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Qualitative analysis of a Lotka-Volterra competition system with advection. (English) Zbl 1327.92050
Summary: 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.

92D25 Population dynamics (general)
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35B36 Pattern formations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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[1] N. D. Alikakos, \(L^p\) bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4, 827, (1979) · Zbl 0421.35009
[2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3, 13, (1990) · Zbl 0729.35062
[3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, Function Spaces, 133, 9, (1993)
[4] A. Chertock, On a Chemotaxis Model with Saturated Chemotactic Flux,, Kinet. Relat. Models, 5, 51, (2012) · Zbl 1398.92033
[5] Y. S. Choi, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10, 719, (2004) · Zbl 1047.35054
[6] E. Conway, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations,, SIAM J. Appl. Math., 35, 1, (1978) · Zbl 0383.35035
[7] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal,, Discrete Contin. Dyn. Syst., 34, 1701, (2014) · Zbl 1277.35002
[8] M. G. Crandall, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8, 321, (1971) · Zbl 0219.46015
[9] M. G. Crandall, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. and Anal., 52, 161, (1973) · Zbl 0275.47044
[10] P. De Mottoni, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion,, SIAM J. Appl. Math., 37, 648, (1979) · Zbl 0425.35055
[11] S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems,, Hiroshima Math. J., 18, 127, (1988) · Zbl 0703.35092
[12] P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54, 497, (1976) · Zbl 0345.34044
[13] J. K. Hale, Existence and stability of transition layers,, Japan J. Appl. Math., 5, 367, (1988) · Zbl 0669.34027
[14] D. Henry, <em>Geometric Theory of Semilinear Parabolic Equations</em>,, Springer-Verlag, (1981) · Zbl 0456.35001
[15] M. A. Herrero, Chemotactic collapse for the Keller-Segel model,, J. Math. Biol., 35, 177, (1996) · Zbl 0866.92009
[16] T. Hillen, A user’s guidence to PDE models for chemotaxis,, J. Math. Biol., 58, 183, (2009) · Zbl 1161.92003
[17] D. Horstmann, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215, 52, (2005) · Zbl 1085.35065
[18] Y. Kan-on, Existence of nonconstant stable equilibria in competition-diffusion equations,, Hiroshima Math. J., 23, 193, (1993) · Zbl 0823.35090
[19] T. Kato, <em>Functional Analysis</em>,, Springer Classics in Mathematics, (1996)
[20] K. Kishimoto, The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains,, J. Differential Equations, 58, 15, (1985) · Zbl 0599.35080
[21] T. Kolokolnikov, Stability of spiky solutions in a competition model with cross-diffusion,, SIAM J. Appl. Math., 71, 1428, (2011) · Zbl 1259.35018
[22] Y. Lou, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131, 79, (1996) · Zbl 0867.35032
[23] Y. Lou, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154, 157, (1999) · Zbl 0934.35040
[24] Y. Lou, On the global existence of a cross-diffusion system,, Discret Contin. Dynam. Systems, 4, 193, (1998) · Zbl 0960.35049
[25] Y. Lou, On a limiting system in the Lotka-Volterra competition with cross-diffusion,, Discrete Contin. Dyn. Syst., 10, 435, (2004) · Zbl 1174.35360
[26] O. A. Ladyzenskaja, <em>Linear and Quasi-Linear Equations of Parabolic Type</em>,, American Mathematical Society, (1968)
[27] H. Matano, Pattern formation in competition-diffusion systems in nonconvex domains,, Publ. Res. Inst. Math. Sci., 19, 1049, (1983) · Zbl 0548.35063
[28] M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11, 621, (1981) · Zbl 0483.35045
[29] M. Mimura, Effect of domain-shape on coexistence problems in a competition-diffusion system,, J. Math. Biol., 29, 219, (1991) · Zbl 0737.92024
[30] M. Mimura, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9, 49, (1980) · Zbl 0425.92010
[31] M. Mimura, Coexistence problem for two competing species models with density-dependent diffusion,, Hiroshima Math. J., 14, 425, (1984) · Zbl 0562.92011
[32] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, Journal. Theor. Biol., 42, 63, (1973)
[33] W.-M. Ni, <em>The Mathematics of Diffusion</em>,, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, 978, (2011)
[34] W.-M. Ni, Monotonicity of stable solutions in shadow systems,, Trans. Amer. Math. Soc., 353, 5057, (2001) · Zbl 0981.35018
[35] W.-M. Ni, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion,, Discret Cotin Dyn. Syst., 34, 5271, (2014) · Zbl 1326.35019
[36] J. Shi, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246, 2788, (2009) · Zbl 1165.35358
[37] N. Shigesada, Spatial segregation of interacting species,, J. Theoret. Biol., 79, 83, (1979)
[38] Q. Wang, On the steady state of a shadow system to the SKT competition model,, Discrete Contin. Dyn. Syst.-Series B, 19, 2941, (2014) · Zbl 1306.35029
[39] X. Wang, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly’s compactness theorem,, J. Math. Biol., 66, 1241, (2013) · Zbl 1301.92006
[40] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248, 2889, (2010) · Zbl 1190.92004
[41] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction,, J. Math. Anal. Appl., 384, 261, (2011) · Zbl 1241.35028
[42] Y. Wu, The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 29, 367, (2011) · Zbl 1209.35144
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