doi: 10.3934/dcdss.2022120
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Hopf bifurcations in the full SKT model and where to find them

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

* Corresponding author: Cinzia Soresina

Received  April 2021 Revised  March 2022 Early access May 2022

In this paper, we consider the Shigesada–Kawasaki–Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator–inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies as the interspecific competition pressure increases. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.

Citation: Cinzia Soresina. Hopf bifurcations in the full SKT model and where to find them. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022120
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.

[2]

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

[3]

M. BeckJ. KnoblochD. J. B. LloydB. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972.  doi: 10.1137/080713306.

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M. Breden, Computer-assisted proofs for some nonlinear diffusion problems, Commun. Nonlinear Sci. Numer. Simul., 109 (2022), Paper No. 106292, 22 pp. doi: 10.1016/j.cnsns.2022.106292.

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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.

[6]

M. BredenC. Kuehn and C. Soresina, On the influence of cross-diffusion in pattern formation, J. Comput. Dyn., 8 (2021), 213-240.  doi: 10.3934/jcd.2021010.

[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.

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J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E, 73 (2006), 056211, 15 pp. doi: 10.1103/PhysRevE.73.056211.

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J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift–Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688.  doi: 10.1016/j.physleta.2006.08.072.

[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), Paper No. 24, 39 pp. doi: 10.1007/s00030-018-0515-9.

[11]

P. CoulletC. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069.  doi: 10.1103/PhysRevLett.84.3069.

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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.

[13]

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.

[14]

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.

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J. Diamond, Assembly of species communities, in Ecology and Evolution of Communities (eds. M. Cody and D. J.M.), Cambridge, Mass: Harvard Univ Press, 1975,342–444.

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T. Dohnal, J. Rademacher, H. Uecker and D. Wetzel, $ {\texttt{pde2path}} $ 2.0: Multi-parameter continuation and periodic domains, in Proceedings of the 8th European Nonlinear Dynamics Conference, ENOC, vol. 2014, 2014.

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N. Ehstand, C. Kuehn and C. Soresina, Numerical continuation for fractional PDEs: Sharp teeth and bloated snakes, Commun. Nonlinear Sci. Numer. Simul., 98 (2021), Paper No. 105762, 23 pp. doi: 10.1016/j.cnsns.2021.105762.

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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.

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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.

[20]

G. GambinoM. C. LombardoS. Lupo and M. Sammartino, Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion, Ric. Mat., 65 (2016), 449-467.  doi: 10.1007/s11587-016-0267-y.

[21]

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.

[22]

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.

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin Heidelberg, Germany, 1981.

[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.

[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.

[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.

[27]

H. Izuhara and M. Mimura, Reaction-diffusion system approximation to the cross-diffusion competition system, Hiroshima Math. J., 38 (2008), 315-347. 

[28]

A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.

[29]

A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, Springer, 2016. doi: 10.1007/978-3-319-34219-1.

[30]

Y. Kan-On, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. 

[31]

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.

[32]

C. Kuehn, PDE Dynamics: An Introduction, SIAM, 2019.

[33]

C. Kuehn, N. Berglund, C. Bick, M. Engel, T. Hurth, A. Iuorio and C. Soresina, A general view on double limits in differential equations, Phys. D, 431 (2022), Paper No. 133105, 26 pp. doi: 10.1016/j.physd.2021.133105.

[34]

C. Kuehn and C. Soresina, Cross-diffusion induced instability on networks, in preparation.

[35]

C. Kuehn and C. Soresina, Numerical continuation for a fast reaction system and its cross-diffusion limit, Partial Differ. Equ. Appl., 1 (2020), Paper No. 7, 26 pp. doi: 10.1007/s42985-020-0008-7.

[36]

S. A. Levin, Dispersion and population interactions, The American Naturalist, 108 (1974), 207-228.  doi: 10.1086/282900.

[37]

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.

[38]

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.

[39]

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.

[40]

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.

[41]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. 

[42]

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.

[43]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[44]

U. Prüfert, $ {\texttt{OOPDE}} $ - an object oriented approach to finite elements in MATLAB, 2014, Quickstart Guide.

[45]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[46]

C. Soresina, Supplementary material, Matlab scripts for the bifurcation diagrams at https://github.com/soresina/fullSKT, 2021, Accessed March 16, 2021.

[47]

C. Soresina, Supplementary material, Matlab scripts for the Stuart–Landau and Hopf coefficients at https://github.com/soresina/fullSKT-SL-H, 2021, Accessed April 22, 2021.

[48]

H. Uecker, Hopf bifurcation and time periodic orbits with $ {\texttt{pde2path}} $ – algorithms and applications, Commun. Comput. Phys., 25 (2019), 812-852.  doi: 10.4208/cicp.oa-2017-0181.

[49]

H. Uecker, Continuation and bifurcation in nonlinear PDEs–Algorithms, applications, and experiments, Jahresber. Dtsch. Math.-Ver., 124 (2022), 43-80.  doi: 10.1365/s13291-021-00241-5.

[50]

H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2021. doi: 10.1137/1.9781611976618.

[51]

H. UeckerD. Wetzel and J. D. M. Rademacher, $ {\texttt{pde2path}} $ - A Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.

[52]

E. Wilson, Sociobiology: The New Synthesis, Cambridge: Harvard, 1975.

[53]

D. J. WollkindV. S. Manoranjan and L. Zhang, Weakly nonlinear stability analyses of prototype reaction-diffusion model equations, SIAM Rev., 36 (1994), 176-214.  doi: 10.1137/1036052.

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.

[2]

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

[3]

M. BeckJ. KnoblochD. J. B. LloydB. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972.  doi: 10.1137/080713306.

[4]

M. Breden, Computer-assisted proofs for some nonlinear diffusion problems, Commun. Nonlinear Sci. Numer. Simul., 109 (2022), Paper No. 106292, 22 pp. doi: 10.1016/j.cnsns.2022.106292.

[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.

[6]

M. BredenC. Kuehn and C. Soresina, On the influence of cross-diffusion in pattern formation, J. Comput. Dyn., 8 (2021), 213-240.  doi: 10.3934/jcd.2021010.

[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.

[8]

J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E, 73 (2006), 056211, 15 pp. doi: 10.1103/PhysRevE.73.056211.

[9]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift–Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688.  doi: 10.1016/j.physleta.2006.08.072.

[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), Paper No. 24, 39 pp. doi: 10.1007/s00030-018-0515-9.

[11]

P. CoulletC. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069.  doi: 10.1103/PhysRevLett.84.3069.

[12]

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.

[13]

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.

[14]

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.

[15]

J. Diamond, Assembly of species communities, in Ecology and Evolution of Communities (eds. M. Cody and D. J.M.), Cambridge, Mass: Harvard Univ Press, 1975,342–444.

[16]

T. Dohnal, J. Rademacher, H. Uecker and D. Wetzel, $ {\texttt{pde2path}} $ 2.0: Multi-parameter continuation and periodic domains, in Proceedings of the 8th European Nonlinear Dynamics Conference, ENOC, vol. 2014, 2014.

[17]

N. Ehstand, C. Kuehn and C. Soresina, Numerical continuation for fractional PDEs: Sharp teeth and bloated snakes, Commun. Nonlinear Sci. Numer. Simul., 98 (2021), Paper No. 105762, 23 pp. doi: 10.1016/j.cnsns.2021.105762.

[18]

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.

[19]

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.

[20]

G. GambinoM. C. LombardoS. Lupo and M. Sammartino, Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion, Ric. Mat., 65 (2016), 449-467.  doi: 10.1007/s11587-016-0267-y.

[21]

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.

[22]

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.

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin Heidelberg, Germany, 1981.

[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.

[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.

[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.

[27]

H. Izuhara and M. Mimura, Reaction-diffusion system approximation to the cross-diffusion competition system, Hiroshima Math. J., 38 (2008), 315-347. 

[28]

A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15.

[29]

A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, Springer, 2016. doi: 10.1007/978-3-319-34219-1.

[30]

Y. Kan-On, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. 

[31]

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.

[32]

C. Kuehn, PDE Dynamics: An Introduction, SIAM, 2019.

[33]

C. Kuehn, N. Berglund, C. Bick, M. Engel, T. Hurth, A. Iuorio and C. Soresina, A general view on double limits in differential equations, Phys. D, 431 (2022), Paper No. 133105, 26 pp. doi: 10.1016/j.physd.2021.133105.

[34]

C. Kuehn and C. Soresina, Cross-diffusion induced instability on networks, in preparation.

[35]

C. Kuehn and C. Soresina, Numerical continuation for a fast reaction system and its cross-diffusion limit, Partial Differ. Equ. Appl., 1 (2020), Paper No. 7, 26 pp. doi: 10.1007/s42985-020-0008-7.

[36]

S. A. Levin, Dispersion and population interactions, The American Naturalist, 108 (1974), 207-228.  doi: 10.1086/282900.

[37]

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.

[38]

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.

[39]

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.

[40]

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.

[41]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. 

[42]

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.

[43]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[44]

U. Prüfert, $ {\texttt{OOPDE}} $ - an object oriented approach to finite elements in MATLAB, 2014, Quickstart Guide.

[45]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[46]

C. Soresina, Supplementary material, Matlab scripts for the bifurcation diagrams at https://github.com/soresina/fullSKT, 2021, Accessed March 16, 2021.

[47]

C. Soresina, Supplementary material, Matlab scripts for the Stuart–Landau and Hopf coefficients at https://github.com/soresina/fullSKT-SL-H, 2021, Accessed April 22, 2021.

[48]

H. Uecker, Hopf bifurcation and time periodic orbits with $ {\texttt{pde2path}} $ – algorithms and applications, Commun. Comput. Phys., 25 (2019), 812-852.  doi: 10.4208/cicp.oa-2017-0181.

[49]

H. Uecker, Continuation and bifurcation in nonlinear PDEs–Algorithms, applications, and experiments, Jahresber. Dtsch. Math.-Ver., 124 (2022), 43-80.  doi: 10.1365/s13291-021-00241-5.

[50]

H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2021. doi: 10.1137/1.9781611976618.

[51]

H. UeckerD. Wetzel and J. D. M. Rademacher, $ {\texttt{pde2path}} $ - A Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.

[52]

E. Wilson, Sociobiology: The New Synthesis, Cambridge: Harvard, 1975.

[53]

D. J. WollkindV. S. Manoranjan and L. Zhang, Weakly nonlinear stability analyses of prototype reaction-diffusion model equations, SIAM Rev., 36 (1994), 176-214.  doi: 10.1137/1036052.

Figure 1.  Neutral stability curves for $ \lambda_k, \, k = 1, \dots, 5 $ for different values of the cross-diffusion coefficient $ d_{21} $. The white area denotes the stability of the homogenous steady-state $ (u_*, v_*) $, while in the grey region the homogeneous steady state is destabilized and stable non-homogeneous stationary solutions appear. The green dotted horizontal line marks the "usual" value of parameter $ d_{12} $. The doubly-degenerate point $ (\hat{d}, \hat{d}_{12}) $ corresponds to the intersection of the neutral stability curves associated to the 1- and 2-modes
Figure 2.  Bifurcation diagrams for different values of the cross-diffusion coefficient $ d_{21} $. The bifurcation parameter is the standard diffusion coefficient $ d $, while on the $ y $-axis we have $ v(0) $. Thick/thin lines denotes stable/unstable stationary solutions. Circles/crosses/diamonds mark pitchfork/fold/Hopf bifurcations points
Figure 3.  Qualitative representation of the bifurcation structure at the first and second bifurcation points. Numbers along the branches indicate the number of eigenvalues with positive real part detected by the continuation software $ {\texttt{pde2path}}$
Figure 4.  Qualitative representation of the bifurcation structure close to the bifurcation point, predicted by the Stuart–Landau equation (15)
Figure 5.  Sign of $ L $ along the neutral stability curves of $ \lambda_1 $ and $ \lambda_2 $ in the $ (d, d_{21}) $-plane, with $ d_{12} = 2 $ (left) and $ d_{12} = 3 $ (right). The remain parameter values are listed in Table 1
Figure 6.  Sign of $ L $ for $ \lambda_1 $ (left) and $ \lambda_2 $ (right) in the $ (d_{12}, d_{21}) $-plane
Figure 7.  Sign of the coefficient $ L $ along the neutral stability curves for $ \lambda_k, \, k = 1, \dots, 6 $ for different values of the cross-diffusion coefficient $ d_{21} $. Colours appears on the curves related to the first two modes, ( $ L>0 $, $ L<0 $), while the other modes are marked in gray
Figure 8.  Qualitative representation of sign of $ L $ ( $ L>0 $, $ L<0 $), predicted by the Stuart–Landau equation (15), along the neutral stability curves close to the doubly degenerate point. The region in which time-periodic spatial patter may appear is marked in yellow (solid line denotes stable solutions, dotted line unstable ones)
Table 1.  The parameter sets used in the numerical simulations, relevant to the weak competition regime. The sign of the quantity $ \alpha $ and $ \beta $ in (5) is reported
$ \Omega $ $ r_1 $ $ r_2 $ $ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \alpha $ $ \beta $
(0, 1) 5 2 3 3 1 1 $ + $ $ - $
$ \Omega $ $ r_1 $ $ r_2 $ $ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \alpha $ $ \beta $
(0, 1) 5 2 3 3 1 1 $ + $ $ - $
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