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On the local behavior of non-negative solutions to a logarithmically singular equation
Infinite dimensional relaxation oscillation in aggregation-growth systems
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 |
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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