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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 |
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.
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] |
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. |
[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. |
[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. 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. Breden, J.-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. 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. |
[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. |
[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. |
[11] |
L. Desvillettes, T. Lepoutre, A. 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. |
[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. |
[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. |
[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. |
[17] |
G. Galiano, M. 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. |
[18] |
G. Gambino, M. 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. |
[19] |
G. Gambino, M. 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. |
[20] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
doi: 10.1007/BFb0089647. |
[21] |
L. T. Hoang, T. 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. |
[22] |
R. Hoffman, G. 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. |
[23] |
H. Hoi, T. Eichler and J. Dittami,
Territorial spacing and interspecific competition in three species of reed warblers, Oecologia, 87 (1991), 443-448.
doi: 10.1007/BF00634604. |
[24] |
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. |
[25] |
M. Iida, H. 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.
doi: 10.32917/hmj/1220619462. |
[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. |
[29] |
A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, SpringerBriefs in Mathematics, Springer, Cham, 2016.
doi: 10.1007/978-3-319-34219-1. |
[30] |
A. Jüngel, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
C. Kuehn, PDE Dynamics. An Introduction, Mathematical Modeling and Computation, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. |
[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. |
[38] |
S. Levin,
Dispersion and population interactions, Amer. Naturalist, 108 (1974), 207-228.
doi: 10.1086/282900. |
[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. |
[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. |
[41] |
Y. Lou, W.-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. |
[42] |
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. |
[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. |
[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. |
[45] |
M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[46] |
T. Mori, T. 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. |
[47] |
W.-M. Ni,
Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[48] |
W.-M. Ni, Y. 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. |
[49] |
F. Palomares, N. Fernández, S. Roques, C. Chávez, L. Silveira, C. Keller and B. Adrados, Fine-scale habitat segregation between two ecologically similar top predators, PLoS one, 11 (2016).
doi: 10.1371/journal.pone.0155626. |
[50] |
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. |
[51] |
U. Suwanvecho and W. Brockelman,
Interspecific territoriality in gibbons (Hylobates lar and H. pileatus) and its effects on the dynamics of interspecies contact zones, Primates, 53 (2012), 97-108.
doi: 10.1007/s10329-011-0284-0. |
[52] |
C. Tian, Z. Lin and M. Pedersen,
Instability induced by cross-diffusion in reaction-diffusion systems, Nonlinear Anal. Real World Appl., 11 (2010), 1036-1045.
doi: 10.1016/j.nonrwa.2009.01.043. |
[53] |
H. Uecker,
Hopf bifurcation and time periodic orbits with pde2path – Algorithms and applications, Commun. Comput. Phys., 25 (2019), 812-852.
doi: 10.4208/cicp.oa-2017-0181. |
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H. Uecker, D. Wetzel and J. D. M. Rademacher,
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. |
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L. Wauters, G. Tosi and J. Gurnell, A review of the competitive effects of alien grey squirrels on behaviour, activity and habitat use of red squirrels in mixed, deciduous woodland in Italy, Hystrix Italian J. Mammalogy, 16 (2005).
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E. Wilson, Sociobiology: The New Synthesis, Cambridge, 1975. Google Scholar |
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] |
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. |
[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. |
[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. 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. Breden, J.-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. 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. |
[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. |
[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. |
[11] |
L. Desvillettes, T. Lepoutre, A. 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. |
[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. |
[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. |
[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. |
[17] |
G. Galiano, M. 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. |
[18] |
G. Gambino, M. 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. |
[19] |
G. Gambino, M. 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. |
[20] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
doi: 10.1007/BFb0089647. |
[21] |
L. T. Hoang, T. 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. |
[22] |
R. Hoffman, G. 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. |
[23] |
H. Hoi, T. Eichler and J. Dittami,
Territorial spacing and interspecific competition in three species of reed warblers, Oecologia, 87 (1991), 443-448.
doi: 10.1007/BF00634604. |
[24] |
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. |
[25] |
M. Iida, H. 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.
doi: 10.32917/hmj/1220619462. |
[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. |
[29] |
A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, SpringerBriefs in Mathematics, Springer, Cham, 2016.
doi: 10.1007/978-3-319-34219-1. |
[30] |
A. Jüngel, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
C. Kuehn, PDE Dynamics. An Introduction, Mathematical Modeling and Computation, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. |
[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. |
[38] |
S. Levin,
Dispersion and population interactions, Amer. Naturalist, 108 (1974), 207-228.
doi: 10.1086/282900. |
[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. |
[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. |
[41] |
Y. Lou, W.-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. |
[42] |
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. |
[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. |
[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. |
[45] |
M. Mimura and K. Kawasaki,
Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[46] |
T. Mori, T. 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. |
[47] |
W.-M. Ni,
Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[48] |
W.-M. Ni, Y. 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. |
[49] |
F. Palomares, N. Fernández, S. Roques, C. Chávez, L. Silveira, C. Keller and B. Adrados, Fine-scale habitat segregation between two ecologically similar top predators, PLoS one, 11 (2016).
doi: 10.1371/journal.pone.0155626. |
[50] |
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. |
[51] |
U. Suwanvecho and W. Brockelman,
Interspecific territoriality in gibbons (Hylobates lar and H. pileatus) and its effects on the dynamics of interspecies contact zones, Primates, 53 (2012), 97-108.
doi: 10.1007/s10329-011-0284-0. |
[52] |
C. Tian, Z. Lin and M. Pedersen,
Instability induced by cross-diffusion in reaction-diffusion systems, Nonlinear Anal. Real World Appl., 11 (2010), 1036-1045.
doi: 10.1016/j.nonrwa.2009.01.043. |
[53] |
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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 |
0.1267 | 0.0317 | 0.0141 |
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 |
0.1267 | 0.0317 | 0.0141 |
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