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September  2012, 17(6): 1859-1887. doi: 10.3934/dcdsb.2012.17.1859

Infinite dimensional relaxation oscillation in aggregation-growth systems

Shin-Ichiro Ei 1, , Hirofumi Izuhara 2, and  Masayasu Mimura 2,

1. 

Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
2. 

Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, 214-8571, Japan, Japan

Received  October 2011 Revised  January 2012 Published  May 2012

Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.
Keywords: pattern formation, Relaxation oscillation, periodic solution, aggregation-growth system, bifurcation..
Mathematics Subject Classification: Primary: 35K57, 35B10; Secondary: 35R1.
Citation: Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura. Infinite dimensional relaxation oscillation in aggregation-growth systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1859-1887. doi: 10.3934/dcdsb.2012.17.1859
References:
[1]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.

[2]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invations and pulsating travelling fronts, J. Math. Pures Appl. (9), 84 (2005), 1101-1146.

[3]

, E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang,, \emph{AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HomCont)}., (). 

[4]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S.-I. Ei and M. Mimura, Relaxation oscillations in combustion models of thermal self-ignition, J. Dynam. Differential Equations, 4 (1992), 191-229.

[6]

T. Funaki, H. Izuhara, M. Mimura and C. Urabe, A link between microscopic and macroscopic models of self-organized aggregation,, in preparetion., (). 

[7]

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

[8]

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.

[9]

R. Jeanson, C. Rivault, J.-L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroach, Animal Behaviour, 69 (2005), 169-180. doi: 10.1016/j.anbehav.2004.02.009.

[10]

S. R. Kay and S. K. Scott, Oscillations of simple exothermic reaction in a closed system. II. Exact Arrhenius kinetics, Proc. R. Soc. Lond. A, 416 (1988), 343-359. doi: 10.1098/rspa.1988.0038.

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[12]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.

[13]

M. Mimura and T. Tsujikawa, Aggregation pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9.

[14]

H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete Contin. Dyn. Syst. S, 5 (2012), 147-158.

[15]

J. E. Pearson, Complex patterns in a simple system, Science, 216 (1993), 189-192. doi: 10.1126/science.261.5118.189.

[16]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.

[17]

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.

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments, Theor. Popul. Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.

[19]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[20]

A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London Series B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

show all references

References:
[1]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.

[2]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invations and pulsating travelling fronts, J. Math. Pures Appl. (9), 84 (2005), 1101-1146.

[3]

, E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang,, \emph{AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HomCont)}., (). 

[4]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S.-I. Ei and M. Mimura, Relaxation oscillations in combustion models of thermal self-ignition, J. Dynam. Differential Equations, 4 (1992), 191-229.

[6]

T. Funaki, H. Izuhara, M. Mimura and C. Urabe, A link between microscopic and macroscopic models of self-organized aggregation,, in preparetion., (). 

[7]

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

[8]

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.

[9]

R. Jeanson, C. Rivault, J.-L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroach, Animal Behaviour, 69 (2005), 169-180. doi: 10.1016/j.anbehav.2004.02.009.

[10]

S. R. Kay and S. K. Scott, Oscillations of simple exothermic reaction in a closed system. II. Exact Arrhenius kinetics, Proc. R. Soc. Lond. A, 416 (1988), 343-359. doi: 10.1098/rspa.1988.0038.

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[12]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.

[13]

M. Mimura and T. Tsujikawa, Aggregation pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9.

[14]

H. Murakawa, A relation between cross-diffusion and reaction-diffusion, Discrete Contin. Dyn. Syst. S, 5 (2012), 147-158.

[15]

J. E. Pearson, Complex patterns in a simple system, Science, 216 (1993), 189-192. doi: 10.1126/science.261.5118.189.

[16]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. AMS, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.

[17]

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.

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic-waves in heterogeneous environments, Theor. Popul. Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.

[19]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[20]

A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London Series B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.

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