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A hydrodynamic limit for chemotaxis in a given heterogeneous environment. (English) Zbl 1370.92030

Summary: In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by A. Koukkous [Stochastic Processes Appl. 84, No. 2, 297–312 (1999; Zbl 0996.60108)] and C. Kipnis and C. Landim [Scaling limits of interacting particle systems. Berlin: Springer (1999; Zbl 0927.60002)]. Numerical simulations extend and illustrate the theoretical findings.

MSC:

92C17 Cell movement (chemotaxis, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F25 \(L^p\)-limit theorems
35K55 Nonlinear parabolic equations
35Q82 PDEs in connection with statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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References:

[1] Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147-177 (1980) · Zbl 0434.92001 · doi:10.1007/BF00275919
[2] Avena, L., Franco, T., Jara, M., Völlering, F.: Symmetric exclusion as a random environment: hydrodynamic limits. Ann. Inst. H. Poincaré Probab. Statist. 51, 901-916 (2015) · Zbl 1359.82012 · doi:10.1214/14-AIHP607
[3] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34, 339-1369 (2006) · Zbl 1101.60075 · doi:10.1214/009117906000000115
[4] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics for attractive particle systems in random environment. Ann. Inst. H. Poincaré Probab. Statist. 50, 403-424 (2014) · Zbl 1294.60116 · doi:10.1214/12-AIHP510
[5] Billingsley, P.: Convergence of Probability Measures. 2nd ed. Wiley, New York (1999) · Zbl 0944.60003
[6] Braxmeier-Even, N., Olla, S.: Hydrodynamic limit for an Hamiltonian system with boundary conditions and conservative noise. Arch. Ration. Mech. Anal. 213, 561-585 (2014) · Zbl 1307.35204 · doi:10.1007/s00205-014-0741-1
[7] Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217-237 (1981) · Zbl 0481.92010 · doi:10.1016/0025-5564(81)90055-9
[8] Covert, P., Rezakhanlou, F.: Hydrodynamic limit for particle systems with nonconstant speed parameter. J. Statist. Phys. 88, 383-426 (1997) · Zbl 0939.82032 · doi:10.1007/BF02508477
[9] De Masi, A., Luckhaus, S., Presutti, E.: Two scales hydrodynamic limit for a model of Malignant tumor cells. Ann. Inst. H. Poincare Probab. Statist. 43, 257-297 (2007) · Zbl 1122.60086 · doi:10.1016/j.anihpb.2006.03.003
[10] Fritz, J., Tóth, B.: Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas. Commun. Math. Phys. 249, 1-27 (2004) · Zbl 1126.82015 · doi:10.1007/s00220-004-1103-x
[11] Gonçalves, P., Jara, M.: Scaling limits for gradient systems in random environment. J. Stat. Phys. 131, 691-716 (2008) · Zbl 1144.82043 · doi:10.1007/s10955-008-9509-z
[12] Grosskinsky, S., Spohn, H.: Stationary measures and hydrodynamics of zero range processes with several species of particles. Bull. Braz. Math. Soc. (N.S.) 34, 489-507 (2003) · Zbl 1083.82019 · doi:10.1007/s00574-003-0026-z
[13] Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31-59 (1988) · Zbl 0652.60107 · doi:10.1007/BF01218476
[14] Jäger, W., Luckhaus, S.: On explosion of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819-824 (1992) · Zbl 0746.35002 · doi:10.1090/S0002-9947-1992-1046835-6
[15] Jara, M.: Hydrodynamic limit of the exclusion process in inhomogeneous media. In: Dynamics, Games and Science (II), Springer Proc. Math. 2, pp 449-465. Springer, Heidelberg (2011) · Zbl 1432.60091
[16] Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26, 399-415 (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[17] Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer (1999) · Zbl 0927.60002
[18] Koukkous, A.: Hydrodynamic behavior of symmetric zero-range processes with random rates. Stoch. Process. Appl. 84, 297-312 (1999) · Zbl 0996.60108 · doi:10.1016/S0304-4149(99)00054-X
[19] Koukkous, A., Guiol, H.: Large deviations for a zero mean asymmetric zero range process in random media (2010). arXiv:math/0009110 · Zbl 1307.35204
[20] Krug, J., Ferrari, P.A.: Phase transitions in driven diffusive systems with random rates. J. Phys. A 29, L465-L471 (1996) · Zbl 0906.60075
[21] Landim, C.: Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes. Ann. Probab. 24, 599-638 (1996) · Zbl 0862.60095 · doi:10.1214/aop/1039639356
[22] Luckhaus, S., Triolo, L.: The continuum reaction-diffusion limit of a stochastic cellular growth model. Atti Accad. Naz. Lincei CI. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15, 215-223 (2004) · Zbl 1162.60346
[23] Marahrens, D.: A Hydrodynamic Limit for a Zero Range Process in a Random Medium with Slowly Varying Average. Diplomathesis, University of Heidelberg (2008) · Zbl 1359.82012
[24] Oelschläger, K.: A law of large numbers for moderately interacting diffusion processes. Z. Wahrsch. Verw. Gebiete 69, 279-322 (1985) · Zbl 0549.60071 · doi:10.1007/BF02450284
[25] Oelschläger, K.: On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab. Theory Rel. Fields 82, 565-586 (1989) · Zbl 0673.60110 · doi:10.1007/BF00341284
[26] Othmer, H.G., Stevens, A.: Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044-1081 (1997) · Zbl 0990.35128 · doi:10.1137/S0036139995288976
[27] Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Random Fields, Vols. I-II (Esztergom, 1979). Colloq. Math. Soc. JÁnos Bolyai, vol. 27, pp 835-873 (1981) · Zbl 0499.60059
[28] Rafferty, T., Chleboun, P., Grosskinsky, S.: Monotonicity and condensation in homogeneous stochastic particle systems (2015). arXiv:1505.02049 · Zbl 1391.60241
[29] Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on Zd. Commun. Math. Phys. 140, 417-448 (1991) · Zbl 0738.60098 · doi:10.1007/BF02099130
[30] Schaaf, R.: Stationary solutions of chemotaxis systems. Trans. Am. Math. Soc. 292, 531-556 (1985) · Zbl 0637.35007 · doi:10.1090/S0002-9947-1985-0808736-1
[31] Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246-290 (1970) · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4
[32] Stevens, A.: Trail following and aggregation of myxobacteria. J. Biol. Syst. 3, 1059-1068 (1995) · doi:10.1142/S0218339095000952
[33] Stevens, A.: A stochastic cellular automaton modeling gliding and aggregation of myxobacteria. SIAM J. Appl. Math. 61, 172-182 (2000) · Zbl 0992.92005 · doi:10.1137/S0036139998342053
[34] Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183-212 (2000) · Zbl 0963.60093 · doi:10.1137/S0036139998342065
[35] Toth, B., Valko, B.: Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit. Commun. Math. Phys. 256, 111-157 (2005) · Zbl 1088.82019 · doi:10.1007/s00220-005-1314-9
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