# American Institute of Mathematical Sciences

December  2014, 7(6): 1181-1191. doi: 10.3934/dcdss.2014.7.1181

## Alternate steady states for classes of reaction diffusion models on exterior domains

 1 Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, United States 2 TIFR Center for Applicable Mathematics, Yelahanka, Bangalore 560065, India 3 Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412

Received  January 2013 Revised  November 2013 Published  June 2014

We study positive radial solutions to the problem \begin{equation*} \left\{ \begin{split} -\Delta u &= \lambda K(|x|)f(u), \quad x \in \Omega, \\u(x) &= 0 \qquad

\mbox{ if } |x|=r_0, \\u(x) &\rightarrow 0 \qquad

\mbox{ as } |x|\rightarrow\infty, \end{split} \right. \end{equation*} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x\in\mathbb{R}^N: |x|>r_0\}$, $r_0>0$, and $N>2$. Here, $f\in C^2[0,\infty)$ and $f(u)>0$ on $(0,\sigma)$ and $f(u)<0$ for $u>\sigma$. Furthermore, $K:[r_0, \infty)\rightarrow(0,\infty)$ is continuous and $\lim_{r\rightarrow\infty}K(r)=0$. We discuss the existence of multiple positive solutions for a certain range of $\lambda$ leading to the occurrence of an S-shaped bifurcation curve when $f$ satisfies some additional assumptions. In particular, the two models we consider are $f_1(u)=u-\frac{u^2}{K}-c\frac{u^2}{1+u^2}$ and $f_2(u)=\tilde{K}-u+\tilde{c}\frac{u^4}{1+u^4}$. We prove our results by the method of sub-super solutions.
Citation: Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181
##### References:
 [1] H. Asakawa, Nonresonant singular two-point boundary value problems, Nonlinear Anal., 44 (2001), 791-809. doi: 10.1016/S0362-546X(99)00308-9. [2] A. K. Ben-Naoum and C. D. Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem, Differential and Integral Equations, 10 (1997), 1093-1112. [3] D. Butler, S. Sasi and R. Shivaji, Existence of alternate steady states in a phosphorous cycling model, ISRN Mathematical Analysis, (2012), Art. ID 869147, 11 pp. [4] S. R. Carpenter, D. Ludwig and W. A. Brock, Management of eutrophication for lakes subject to potentially irreversible change, Ecological Applications, 9 (1999), 751-771. [5] E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differential Integral Equations, 24 (2011), 861-875. [6] E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741. doi: 10.1016/j.jmaa.2011.03.048. [7] M. Scheffer, W. Brock and F. Westley, Socioeconomic mechanisms preventing optimum use of ecosystem services: An interdisciplinary theoretical analysis, Ecosystems, 3 (2000), 451-471. doi: 10.1007/s100210000040. [8] E. H. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems, Ecology, 86 (2005), 1797-1807.

show all references

##### References:
 [1] H. Asakawa, Nonresonant singular two-point boundary value problems, Nonlinear Anal., 44 (2001), 791-809. doi: 10.1016/S0362-546X(99)00308-9. [2] A. K. Ben-Naoum and C. D. Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem, Differential and Integral Equations, 10 (1997), 1093-1112. [3] D. Butler, S. Sasi and R. Shivaji, Existence of alternate steady states in a phosphorous cycling model, ISRN Mathematical Analysis, (2012), Art. ID 869147, 11 pp. [4] S. R. Carpenter, D. Ludwig and W. A. Brock, Management of eutrophication for lakes subject to potentially irreversible change, Ecological Applications, 9 (1999), 751-771. [5] E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differential Integral Equations, 24 (2011), 861-875. [6] E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741. doi: 10.1016/j.jmaa.2011.03.048. [7] M. Scheffer, W. Brock and F. Westley, Socioeconomic mechanisms preventing optimum use of ecosystem services: An interdisciplinary theoretical analysis, Ecosystems, 3 (2000), 451-471. doi: 10.1007/s100210000040. [8] E. H. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems, Ecology, 86 (2005), 1797-1807.
 [1] Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049 [2] Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133 [3] Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417 [4] Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216 [5] María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024 [6] Kimun Ryu, Inkyung Ahn. Positive steady--states for two interacting species models with linear self-cross diffusions. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1049-1061. doi: 10.3934/dcds.2003.9.1049 [7] Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations and Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81 [8] Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701 [9] Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819 [10] Kwangjoong Kim, Wonhyung Choi, Inkyung Ahn. Reaction-advection-diffusion competition models under lethal boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021250 [11] Robert E. Beardmore, Rafael Peña-Miller. Antibiotic cycling versus mixing: The difficulty of using mathematical models to definitively quantify their relative merits. Mathematical Biosciences & Engineering, 2010, 7 (4) : 923-933. doi: 10.3934/mbe.2010.7.923 [12] Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039 [13] Manuela Caratozzolo, Santina Carnazza, Luigi Fortuna, Mattia Frasca, Salvatore Guglielmino, Giovanni Gurrieri, Giovanni Marletta. Self-organizing models of bacterial aggregation states. Mathematical Biosciences & Engineering, 2008, 5 (1) : 75-83. doi: 10.3934/mbe.2008.5.75 [14] Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1 [15] Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053 [16] Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11 [17] Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259 [18] Guo Lin, Wan-Tong Li, Mingju Ma. Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 393-414. doi: 10.3934/dcdsb.2010.13.393 [19] Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545 [20] Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415

2020 Impact Factor: 2.425