Martingale solutions of stochastic nonlocal cross-diffusion systems

Mostafa Bendahmane; Kenneth H. Karlsen; Mostafa Bendahmane; Kenneth H. Karlsen

doi:10.3934/nhm.2022024

Networks and Heterogeneous Media

2022, Volume 17, Issue 5: 719-752. doi: 10.3934/nhm.2022024

Previous Article Next Article

Martingale solutions of stochastic nonlocal cross-diffusion systems

Mostafa Bendahmane ^{1
,},
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: 01 March 2022 Published: 16 June 2022
Primary: 60H15, 35K57; Secondary: 35M10, 35A05

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.
- Stochastic partial differential equation,
- cross-diffusion system,
- existence,
- martingale solution,
- population dynamics
Citation: Mostafa Bendahmane, Kenneth H. Karlsen. Martingale solutions of stochastic nonlocal cross-diffusion systems[J]. Networks and Heterogeneous Media, 2022, 17(5): 719-752. doi: 10.3934/nhm.2022024

Related Papers:

Abstract

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.

References

[1]	A reaction-diffusion system modeling predator-prey with prey-taxis. Nonlinear Anal. Real World Appl. (2008) 9: 2086-2105.
[2]	A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion. Comput. Math. Appl. (2015) 70: 132-157.
[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]	Remarks about spatially structured SI model systems with cross diffusion. Contributions to Partial Differential Equations and Applications (2019) 47: 43-64.
[5]	Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Netw. Heterog. Media (2008) 3: 863-879.
[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.
[7]	Conservative cross diffusions and pattern formation through relaxation. J. Math. Pures Appl. (2009) 92: 651-667.
[8]	Stochastic Navier-Stokes equations. Acta Appl. Math. (1995) 38: 267-304.
[9]	Martingale solutions for stochastic Euler equations. Stochastic Anal. Appl. (1999) 17: 713-725.
[10]	The Stampacchia maximum principle for stochastic partial differential equations and applications. J. Differential Equations (2016) 260: 2926-2972.
[11]	(2014) Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press.
[12]	Local martingale and pathwise solutions for an abstract fluids model. Phys. D (2011) 240: 1123-1144.
[13]	Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. (2016) 44: 1916-1955.
[14]	Global martingale solutions for a stochastic population cross-diffusion system. Stochastic Process. Appl. (2019) 129: 3792-3820.
[15]	F. Flandoli, An introduction to 3D stochastic fluid dynamics,, In SPDE in Hydrodynamic: Recent Progress and Prospects, 1942 (2008), 51–150.
[16]	Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields (1995) 102: 367-391.
[17]	Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. AMat. (2001) 95: 281-295.
[18]	Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model. Numer. Math. (2003) 93: 655-673.
[19]	Global weak solutions and asymptotic limits of a cahn–hilliard–darcy system modelling tumour growth. AIMS Math. (2016) 1: 318-360.
[20]	Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete Contin. Dyn. Syst. Ser. B (2014) 19: 1047-1085.
[21]	Martingale solution to equations for differential type fluids of grade two driven by random force of Lévy type. Potential Anal. (2013) 38: 1291-1331.
[22]	Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl. (2013) 123: 4294-4336.
[23]	G. Leoni, A First Course in Sobolev Spaces, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, second edition, 2017.
[24]	A more functional response to predator-prey stability. The American Naturalist (1977) 111: 381-383.
[25]	Hypothesis for origin of planktonic patchiness. Nature (1976) 259: 659-659.
[26]	Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. (1980) 9: 49-64.
[27]	On a diffusive prey-predator model which exhibits patchiness. J. Theoret. Biol. (1978) 75: 249-262.
[28]	Pattern formation in interacting and diffusing systems in population biology. Advances in Biophysics (1982) 15: 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.
[32]	Stochastic nonlinear wave equations in local Sobolev spaces. Electron. J. Probab. (2010) 15: 1041-1091.
[33]	C. Prévôt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer, Berlin, 2007.
[34]	Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations. Z. Angew. Math. Phys. (2015) 66: 2197-2235.
[35]	Density dependent stochastic Navier-Stokes equations with non-Lipschitz random forcing. Rev. Math. Phys. (2010) 22: 669-697.
[36]	Compact sets in the space $L^ p(0, T;B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)