Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts

Abstract. We explore several hypotheses for the swarming behaviour in locusts, with a goal of understanding how swarm cohesion can be maintained by the huge population of insects (up to 109 individuals) over long distances (up to thousands of miles) and long periods of time (over a week). The mathematical models that correspond to such hypotheses are generally partial differential equations that can be analysed for travelling wave solutions. The nature of a swarm (and the fact that it contains a finite number of individuals) mandates that we seek travelling band (pulse) solutions. However, most biologically reasonable models fail to produce such ideal behaviour unless unusual and unrealistic assumptions are made. The failure of such models, general difficulties encountered with similar models of other migratory phenomena, and possible approaches to alleviate these problems are described and discussed.

[1]  S. Gueron,et al.  A model of herd grazing as a travelling wave, chemotaxis and stability , 1989, Journal of mathematical biology.

[2]  J. Rinzel,et al.  Waves in a simple, excitable or oscillatory, reaction-diffusion model , 1981 .

[3]  W. Alt Biased random walk models for chemotaxis and related diffusion approximations , 1980, Journal of mathematical biology.

[4]  Frédéric O. Albrecht,et al.  Polymorphisme phasaire et biologie des acridiens migrateurs , 1967 .

[5]  J. G. Skellam Random dispersal in theoretical populations , 1951, Biometrika.

[6]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[7]  Daniel Gru,et al.  Advection—di⁄usion equations for generalized tactic searching behaviors , 1999 .

[8]  G B Ermentrout,et al.  Selecting a common direction. II. Peak-like solutions representing total alignment of cell clusters. , 1996, Journal of mathematical biology.

[9]  T. Nagai,et al.  Traveling waves in a chemotactic model , 1991, Journal of mathematical biology.

[10]  Hans F. Weinberger,et al.  Spatial patterning of the spruce budworm , 1979 .

[11]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[12]  J. Kennedy,et al.  The migration of the Desert Locust (Schistocerca gregaria Forsk.) I. The behaviour of swarms. II. A theory of long-range migrations , 1951, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[13]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[14]  J. G. Skellam Random dispersal in theoretical populations , 1951, Biometrika.

[15]  明 大久保,et al.  Diffusion and ecological problems : mathematical models , 1980 .

[16]  R. C. Rainey,et al.  Migration and meteorology. Flight behaviour and the atmospheric environment of locusts and other migrant pests. , 1989 .

[17]  Daniel Grünbaum,et al.  Advection–diffusion equations for generalized tactic searching behaviors , 1999 .

[18]  Beate Pfistner A One Dimensional Model for the Swarming Behavior of Myxobacteria , 1990 .

[19]  Steven R. Dunbar,et al.  Travelling wave solutions of diffusive Lotka-Volterra equations , 1983 .

[20]  M E Gurtin,et al.  On interacting populations that disperse to avoid crowding: preservation of segregation , 1985, Journal of mathematical biology.

[21]  Alexander Mogilner Modelling spatio-angular patterns in cell biology , 1995 .

[22]  L. A. Segel,et al.  A gradually slowing travelling band of chemotactic bacteria , 1984, Journal of mathematical biology.

[23]  W. Alt Degenerate diffusion equations with drift functionals modelling aggregation , 1985 .

[24]  Inge S. Helland,et al.  Diffusion models for the dispersal of insects near an attractive center , 1983 .

[25]  James D. Murray,et al.  A generalized diffusion model for growth and dispersal in a population , 1981 .

[26]  Z. Waloff Observations on the airspeeds of freely flying locusts , 1972 .

[27]  G. M. Odell,et al.  Traveling bands of chemotactic bacteria revisited , 1976 .

[28]  C. Conley,et al.  Critical manifolds, travelling waves, and an example from population genetics , 1982, Journal of mathematical biology.

[29]  K. Aoki,et al.  Gene-culture waves of advance , 1987, Journal of mathematical biology.

[30]  Peggy E. Ellis The Gregarious Behaviour of Marching Locusta Migratoria Migratorioides (R. & F.) Hoppers , 1953 .

[31]  Lee A. Segel,et al.  Mathematical models in molecular and cellular biology , 1982, The Mathematical Gazette.

[32]  F L Ochoa,et al.  A generalized reaction diffusion model for spatial structure formed by motile cells. , 1984, Bio Systems.

[33]  Morton E. Gurtin,et al.  On interacting populations that disperse to avoid crowding: The effect of a sedentary colony , 1984 .

[34]  Wolfgang Alt,et al.  Stability results for a diffusion equation with functional drift approximating a chemotaxis model , 1987 .

[35]  L. Segel,et al.  Traveling bands of chemotactic bacteria: a theoretical analysis. , 1971, Journal of theoretical biology.

[36]  Evelyn Fox Keller,et al.  Necessary and sufficient conditions for chemotactic bands , 1975 .

[37]  D. Grünbaum Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming , 1994, Journal of mathematical biology.

[38]  V. Lakshmikantham,et al.  Nonlinear Analysis: Theory, Methods and Applications , 1978 .