Abstract
The asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.
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References
Cantrell, R.S., Cosner, C.: On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol. 37, 103–145 (1998)
Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology. John Wiley and Sons, Chichester (2003)
Dancer, E.N.: On the principal eigenvalue of linear cooperating elliptic system with small diffusion. J. Evolut. Eqns. 9, 419–428 (2009)
de Figueiredo, D., Mitidieri, E.: Maximum principles for linear elliptic systems. Rend. Istit. Mat. Univ. Trieste 22, 36–66 (1990)
Gantmacher, F.: Theory of Matrices. AMS Chelsea publishing, New York (1959)
He, X., Ni, W.-M.: The effects of diffusion and spatial variation in Lotka–Volterra competition-diffusion system I: Heterogeneity vs. homogeneity. J. Diff. Eqns. 254, 528–546 (2013)
He, X., Ni, W.-M.: The effects of diffusion and spatial variation in Lotka–Volterra competition-diffusion system II: the general case. J. Diff. Eqns. 254, 4088–4108 (2013)
He, X., Ni, W.-M.: Global dynamics of the Lotka–Volterra competition-diffusion system: diffusion and spatial heterogeneity, I. Comm. Pure Appl. Math. (to appear)
Hess, P.: Periodic Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series 247. Longman Scientific & Technical, Harlow (1991)
Hutson, V.: Private communication
Hutson, V., Lou, Y., Mischaikow, K.: Convergence in competition models with small diffusion coefficients. J. Diff. Eqns. 211, 135–161 (2005)
Hsu, S.-B., Smith, H., Waltman, P.: Competitive exclusion and coexistence for competitive systems on ordered Banach spaces. Trans. Am. Math. Soc. 348, 4083–4094 (1996)
Iida, M., Tatsuya, M., Ninomiya, H., Yanagida, E.: Diffusion-induced extinction of a superior species in a competition system. Jpn. J. Ind. Appl. Math. 15, 223–252 (1998)
Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Uspehi Mat. Nauk (N.S.) 3(1(23)), 3–95 (1948)
Lam, K.-Y., Ni, W.-M.: Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems. SIAM J. Appl. Math. 72, 1695–1712 (2012)
Lou, Y.: On the effects of migration and spatial inhomogeneity on single and multiple species. J. Diff. Eqns. 223, 400–426 (2006)
Nagel, R.: Operator matrices and reaction-diffusion systems. Rend. Semin. Mat. Fis. Milano 59, 185–196 (1989)
Ninomiya, H.: Separatrices of competition-diffusion equations. J. Math. Kyoto Univ. 35, 539–567 (1995)
Pacala, S., Roughgarden, J.: Spatial heterogeneity and interspecific competition. Theor. Pop. Biol. 21, 92–113 (1982)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations, 2nd edn. Springer, Berlin (1984)
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1992)
Sattinger, D.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1971)
Smith, H.: Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. American Mathematical Society, Providence (1995)
Sweers, G.: Strong positivity in \(C(\bar{\Omega })\) for elliptic systems. Math. Z. 209, 251–271 (1992)
Acknowledgments
This research was partially supported by the NSF Grants DMS-1021179 and DMS-1411476, and has been supported in part by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS-0931642.
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Lam, KY., Lou, Y. Asymptotic Behavior of the Principal Eigenvalue for Cooperative Elliptic Systems and Applications. J Dyn Diff Equat 28, 29–48 (2016). https://doi.org/10.1007/s10884-015-9504-4
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DOI: https://doi.org/10.1007/s10884-015-9504-4