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March  2021, 14(3): 919-933. doi: 10.3934/dcdss.2020228

Spatio-temporal coexistence in the cross-diffusion competition system

Hirofumi Izuhara 1, and  Shunsuke Kobayashi 2,,

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

* Corresponding author: Shunsuke Kobayashi

Received  January 2019 Revised  September 2019 Published  March 2021 Early access  December 2019

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.

Keywords: Periodic solution, cross-diffusion, center manifold theory, spatial segregation.
Mathematics Subject Classification: Primary: 37G15, 37G05, 37L10; Secondary: 35B10, 35B36, 35K59, 35Q92.
Citation: Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
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.,

[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.

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.,

[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.

Figure 1.  Neutral stability curves in $ (d,\gamma) $-plane. The horizontal axis and the vertical axis mean the value $ d $ and the value $ \gamma $, respectively. The parameter values are $ r_1 = 5 $, $ r_2 = 2 $, $ a_1 = 3 $, $ a_2 = 1 $, $ b_1 = 1 $, $ b_2 = 3 $ and $ L = 1 $
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Figure 2.  Global bifurcation diagrams for (3) with (4) when the value of $ \gamma $ varies. The horizontal axis and the vertical axis mean the value $ d $ and the boundary value $ u(0) $, respectively. Solid curves and dashed curves mean stable branches and unstable ones, respectively. The marks $ \square $ and $ \blacksquare $ indicate a pitchfork bifurcation point and a Hopf bifurcation point, respectively. The parameter values are the same as the ones in Figure 1
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Figure 3.  (a) Enlarged view of the bifurcation diagram for $ \gamma = 1.7 $ in Figure 2 in the neighborhood of the Hopf bifurcation points. The marks $ \bullet $ indicate stable periodic solution branch. The parameter values are the same as the ones in Figure 2. (b) A periodic solution at $ d = 0.01528 $
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