# American Institute of Mathematical Sciences

March  2015, 35(3): 1239-1284. doi: 10.3934/dcds.2015.35.1239

## Qualitative analysis of a Lotka-Volterra competition system with advection

 1 Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130 2 Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, Halifax, Nova Scotia, Canada B3H 4R2, Canada 3 Hanqing Advanced Institute of Economics and Finance, Renmin University of China, No. 59 Zhongguancun Street, Haidian District, Beijing 100872, China

Received  January 2014 Revised  August 2014 Published  October 2014

We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competition. We establish the global existence of bounded classical solutions to the system over one-dimensional finite domains. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bounded steady states in order to model species segregation phenomenon. We then investigate the one-dimensional stationary problem. Through bifurcation theory, we obtain the existence of nonconstant positive steady states, which are small perturbations from the positive equilibrium; we also rigourously study the stability of these bifurcating solutions when diffusion coefficients of the escaper and its competitor are large and small respectively. In the limit of large advection rate, we show that the reaction-advection-diffusion system converges to a shadow system involving the competitor population density and an unknown positive constant. Existence and stability of positive nonconstant solutions to the shadow system have also been obtained through bifurcation theories. Finally, we construct infinitely many single interior transition layers to the shadow system when crowding rate of the escapers and diffusion rate of their interspecific competitors are sufficiently small. The transition-layer solutions can be used to model the interspecific segregation phenomenon.
Citation: Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
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Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system,, J. Math. Biol., 29 (1991), 219. doi: 10.1007/BF00160536. Google Scholar [30] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035. Google Scholar [31] M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion,, Hiroshima Math. J., 14 (1984), 425. Google Scholar [32] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, Journal. Theor. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar [33] W.-M. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (2011), 978. doi: 10.1137/1.9781611971972. Google Scholar [34] W.-M. Ni, P. Polascik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, Trans. Amer. Math. Soc., 353 (2001), 5057. doi: 10.1090/S0002-9947-01-02880-X. Google Scholar [35] W.-M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion,, Discret Cotin Dyn. Syst., 34 (2014), 5271. doi: 10.3934/dcds.2014.34.5271. Google Scholar [36] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar [37] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar [38] Q. Wang, On the steady state of a shadow system to the SKT competition model,, Discrete Contin. Dyn. Syst.-Series B, 19 (2014), 2941. doi: 10.3934/dcdsb.2014.19.2941. Google Scholar [39] X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem,, J. Math. Biol., 66 (2013), 1241. doi: 10.1007/s00285-012-0533-x. Google Scholar [40] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [41] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction,, J. Math. Anal. Appl., 384 (2011), 261. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar [42] Y. Wu and Q. Xu, The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 29 (2011), 367. doi: 10.3934/dcds.2011.29.367. Google Scholar

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##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13. Google Scholar [3] _______, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, Function Spaces, 133 (1993), 9. doi: 10.1007/978-3-663-11336-2_1. Google Scholar [4] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux,, Kinet. Relat. Models, 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar [5] Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 719. doi: 10.3934/dcds.2004.10.719. Google Scholar [6] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations,, SIAM J. Appl. Math., 35 (1978), 1. doi: 10.1137/0135001. Google Scholar [7] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal,, Discrete Contin. Dyn. Syst., 34 (2014), 1701. doi: 10.3934/dcds.2014.34.1701. Google Scholar [8] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar [9] _______, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. and Anal., 52 (1973), 161. Google Scholar [10] P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion,, SIAM J. Appl. Math., 37 (1979), 648. doi: 10.1137/0137048. Google Scholar [11] S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems,, Hiroshima Math. J., 18 (1988), 127. Google Scholar [12] P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497. doi: 10.1016/0022-247X(76)90218-3. Google Scholar [13] J. K. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367. doi: 10.1007/BF03167908. Google Scholar [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981). Google Scholar [15] M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model,, J. Math. Biol., 35 (1996), 177. doi: 10.1007/s002850050049. Google Scholar [16] T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [17] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [18] Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations,, Hiroshima Math. J., 23 (1993), 193. Google Scholar [19] T. Kato, Functional Analysis,, Springer Classics in Mathematics, (1996). Google Scholar [20] K. Kishimoto and H. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains,, J. Differential Equations, 58 (1985), 15. doi: 10.1016/0022-0396(85)90020-8. Google Scholar [21] T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion,, SIAM J. Appl. Math., 71 (2011), 1428. doi: 10.1137/100808381. Google Scholar [22] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79. doi: 10.1006/jdeq.1996.0157. Google Scholar [23] ________, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154 (1999), 157. doi: 10.1006/jdeq.1998.3559. Google Scholar [24] Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discret Contin. Dynam. Systems, 4 (1998), 193. doi: 10.3934/dcds.1998.4.193. Google Scholar [25] Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 435. doi: 10.3934/dcds.2004.10.435. Google Scholar [26] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968). Google Scholar [27] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains,, Publ. Res. Inst. Math. Sci., 19 (1983), 1049. doi: 10.2977/prims/1195182020. Google Scholar [28] M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11 (1981), 621. Google Scholar [29] M. Mimura, S.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system,, J. Math. Biol., 29 (1991), 219. doi: 10.1007/BF00160536. Google Scholar [30] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035. Google Scholar [31] M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion,, Hiroshima Math. J., 14 (1984), 425. Google Scholar [32] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, Journal. Theor. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar [33] W.-M. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (2011), 978. doi: 10.1137/1.9781611971972. Google Scholar [34] W.-M. Ni, P. Polascik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, Trans. Amer. Math. Soc., 353 (2001), 5057. doi: 10.1090/S0002-9947-01-02880-X. Google Scholar [35] W.-M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion,, Discret Cotin Dyn. Syst., 34 (2014), 5271. doi: 10.3934/dcds.2014.34.5271. Google Scholar [36] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar [37] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar [38] Q. Wang, On the steady state of a shadow system to the SKT competition model,, Discrete Contin. Dyn. Syst.-Series B, 19 (2014), 2941. doi: 10.3934/dcdsb.2014.19.2941. Google Scholar [39] X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem,, J. Math. Biol., 66 (2013), 1241. doi: 10.1007/s00285-012-0533-x. Google Scholar [40] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [41] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction,, J. Math. Anal. Appl., 384 (2011), 261. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar [42] Y. Wu and Q. Xu, The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 29 (2011), 367. doi: 10.3934/dcds.2011.29.367. Google Scholar
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