doi: 10.3934/jcd.2021010

On the influence of cross-diffusion in pattern formation

1. 

CMAP, École Polytechnique, route de Saclay, 91120 Palaiseau, France

2. 

Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching bei München, Germany

3. 

Institut für Mathematik und Wissenschaftliches Rechnen, Karl–Franzens Universität Graz, Heinrichstr. 36, 8010 Graz, Austria

* Corresponding author: Maxime Breden

Received  April 2020 Revised  March 2021 Published  April 2021

Fund Project: MB and CK have been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CS has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement No. 754462. Support by INdAM-GNFM is gratefully acknowledged by CS

In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software $\mathtt{pde2path}$. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

Citation: Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, doi: 10.3934/jcd.2021010
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅰ. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.  Google Scholar

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H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction–diffusion systems, Differential and Integral Equations, 3 (1990), 13-75.   Google Scholar

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J. Benson and B. Patterson, Inter-specific territoriality in a Canis hybrid zone: Spatial segregation between wolves, coyotes, and hybrids, Oecologia, 173 (2013), 1539-1550.  doi: 10.1007/s00442-013-2730-8.  Google Scholar

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M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, J. Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033.  Google Scholar

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M. Breden, C. Kuehn and C. Soresina, On the influence of cross-diffusion in pattern formation, supplementary material., Available from: https://github.com/soresina/fullSKT. Google Scholar

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M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction–diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

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J. Cecere, S. Bondì, S. Podofillini, S. Imperio and M. Griggio, et al., Spatial segregation of home ranges between neighbouring colonies in a diurnal raptor, Scientific Reports, 8 (2018). doi: 10.1038/s41598-018-29933-2.  Google Scholar

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L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.  doi: 10.1016/j.jde.2005.08.002.  Google Scholar

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F. Conforto, L. Desvillettes and C. Soresina, About reaction–diffusion systems involving the Holling-type Ⅱ and the Beddington–DeAngelis functional responses for predator–prey models, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp. doi: 10.1007/s00030-018-0515-9.  Google Scholar

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L. DesvillettesT. LepoutreA. Moussa and A. Trescases, On the entropic structure of reaction-cross diffusion systems, Comm. Partial Differential Equations, 40 (2015), 1705-1747.  doi: 10.1080/03605302.2014.998837.  Google Scholar

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L. Desvillettes and C. Soresina, Non-triangular cross-diffusion systems with predator–prey reaction terms, Ric. Mat., 68 (2019), 295-314.  doi: 10.1007/s11587-018-0403-y.  Google Scholar

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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.  doi: 10.1016/j.jmaa.2015.03.078.  Google Scholar

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J. Diamond, Assembly of species communities, in Ecology and Evolution of Communities, Harvard Univ Press, Cambridge, MA, 1975, 342-444. Google Scholar

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T. Dohnal, J. Rademacher, H. Uecker and D. Wetzel, pde2path 2.0: Multi-parameter continuation and periodic domains, in Proceedings of the 8th European Nonlinear Dynamics Conference, ENOC, 2014 (2014). Google Scholar

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S.-I. Ei and M. Mimura, Pattern formation in heterogeneous reaction–diffusion–advection systems with an application to population dynamics, SIAM J. Math. Anal., 21 (1990), 346-361.  doi: 10.1137/0521019.  Google Scholar

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G. GalianoM. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673.  doi: 10.1007/s002110200406.  Google Scholar

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G. GambinoM. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in a 2D domain, Nonlinear Anal. Real World Appl., 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.  Google Scholar

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G. GambinoM. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion, Math. Comput. Simulation, 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.  Google Scholar

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L. T. HoangT. V. Nguyen and T. V. Phan, Gradient estimates and global existence of smooth solutions to a cross-diffusion system, SIAM J. Math. Anal., 47 (2015), 2122-2177.  doi: 10.1137/140981447.  Google Scholar

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R. HoffmanG. Larson and B. Brokes, Habitat segregation of Ambystoma gracile and Ambystoma macrodactylum in mountain ponds and lakes, Mount Rainier National Park, Washington, USA, J. Herpetology, 37 (2003), 24-34.  doi: 10.1670/0022-1511(2003)037[0024:HSOAGA]2.0.CO;2.  Google Scholar

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H. HoiT. Eichler and J. Dittami, Territorial spacing and interspecific competition in three species of reed warblers, Oecologia, 87 (1991), 443-448.  doi: 10.1007/BF00634604.  Google Scholar

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M. IidaM. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

[25]

M. IidaH. Ninomiya and H. Yamamoto, A review on reaction–diffusion approximation, J. Elliptic Parabol. Equ., 4 (2018), 565-600.  doi: 10.1007/s41808-018-0029-y.  Google Scholar

[26]

H. Izuhara and S. Kobayashi, Spatio-temporal coexistence in the cross-diffusion competition system, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 919-933.  doi: 10.3934/dcdss.2020228.  Google Scholar

[27]

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.  Google Scholar

[28]

A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.  Google Scholar

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A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-34219-1.  Google Scholar

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A. JüngelC. Kuehn and L. Trussardi, A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.  doi: 10.1017/S0956792516000346.  Google Scholar

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Y. Kan-On, On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates, Discrete Contin. Dyn. Syst., 40 (2020), 3561-3570.  doi: 10.3934/dcds.2020161.  Google Scholar

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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.  Google Scholar

[33]

C. Kennedy, Site segregation by species of Acanthocephala in fish, with special reference to eels, Anguilla anguilla, Parasitology, 90 (1985), 375-390.  doi: 10.1017/S0031182000051076.  Google Scholar

[34]

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.  Google Scholar

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C. Kuehn, Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs, Appl. Math. Comput., 266 (2015), 656-674.  doi: 10.1016/j.amc.2015.05.120.  Google Scholar

[36]

C. Kuehn, PDE Dynamics. An Introduction, Mathematical Modeling and Computation, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019.  Google Scholar

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C. Kuehn and C. Soresina, Numerical continuation for a fast reaction system and its cross-diffusion limit, SN Partial Differential Equations Appl., 1 (2020). doi: 10.1007/s42985-020-0008-7.  Google Scholar

[38]

S. Levin, Dispersion and population interactions, Amer. Naturalist, 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

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Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka–Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

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Y. LouW.-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.  Google Scholar

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show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅰ. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.  Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction–diffusion systems, Differential and Integral Equations, 3 (1990), 13-75.   Google Scholar

[3]

J. Benson and B. Patterson, Inter-specific territoriality in a Canis hybrid zone: Spatial segregation between wolves, coyotes, and hybrids, Oecologia, 173 (2013), 1539-1550.  doi: 10.1007/s00442-013-2730-8.  Google Scholar

[4]

V. Biktashev and M. Tsyganov, Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law, Scientific Reports, 6 (2016), 1-8.  doi: 10.1038/srep30879.  Google Scholar

[5]

M. Breden and R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof, J. Differential Equations, 264 (2018), 6418-6458.  doi: 10.1016/j.jde.2018.01.033.  Google Scholar

[6]

M. Breden, C. Kuehn and C. Soresina, On the influence of cross-diffusion in pattern formation, supplementary material., Available from: https://github.com/soresina/fullSKT. Google Scholar

[7]

M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction–diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[8]

J. Cecere, S. Bondì, S. Podofillini, S. Imperio and M. Griggio, et al., Spatial segregation of home ranges between neighbouring colonies in a diurnal raptor, Scientific Reports, 8 (2018). doi: 10.1038/s41598-018-29933-2.  Google Scholar

[9]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.  doi: 10.1016/j.jde.2005.08.002.  Google Scholar

[10]

F. Conforto, L. Desvillettes and C. Soresina, About reaction–diffusion systems involving the Holling-type Ⅱ and the Beddington–DeAngelis functional responses for predator–prey models, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp. doi: 10.1007/s00030-018-0515-9.  Google Scholar

[11]

L. DesvillettesT. LepoutreA. Moussa and A. Trescases, On the entropic structure of reaction-cross diffusion systems, Comm. Partial Differential Equations, 40 (2015), 1705-1747.  doi: 10.1080/03605302.2014.998837.  Google Scholar

[12]

L. Desvillettes and C. Soresina, Non-triangular cross-diffusion systems with predator–prey reaction terms, Ric. Mat., 68 (2019), 295-314.  doi: 10.1007/s11587-018-0403-y.  Google Scholar

[13]

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.  Google Scholar

[14]

J. Diamond, Assembly of species communities, in Ecology and Evolution of Communities, Harvard Univ Press, Cambridge, MA, 1975, 342-444. Google Scholar

[15]

T. Dohnal, J. Rademacher, H. Uecker and D. Wetzel, pde2path 2.0: Multi-parameter continuation and periodic domains, in Proceedings of the 8th European Nonlinear Dynamics Conference, ENOC, 2014 (2014). Google Scholar

[16]

S.-I. Ei and M. Mimura, Pattern formation in heterogeneous reaction–diffusion–advection systems with an application to population dynamics, SIAM J. Math. Anal., 21 (1990), 346-361.  doi: 10.1137/0521019.  Google Scholar

[17]

G. GalianoM. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673.  doi: 10.1007/s002110200406.  Google Scholar

[18]

G. GambinoM. C. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in a 2D domain, Nonlinear Anal. Real World Appl., 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.  Google Scholar

[19]

G. GambinoM. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion, Math. Comput. Simulation, 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.  Google Scholar

[20]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[21]

L. T. HoangT. V. Nguyen and T. V. Phan, Gradient estimates and global existence of smooth solutions to a cross-diffusion system, SIAM J. Math. Anal., 47 (2015), 2122-2177.  doi: 10.1137/140981447.  Google Scholar

[22]

R. HoffmanG. Larson and B. Brokes, Habitat segregation of Ambystoma gracile and Ambystoma macrodactylum in mountain ponds and lakes, Mount Rainier National Park, Washington, USA, J. Herpetology, 37 (2003), 24-34.  doi: 10.1670/0022-1511(2003)037[0024:HSOAGA]2.0.CO;2.  Google Scholar

[23]

H. HoiT. Eichler and J. Dittami, Territorial spacing and interspecific competition in three species of reed warblers, Oecologia, 87 (1991), 443-448.  doi: 10.1007/BF00634604.  Google Scholar

[24]

M. IidaM. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.  doi: 10.1007/s00285-006-0013-2.  Google Scholar

[25]

M. IidaH. Ninomiya and H. Yamamoto, A review on reaction–diffusion approximation, J. Elliptic Parabol. Equ., 4 (2018), 565-600.  doi: 10.1007/s41808-018-0029-y.  Google Scholar

[26]

H. Izuhara and S. Kobayashi, Spatio-temporal coexistence in the cross-diffusion competition system, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 919-933.  doi: 10.3934/dcdss.2020228.  Google Scholar

[27]

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.  Google Scholar

[28]

A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.  Google Scholar

[29]

A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-34219-1.  Google Scholar

[30]

A. JüngelC. Kuehn and L. Trussardi, A meeting point of entropy and bifurcations in cross-diffusion herding, European J. Appl. Math., 28 (2017), 317-356.  doi: 10.1017/S0956792516000346.  Google Scholar

[31]

Y. Kan-On, On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates, Discrete Contin. Dyn. Syst., 40 (2020), 3561-3570.  doi: 10.3934/dcds.2020161.  Google Scholar

[32]

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.  Google Scholar

[33]

C. Kennedy, Site segregation by species of Acanthocephala in fish, with special reference to eels, Anguilla anguilla, Parasitology, 90 (1985), 375-390.  doi: 10.1017/S0031182000051076.  Google Scholar

[34]

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.  Google Scholar

[35]

C. Kuehn, Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs, Appl. Math. Comput., 266 (2015), 656-674.  doi: 10.1016/j.amc.2015.05.120.  Google Scholar

[36]

C. Kuehn, PDE Dynamics. An Introduction, Mathematical Modeling and Computation, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019.  Google Scholar

[37]

C. Kuehn and C. Soresina, Numerical continuation for a fast reaction system and its cross-diffusion limit, SN Partial Differential Equations Appl., 1 (2020). doi: 10.1007/s42985-020-0008-7.  Google Scholar

[38]

S. Levin, Dispersion and population interactions, Amer. Naturalist, 108 (1974), 207-228.  doi: 10.1086/282900.  Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka–Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

[42]

Y. LouW.-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.  Google Scholar

[43]

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.  Google Scholar

[44]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  doi: 10.32917/hmj/1206133994.  Google Scholar

[45]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  doi: 10.1007/BF00276035.  Google Scholar

[46]

T. MoriT. Suzuki and S. Yotsutani, Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation, Math. Models Methods Appl. Sci., 28 (2018), 2191-2210.  doi: 10.1142/S0218202518400122.  Google Scholar

[47]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[48]

W.-M. NiY. 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.  Google Scholar

[49]

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Figure 1.  Sign of the quantities $ \alpha $ and $ \beta $ in the weak competition regime (2), depending on the value of $ r_1/r_2 $
Figure 2.  Sign of the quantities $ \alpha $ and $ \beta $ in the strong competition casek__ge (3), depending on the value of $ r_1/r_2 $
Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $). (A) "Usual" bifurcation diagram with respect to $ v(0) $. (B) Bifurcation diagram with respect to $ ||u||_{L^2} $">Figure 3.  Bifurcation diagrams represented with different quantities in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $). (A) "Usual" bifurcation diagram with respect to $ v(0) $. (B) Bifurcation diagram with respect to $ ||u||_{L^2} $
Figure 4.  Different types of stable solutions on the blue and red branches of the bifurcation diagram 3. The densities of $ u $ and $ v $ on the domain are denoted with black and blue lines respectively
Table 1, with $ d_{12} = 3 $). Notice that the range of values of $ d $ for which non-trivial solutions exist is getting smaller and smaller when $ d_{21} $ increases (especially compared to Figure 3b)">Figure 5.  Bifurcation diagrams for different values of the cross-diffusion coefficient $ d_{21} $ in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $). Notice that the range of values of $ d $ for which non-trivial solutions exist is getting smaller and smaller when $ d_{21} $ increases (especially compared to Figure 3b)
Figure 5a. Figures (B) and (D) refer to $ d_{21} = 0.055 $ and correspond to solutions on the pastel blue and pastel red branches of the bifurcation diagram in Figure 5b">Figure 6.  Different types of stable solutions coexisting with the homogeneous one. Species $ u $ and $ v $ on the domain are denoted with black and blue lines respectively. Figures (A) and (C) refer to $ d_{21} = 0.045 $ and correspond to solutions on the pastel blue and red branches of the bifurcation diagram in Figure 5a. Figures (B) and (D) refer to $ d_{21} = 0.055 $ and correspond to solutions on the pastel blue and pastel red branches of the bifurcation diagram in Figure 5b
Table 1, with $ d_{12} = 3 $). We start in Figure 7a with a zoom in of Figure 3b, and then slowly increase $ d_{21} $">Figure 7.  Bifurcation diagrams for different values of the cross-diffusion coefficient $ d_{21} $ in the weak competition case (first parameter set in Table 1, with $ d_{12} = 3 $). We start in Figure 7a with a zoom in of Figure 3b, and then slowly increase $ d_{21} $
Figure 8.  Solutions belonging to different branches when $ d_{12} = 1000 $. The species $ u $ and $ v $ are denoted with black and blue lines respectively, while the red line corresponds to $ uv $. As in the bifurcation diagrams, thin lines indicate unstable solutions, while thicker lines corresponds to stable ones. The red dotted line represents the product $ uv $
Table 1, with $ d_{12} = 3 $)">Figure 9.  Bifurcation diagrams for different values of the cross-diffusion parameter $ d_{21} $ in the strong competition case (second parameter set in Table 1, with $ d_{12} = 3 $)
Table 1, with $ d_{12} = 3 $">Figure 10.  Evolution of the first bifurcation branch for increasing values of $ d_{21}\in [0,1000] $ (darker the blue, greater the value), corresponding to the second parameter set in Table 1, with $ d_{12} = 3 $
Table 1 and $ d_{12}\; = \; d_{21} = 100 $). (A) Bifurcation diagram with respect to the parameter $ d $. Yellow points indicate the positions on the branches at which solutions are shown. (B)–(D) Solutions $ u(x),\;v(x) $ for different values of $ d $ (black lines correspond to species $ u $, while blue lines to species $ v $)">Figure 11.  Bifurcation diagram and some solutions (third parameter set in Table 1 and $ d_{12}\; = \; d_{21} = 100 $). (A) Bifurcation diagram with respect to the parameter $ d $. Yellow points indicate the positions on the branches at which solutions are shown. (B)–(D) Solutions $ u(x),\;v(x) $ for different values of $ d $ (black lines correspond to species $ u $, while blue lines to species $ v $)
Table 1, with $ d_{21} = 0 $">Figure 12.  Evolution of the first bifurcation branch for increasing values of $ d_{12}\in [0,3] $ (darker the blue, greater the value), corresponding to the fourth parameter set in Table 1, with $ d_{21} = 0 $
Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $. (B) Strong competition case with $ d = 0.05 $ and the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $">Figure 13.  Bifurcation diagrams with bifurcation parameter $ r_1 $. (A) Weak competition case with $ d = 0.005 $ and the other parameter values as in the first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $. (B) Strong competition case with $ d = 0.05 $ and the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $
Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (A), (C) stable solutions with $ r_1 = 7.5 $, (E) stable solution with $ r_1 = 15 $. Strong competition case ($ d = 0.05 $, the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (B), (D) stable solutions with $ r_1 = 10 $, (F) stable solution with $ r_1 = 18 $. The colors of the branches refer to Figure 13">Figure 14.  Stable non-homogeneous solutions appearing beyond the usual weak or strong competitions regimes (species $ u $ and $ v $ are denoted with black and blue lines respectively). Weak competition case ($ d = 0.005 $, the other parameter values as in the first parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (A), (C) stable solutions with $ r_1 = 7.5 $, (E) stable solution with $ r_1 = 15 $. Strong competition case ($ d = 0.05 $, the other parameter values as in the second parameter set in Table 1, with $ d_{12} = 3 $ and $ d_{21} = 0 $): (B), (D) stable solutions with $ r_1 = 10 $, (F) stable solution with $ r_1 = 18 $. The colors of the branches refer to Figure 13
Table 1 (weak competition case), with $ d_{12} = 3 $, $ d_{21} = 0 $, $ d_{11} = 0 $ and different values of the self-diffusion coefficient $ d_{22} $">Figure 15.  Bifurcation diagrams with bifurcation parameter $ d $ obtained with the first parameter set in Table 1 (weak competition case), with $ d_{12} = 3 $, $ d_{21} = 0 $, $ d_{11} = 0 $ and different values of the self-diffusion coefficient $ d_{22} $
Table 1.  The parameter sets used in the numerical simulations presented in this section
$ r_1 $ $ r_2 $ $ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ regime α β
5 2 3 3 1 1 weak competition + - [5,24,27]
2 5 1 1 0.5 3 strong competition + 0 [27]
15/2 16/7 4 2 6 1 weak competition - +
5 5 2 3 5 4 strong competition + +
$ r_1 $ $ r_2 $ $ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ regime α β
5 2 3 3 1 1 weak competition + - [5,24,27]
2 5 1 1 0.5 3 strong competition + 0 [27]
15/2 16/7 4 2 6 1 weak competition - +
5 5 2 3 5 4 strong competition + +
Table 2.  Convergence of the first three bifurcation points, numerically detected, to the predicted limited values $ \alpha/(v_*\lambda_k) $ when $ d_{12} $ increases ($ d_{21} = 0 $)
$ d_{12} $ $ d^1_{ \text{bif}} $ $ d^2_{ \text{bif}} $ $ d^3_{ \text{bif}} $
3 0.0328 0.0205 0.0113
10 0.0762 0.0273 0.0131
100 0.1190 0.0311 0.0139
1000 0.1258 0.0315 0.0140
$ d^k_{ \text{bif},\infty} $ 0.1267 0.0317 0.0141
$ d_{12} $ $ d^1_{ \text{bif}} $ $ d^2_{ \text{bif}} $ $ d^3_{ \text{bif}} $
3 0.0328 0.0205 0.0113
10 0.0762 0.0273 0.0131
100 0.1190 0.0311 0.0139
1000 0.1258 0.0315 0.0140
$ d^k_{ \text{bif},\infty} $ 0.1267 0.0317 0.0141
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