American Institute of Mathematical Sciences

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December  2012, 7(4): 705-740. doi: 10.3934/nhm.2012.7.705

A link between microscopic and macroscopic models of self-organized aggregation

Tadahisa Funaki 1, , Hirofumi Izuhara 2, , Masayasu Mimura 2, and  Chiyori Urabe 3,

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

Received  March 2012 Revised  July 2012 Published  December 2012

In some species, one of the roles of pheromones is to influence aggregation behavior. We first propose a macroscopic cross-diffusion model for the self-organized aggregation of German cockroaches that includes directed movement due to an aggregation pheromone. We then propose a microscopic particle model which is set into context with the macroscopic model. Our goal is to link the macroscopic and microscopic descriptions by using the singular and the hydrodynamic limit procedures. A hybrid model related to the macroscopic and microscopic models is also proposed as a cockroach aggregation model. This hybrid model assumes that each individual responds to pheromone concentration and moves by two-mode simple symmetric random walks. It shows that even though the movement of individuals is not directed, two-mode simple symmetric random walks and effect of the pheromone result in self-organized aggregation.
Keywords: hydrodynamic limit, reaction-diffusion system, particle system, Cross-diffusion system, chemotaxis..
Mathematics Subject Classification: Primary: 35K55, 60K35; Secondary: 82C22, 92D2.
Citation: Tadahisa Funaki, Hirofumi Izuhara, Masayasu Mimura, Chiyori Urabe. A link between microscopic and macroscopic models of self-organized aggregation. Networks & Heterogeneous Media, 2012, 7 (4) : 705-740. doi: 10.3934/nhm.2012.7.705
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.  Google Scholar

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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.  Google Scholar

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L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, RI, 1998.  Google Scholar

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber Deutsch Math., 106 (2004), 51-69.  Google Scholar

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S. Ishii and Y. Kuwahara, An aggregation pheromone of the German cockroach Blattella germanica L. (Orthoptera: Blattelidae), Appl. Ent. Zool., 2 (1967), 203-217. Google Scholar

[13]

S. Ishii and Y. Kuwahara, Aggregation of German Cockroach (Blattella germanica) Nymphs, Experientia, 24 (1968), 88-89. Google Scholar

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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. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. Google Scholar

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C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems," Springer, 1999.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[22]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Springer-Verlag, 2001.  Google Scholar

[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.  Google Scholar

[24]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. Google Scholar

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556. doi: 10.2307/2000228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI, 2001.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), ., ().   Google Scholar

[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.  Google Scholar

[6]

L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, RI, 1998.  Google Scholar

[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.  Google Scholar

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Deutsch Math., 105 (2003), 103-165.  Google Scholar

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber Deutsch Math., 106 (2004), 51-69.  Google Scholar

[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.  Google Scholar

[11]

S. Ishii, An aggregation pheromone of the German cockroach, Blattella germanica (L.), Appl. Ent. Zool., 5 (1970), 33-41. Google Scholar

[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. Google Scholar

[13]

S. Ishii and Y. Kuwahara, Aggregation of German Cockroach (Blattella germanica) Nymphs, Experientia, 24 (1968), 88-89. Google Scholar

[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. Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415. Google Scholar

[16]

C. Kipnis and C. Landim, "Scaling Limits of Interacting Particle Systems," Springer, 1999.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Springer-Verlag, 2001.  Google Scholar

[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.  Google Scholar

[24]

J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192. Google Scholar

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556. doi: 10.2307/2000228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI, 2001.  Google Scholar

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