Advanced Search
Article Contents
Article Contents

Infinite dimensional relaxation oscillation in aggregation-growth systems

Abstract / Introduction Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35K57, 35B10; Secondary: 35R15.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


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


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


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


    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.


    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.


    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.


    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.


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


    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.


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


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


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


    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.


    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.


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


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(76) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint