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 and 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.

[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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