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Numerical and mathematical analysis of blow-up problems for a stochastic differential equation
Spatio-temporal coexistence in the cross-diffusion competition system
1. | Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadainishi Miyazaki, 889-2192, Japan |
2. | Graduate School of Science and Technology, Meiji University, Kanagawa 214-8571, Japan |
We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.
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Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D, 29 (1988), 257-282.
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Kuramoto-Sivashinsky dynamics on the center-unstable manifold, SIAM J. Appl. Math., 49 (1988), 676-691.
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Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
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F. Conforto and L. Desvillettes,
Rigorous passage to the limit in a system of reaction-diffusion equations towards a system including cross diffusion, Commun. Math. Sci., 12 (2014), 457-472.
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L. Desvillettes and A. Trescases,
New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59.
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E. J. Doedel, B. E. Oldman, A. R. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, R. Paffenroth, B. Sandstede, X. Wang and C. Zhang, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations., Google Scholar |
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H. Fujii, M. Mimura and Y. Nishiura,
A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Physica D, 5 (1982), 1-42.
doi: 10.1016/0167-2789(82)90048-3. |
[10] |
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimentional Dynamics Systems, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011.
doi: 10.1007/978-0-85729-112-7. |
[11] |
M. W. Hirsch,
Differential equations and convergence almost everywhere in strongly monotone semiflow, Nonlinear Partial Differential Equations (Durham, N.H., 1982), Contemp. Math., Amer. Math. Soc., Providence, 17 (1983), 267-285.
|
[12] |
M. Iida, M. Mimura and H. Ninomiya,
Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[13] |
H. Izuhara and M. Mimura,
Reaction-diffusion system approximation to the cross-diffusion competition system, Hiroshima Math. J., 38 (2008), 315-347.
doi: 10.32917/hmj/1220619462. |
[14] |
Y. Kan-on,
Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.
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[15] |
K. Kishimoto and H. F. Weinberger,
The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential equations, 58 (1985), 15-21.
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[16] |
Y. Kuramoto and T. Tsuzuki,
Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
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[17] |
K. Kuto and Y. Yamada,
Positive solutions for Lotka-Volterra competition systems with large cross-diffusion, Appl. Anal., 89 (2010), 1037-1066.
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Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
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Y. Lou and W.-M. Ni,
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Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999) 157–190.
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[21] |
Y. Lou, W.-M. Ni and Y. P. Wu,
On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[22] |
Y. Lou, W.-M. Ni and S. Yotsutani,
Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.
doi: 10.3934/dcds.2015.35.1589. |
[23] |
Y. Low and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[24] |
H. Matano and M. Mimura,
Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[25] |
M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[26] |
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), 425-449.
doi: 10.32917/hmj/1206133048. |
[27] |
W.-M. Ni, Y. P. Wu and Q. Xu,
The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
Y. P. Wu,
The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[30] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
show all references
References:
[1] |
D. Armbruster, J. Guckenheimer and P. Holmes,
Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D, 29 (1988), 257-282.
doi: 10.1016/0167-2789(88)90032-2. |
[2] |
D. Armbruster, J. Guckenheimer and P. Holmes,
Kuramoto-Sivashinsky dynamics on the center-unstable manifold, SIAM J. Appl. Math., 49 (1988), 676-691.
doi: 10.1137/0149039. |
[3] |
J. Carr, Application of Centre Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981. |
[4] |
L. Chen and A. Jüngel,
Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322.
doi: 10.1137/S0036141003427798. |
[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), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[6] |
F. Conforto and L. Desvillettes,
Rigorous passage to the limit in a system of reaction-diffusion equations towards a system including cross diffusion, Commun. Math. Sci., 12 (2014), 457-472.
doi: 10.4310/CMS.2014.v12.n3.a3. |
[7] |
L. Desvillettes and A. Trescases,
New results for triangular reaction cross diffusion system, J. Math. Anal. Appl., 430 (2015), 32-59.
doi: 10.1016/j.jmaa.2015.03.078. |
[8] |
E. J. Doedel, B. E. Oldman, A. R. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, R. Paffenroth, B. Sandstede, X. Wang and C. Zhang, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations., Google Scholar |
[9] |
H. Fujii, M. Mimura and Y. Nishiura,
A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Physica D, 5 (1982), 1-42.
doi: 10.1016/0167-2789(82)90048-3. |
[10] |
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimentional Dynamics Systems, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011.
doi: 10.1007/978-0-85729-112-7. |
[11] |
M. W. Hirsch,
Differential equations and convergence almost everywhere in strongly monotone semiflow, Nonlinear Partial Differential Equations (Durham, N.H., 1982), Contemp. Math., Amer. Math. Soc., Providence, 17 (1983), 267-285.
|
[12] |
M. Iida, M. Mimura and H. Ninomiya,
Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[13] |
H. Izuhara and M. Mimura,
Reaction-diffusion system approximation to the cross-diffusion competition system, Hiroshima Math. J., 38 (2008), 315-347.
doi: 10.32917/hmj/1220619462. |
[14] |
Y. Kan-on,
Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.
doi: 10.32917/hmj/1206392779. |
[15] |
K. Kishimoto and H. F. Weinberger,
The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential equations, 58 (1985), 15-21.
doi: 10.1016/0022-0396(85)90020-8. |
[16] |
Y. Kuramoto and T. Tsuzuki,
Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
doi: 10.1143/PTP.55.356. |
[17] |
K. Kuto and Y. Yamada,
Positive solutions for Lotka-Volterra competition systems with large cross-diffusion, Appl. Anal., 89 (2010), 1037-1066.
doi: 10.1080/00036811003627534. |
[18] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[19] |
Y. Lou and W.-M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[20] |
Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999) 157–190.
doi: 10.1006/jdeq.1998.3559. |
[21] |
Y. Lou, W.-M. Ni and Y. P. Wu,
On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[22] |
Y. Lou, W.-M. Ni and S. Yotsutani,
Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.
doi: 10.3934/dcds.2015.35.1589. |
[23] |
Y. Low and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[24] |
H. Matano and M. Mimura,
Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[25] |
M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[26] |
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), 425-449.
doi: 10.32917/hmj/1206133048. |
[27] |
W.-M. Ni, Y. P. Wu and Q. Xu,
The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
Y. P. Wu,
The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[30] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |



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