## Topological Methods in Nonlinear Analysis

### Competition systems with strong interaction on a subdomain

#### Abstract

We study the large-interaction limit of an elliptic system modelling the steady states of two species $u$ and $v$ which compete to some extent throughout a domain $\Omega$ but compete strongly on a subdomain $A \subset \Omega$. In the strong-competition limit, $u$ and $v$ segregate on $A$ but not necessarily on $\Omega \setminus A$. The limit problem is a system on $\Omega \setminus A$ and a scalar equation on $A$ and in general admits an interesting range of types of solution, not all of which can be the strong-competition limit of coexistence states of the original system.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 1 (2011), 37-53.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461181487

Mathematical Reviews number (MathSciNet)
MR2839516

Zentralblatt MATH identifier
1234.35091

#### Citation

Crooks, Elaine C.M.; Dancer, E. Norman. Competition systems with strong interaction on a subdomain. Topol. Methods Nonlinear Anal. 37 (2011), no. 1, 37--53. https://projecteuclid.org/euclid.tmna/1461181487

#### References

• S. Cano-Casanova and J. López-Gómez, Permanence under strong aggressions is possible , Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 999–1041 \ref\key 2
• R. S. Cantrell and S. Cosner, Spatial Ecology via Reaction–Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester (2003) \ref\key 3
• E. C. M. Crooks and E. N. Dancer, Highly nonlinear large-competition limits of elliptic systems , Nonlinear Anal., 73 (2010), 1447–1457 \ref\key 4
• E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions , Topol. Methods Nonlinear Anal., 30 (2007), 1–36 \ref\key 5
• E. N. Dancer, On positive solutions of some pairs of differential equations \romII, J. Differential Equations, 60 (1985), 236–258 \ref\key 6
• E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities , J. Differential Equations, 114(1994), 434–475 \ref\key 7
• Y. Du and X. Liang, A diffusive competition model with a protection zone , J. Differential Equations, 244(2008), 61–86 \ref\key 8
• E. B. Fabes, M. Jodeit Jr. and N. M. Rivière, Potential techniques for boundary value problems in $C^1$-domains , Acta Math., 141(1978), 165–186 \ref\key 9
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., rev. 3rd printing, Springer, New York (2001) \ref\key 10
• W. K. Hayman and P. B. Kennedy, Subharmonic Functions, 1 , Academic Press, London (1976) \ref\key 11
• N. Igbida and F. Karami, Localized large reaction for a non-linear reaction-diffusion system , Adv. Differential Equations, 13 (2008), 907–933 \ref\key 12
• J. López-Gómez, Strong competition with refuges , Nonlinear Anal., 30 (1997), 5167–5178 \ref\key 13
• B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schródinger systems with strong competition , Comm. Pure Appl. Math., 63 (2010), 267–302 \ref\key 14
• S. Terracini and G. Verzini, Multiphase in $k$-mixtures of Bose–Einstein condensates , Arch. Rational Mech. Anal., 194 (2009), 717–741 \ref\key 15
• J. Wei and T. Weth, Asymptotic behaviour of solutions of planar elliptic systems with strong competition , Nonlinearity, 21 (2008), 305–317