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October  2022, 17(5): 719-752. doi: 10.3934/nhm.2022024

Martingale solutions of stochastic nonlocal cross-diffusion systems

Mostafa Bendahmane 1, and  Kenneth H. Karlsen 2,

1. 

Institut de Mathématiques de Bordeaux UMR CNRS 525, Université de Bordeaux, F-33076 Bordeaux Cedex, France
2. 

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

The work of Bendahmane is supported by the Fincome project

Received  March 2022 Published  October 2022 Early access  June 2022

Fund Project: The work of Karlsen is supported by the Research Council of Norway ("Stochastic Conservation Laws", 250674/F20)

We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.

Keywords: Stochastic partial differential equation, cross-diffusion system, existence, martingale solution, population dynamics.
Mathematics Subject Classification: Primary: 60H15, 35K57; Secondary: 35M10, 35A05.
Citation: Mostafa Bendahmane, Kenneth H. Karlsen. Martingale solutions of stochastic nonlocal cross-diffusion systems. Networks and Heterogeneous Media, 2022, 17 (5) : 719-752. doi: 10.3934/nhm.2022024
References:
[1]

B. Ainseba, M. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.

[2]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion, Comput. Math. Appl., 70 (2015), 132-157.  doi: 10.1016/j.camwa.2015.04.021.

[3]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, Pattern formation for a reaction diffusion system with constant and cross diffusion, In Numerical Mathematics and Advanced Applications—ENUMATH 2013, 103 (2015), 153–161.

[4]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, Remarks about spatially structured SI model systems with cross diffusion, Contributions to Partial Differential Equations and Applications, 47 (2019), 43-64. 

[5]

M. Bendahmane, Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Netw. Heterog. Media, 3 (2008), 863-879.  doi: 10.3934/nhm.2008.3.863.

[6]

M. Bendahmane, K. H. Karlsen and J. M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. Methods Appl. Sci., 17 (2007), 783–804. doi: 10.1142/S0218202507002108.

[7]

M. Bendahmane, T. Lepoutre, A. Marrocco and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, J. Math. Pures Appl., 92 (2009), 651-667.  doi: 10.1016/j.matpur.2009.05.003.

[8]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[9]

H. Bessaih, Martingale solutions for stochastic Euler equations, Stochastic Anal. Appl., 17 (1999), 713-725.  doi: 10.1080/07362999908809631.

[10]

M. D. Chekroun, E. Park and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926-2972.  doi: 10.1016/j.jde.2015.10.022.

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[12]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[13]

A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab., 44 (2016), 1916-1955.  doi: 10.1214/15-AOP1013.

[14]

G. Dhariwal, A. Jüngel and N. Zamponi, Global martingale solutions for a stochastic population cross-diffusion system, Stochastic Process. Appl., 129 (2019), 3792-3820.  doi: 10.1016/j.spa.2018.11.001.

[15]

F. Flandoli, An introduction to 3D stochastic fluid dynamics,, In SPDE in Hydrodynamic: Recent Progress and Prospects, 1942 (2008), 51–150. doi: 10.1007/978-3-540-78493-7_2.

[16]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[17]

G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. AMat., 95 (2001), 281-295. 

[18]

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.

[19]

H. Garcke and K. Lam, Global weak solutions and asymptotic limits of a cahn–hilliard–darcy system modelling tumour growth, AIMS Math., 1 (2016), 318-360. 

[20]

N. Glatt-Holtz, R. Temam and C. Wang, Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1047-1085.  doi: 10.3934/dcdsb.2014.19.1047.

[21]

E. Hausenblas, P. A. Razafimandimby and M. Sango, Martingale solution to equations for differential type fluids of grade two driven by random force of Lévy type, Potential Anal., 38 (2013), 1291-1331.  doi: 10.1007/s11118-012-9316-7.

[22]

M. Hofmanová, Degenerate parabolic stochastic partial differential equations, Stochastic Process. Appl., 123 (2013), 4294-4336.  doi: 10.1016/j.spa.2013.06.015.

[23]

G. Leoni, A First Course in Sobolev Spaces, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, second edition, 2017. doi: 10.1090/gsm/181.

[24]

S. A. Levin, A more functional response to predator-prey stability, The American Naturalist, 111 (1977), 381-383. 

[25]

S. A. Levin and L. A. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976), 659-659. 

[26]

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

[27]

M. Mimura and J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, J. Theoret. Biol., 75 (1978), 249-262.  doi: 10.1016/0022-5193(78)90332-6.

[28]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Advances in Biophysics, 15 (1982), 19-65. 

[29]

J. D. Murray, Mathematical Biology. I, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[30]

J. D. Murray, Mathematical Biology. II, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[31]

A. Okubo and S. A. Levin., Diffusion and Ecological Problems: Modern Perspectives, 2$^{nd}$ edtion, Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.

[32]

M. Ondreját, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041-1091.  doi: 10.1214/EJP.v15-789.

[33]

C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007.

[34]

P. A. Razafimandimby and M. Sango, Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations, Z. Angew. Math. Phys., 66 (2015), 2197-2235.  doi: 10.1007/s00033-015-0534-x.

[35]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.

[36]

J. Simon, Compact sets in the space $L^ p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

show all references

References:
[1]

B. Ainseba, M. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.

[2]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion, Comput. Math. Appl., 70 (2015), 132-157.  doi: 10.1016/j.camwa.2015.04.021.

[3]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, Pattern formation for a reaction diffusion system with constant and cross diffusion, In Numerical Mathematics and Advanced Applications—ENUMATH 2013, 103 (2015), 153–161.

[4]

V. Anaya, M. Bendahmane, M. Langlais and M. Sepúlveda, Remarks about spatially structured SI model systems with cross diffusion, Contributions to Partial Differential Equations and Applications, 47 (2019), 43-64. 

[5]

M. Bendahmane, Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Netw. Heterog. Media, 3 (2008), 863-879.  doi: 10.3934/nhm.2008.3.863.

[6]

M. Bendahmane, K. H. Karlsen and J. M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. Methods Appl. Sci., 17 (2007), 783–804. doi: 10.1142/S0218202507002108.

[7]

M. Bendahmane, T. Lepoutre, A. Marrocco and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, J. Math. Pures Appl., 92 (2009), 651-667.  doi: 10.1016/j.matpur.2009.05.003.

[8]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[9]

H. Bessaih, Martingale solutions for stochastic Euler equations, Stochastic Anal. Appl., 17 (1999), 713-725.  doi: 10.1080/07362999908809631.

[10]

M. D. Chekroun, E. Park and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926-2972.  doi: 10.1016/j.jde.2015.10.022.

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[12]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[13]

A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab., 44 (2016), 1916-1955.  doi: 10.1214/15-AOP1013.

[14]

G. Dhariwal, A. Jüngel and N. Zamponi, Global martingale solutions for a stochastic population cross-diffusion system, Stochastic Process. Appl., 129 (2019), 3792-3820.  doi: 10.1016/j.spa.2018.11.001.

[15]

F. Flandoli, An introduction to 3D stochastic fluid dynamics,, In SPDE in Hydrodynamic: Recent Progress and Prospects, 1942 (2008), 51–150. doi: 10.1007/978-3-540-78493-7_2.

[16]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[17]

G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. AMat., 95 (2001), 281-295. 

[18]

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.

[19]

H. Garcke and K. Lam, Global weak solutions and asymptotic limits of a cahn–hilliard–darcy system modelling tumour growth, AIMS Math., 1 (2016), 318-360. 

[20]

N. Glatt-Holtz, R. Temam and C. Wang, Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1047-1085.  doi: 10.3934/dcdsb.2014.19.1047.

[21]

E. Hausenblas, P. A. Razafimandimby and M. Sango, Martingale solution to equations for differential type fluids of grade two driven by random force of Lévy type, Potential Anal., 38 (2013), 1291-1331.  doi: 10.1007/s11118-012-9316-7.

[22]

M. Hofmanová, Degenerate parabolic stochastic partial differential equations, Stochastic Process. Appl., 123 (2013), 4294-4336.  doi: 10.1016/j.spa.2013.06.015.

[23]

G. Leoni, A First Course in Sobolev Spaces, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, second edition, 2017. doi: 10.1090/gsm/181.

[24]

S. A. Levin, A more functional response to predator-prey stability, The American Naturalist, 111 (1977), 381-383. 

[25]

S. A. Levin and L. A. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976), 659-659. 

[26]

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

[27]

M. Mimura and J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, J. Theoret. Biol., 75 (1978), 249-262.  doi: 10.1016/0022-5193(78)90332-6.

[28]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Advances in Biophysics, 15 (1982), 19-65. 

[29]

J. D. Murray, Mathematical Biology. I, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[30]

J. D. Murray, Mathematical Biology. II, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[31]

A. Okubo and S. A. Levin., Diffusion and Ecological Problems: Modern Perspectives, 2$^{nd}$ edtion, Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.

[32]

M. Ondreját, Stochastic nonlinear wave equations in local Sobolev spaces, Electron. J. Probab., 15 (2010), 1041-1091.  doi: 10.1214/EJP.v15-789.

[33]

C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007.

[34]

P. A. Razafimandimby and M. Sango, Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations, Z. Angew. Math. Phys., 66 (2015), 2197-2235.  doi: 10.1007/s00033-015-0534-x.

[35]

M. Sango, Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22 (2010), 669-697.  doi: 10.1142/S0129055X10004041.

[36]

J. Simon, Compact sets in the space $L^ p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

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