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PDE problems arising in mathematical biology
A link between microscopic and macroscopic models of self-organized aggregation
1. | Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan |
2. | Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki, Kanagawa 214-8571, Japan, Japan |
3. | FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency, Collaborative Research Center for Innovative Mathematical Modelling, Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan |
References:
[1] |
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. |
[2] |
S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, "Self-Organization in Biological Systems," Princeton University Press, Princeton, NJ, 2003. |
[3] |
A. De Masi, S. Luckhaus and E. Presutti, Two scales hydrodynamic limit for a model of malignant tumor cells, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 257-297.
doi: 10.1016/j.anihpb.2006.03.003. |
[4] |
E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HomCont) . |
[5] |
S.-I. Ei, H. Izuhara and M. Mimura, Infinite dimensional relaxation oscillation in aggregation-growth systems, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 1859-1887.
doi: 10.3934/dcdsb.2012.17.1859. |
[6] |
L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, RI, 1998. |
[7] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Deutsch Math., 105 (2003), 103-165. |
[9] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber Deutsch Math., 106 (2004), 51-69. |
[10] |
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. |
[11] |
S. Ishii, An aggregation pheromone of the German cockroach, Blattella germanica (L.), Appl. Ent. Zool., 5 (1970), 33-41. |
[12] |
S. Ishii and Y. Kuwahara, An aggregation pheromone of the German cockroach Blattella germanica L. (Orthoptera: Blattelidae), Appl. Ent. Zool., 2 (1967), 203-217. |
[13] |
S. Ishii and Y. Kuwahara, Aggregation of German Cockroach (Blattella germanica) Nymphs, Experientia, 24 (1968), 88-89. |
[14] |
R. Jeanson, C. Rivault, J. -L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroaches, Animal Behavior, 69 (2005), 169-180. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. |
[16] |
C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems," Springer, 1999. |
[17] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, 23, Amer. Math. Soc., Providence, R.I. 1967. |
[18] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[19] |
M. Mimura and M. Nagayama, Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability, Chaos, 7 (1997), 817-826.
doi: 10.1063/1.166282. |
[20] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modeling aggregation behavior: from individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[21] |
H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete and Continuous Dynamical Systems, Series S, 5 (2011), 147-158.
doi: 10.3934/dcdss.2012.5.147. |
[22] |
A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Springer-Verlag, 2001. |
[23] |
H. G. Othmer and A. Stevens, Aggregation, blow up and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[24] |
J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. |
[25] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556.
doi: 10.2307/2000228. |
[26] |
A. Stevens, A stochastic cellular automaton modeling gliding and aggregation of myxobacteria, SIAM J. Appl. Math., 61 (2000), 172-182.
doi: 10.1137/S0036139998342053. |
[27] |
A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.
doi: 10.1137/S0036139998342065. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI, 2001. |
show all references
References:
[1] |
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. |
[2] |
S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, "Self-Organization in Biological Systems," Princeton University Press, Princeton, NJ, 2003. |
[3] |
A. De Masi, S. Luckhaus and E. Presutti, Two scales hydrodynamic limit for a model of malignant tumor cells, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 257-297.
doi: 10.1016/j.anihpb.2006.03.003. |
[4] |
E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HomCont) . |
[5] |
S.-I. Ei, H. Izuhara and M. Mimura, Infinite dimensional relaxation oscillation in aggregation-growth systems, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 1859-1887.
doi: 10.3934/dcdsb.2012.17.1859. |
[6] |
L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, RI, 1998. |
[7] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Deutsch Math., 105 (2003), 103-165. |
[9] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber Deutsch Math., 106 (2004), 51-69. |
[10] |
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. |
[11] |
S. Ishii, An aggregation pheromone of the German cockroach, Blattella germanica (L.), Appl. Ent. Zool., 5 (1970), 33-41. |
[12] |
S. Ishii and Y. Kuwahara, An aggregation pheromone of the German cockroach Blattella germanica L. (Orthoptera: Blattelidae), Appl. Ent. Zool., 2 (1967), 203-217. |
[13] |
S. Ishii and Y. Kuwahara, Aggregation of German Cockroach (Blattella germanica) Nymphs, Experientia, 24 (1968), 88-89. |
[14] |
R. Jeanson, C. Rivault, J. -L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroaches, Animal Behavior, 69 (2005), 169-180. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. |
[16] |
C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems," Springer, 1999. |
[17] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs, 23, Amer. Math. Soc., Providence, R.I. 1967. |
[18] |
M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.
doi: 10.1007/BF00276035. |
[19] |
M. Mimura and M. Nagayama, Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability, Chaos, 7 (1997), 817-826.
doi: 10.1063/1.166282. |
[20] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modeling aggregation behavior: from individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[21] |
H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete and Continuous Dynamical Systems, Series S, 5 (2011), 147-158.
doi: 10.3934/dcdss.2012.5.147. |
[22] |
A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Springer-Verlag, 2001. |
[23] |
H. G. Othmer and A. Stevens, Aggregation, blow up and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[24] |
J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. |
[25] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556.
doi: 10.2307/2000228. |
[26] |
A. Stevens, A stochastic cellular automaton modeling gliding and aggregation of myxobacteria, SIAM J. Appl. Math., 61 (2000), 172-182.
doi: 10.1137/S0036139998342053. |
[27] |
A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.
doi: 10.1137/S0036139998342065. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI, 2001. |
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