny Butler et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyze the positive solutions to the steady-state reaction diffusion equation with Dirichlet boundary conditions of the form: βˆ’ Ξ” 𝑒 = πœ† [ 𝐾 βˆ’ 𝑒 + 𝑐 ( 𝑒 4 / ( 1 + 𝑒 4 ) ) ] , π‘₯ ∈ Ξ© , 𝑒 = 0 , π‘₯ ∈ πœ• Ξ© . Here, Ξ” 𝑒 = d i v ( βˆ‡ 𝑒 ) is the Laplacian of 𝑒 , 1 / πœ† is the diffusion coefficient, 𝐾 and 𝑐 are positive constants, and Ξ© βŠ‚ ℝ 𝑁 is a smooth bounded region with πœ• Ξ© in 𝐢 2 . This model describes the steady states of phosphorus cycling in stratified lakes. Also, it describes the colonization of barren soils in drylands by vegetation. In this paper, we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of subsuper solutions.