×

Travelling fronts in the diffusive Nicholson’s blowflies equation with distributed delays. (English) Zbl 0969.35133

The author considers some equations of the form \[ \partial u/\partial t= \partial^2u/\partial x^2- u+\beta(f* u) e^{-(f* u)}, \] where \((f* u)(x, t)= \int^t_{-\infty} f(t- s)u(x, s) ds\) (the kernel \(f: [0,\infty)\to [0,\infty)\) satisfies: \(f(t)\geq 0\), \(\forall t\geq 0\) and \(\int^\infty_0 f(t) dt= 1\)) and \(\beta> 1\) is a parameter. He seeks travelling wave front solutions \(u(x, t)= U(z)\), \(z= x-ct\), \(c>0\), in connection with the steady state solutions \(u=0\) and \(u= \ln\beta\).
The existence of such travelling solutions is proved when \(f(t)\) assumes a special form; some qualitative properties of these solutions are established.

MSC:

35R10 Partial functional-differential equations
92D40 Ecology
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gurney, W. S.C.; Blythe, S. P.; Nisbet, R. M., Nicholson’s blowflies revisited, Nature, 287, 17-21 (1980)
[2] Nicholson, A. J., The self adjustment of populations to change, (Cold Spring Harb. Symp. Quant. Biol., 22 (1957)), 153-173
[3] Yang, Y.; So, J. W.-H., Dynamics for the diffusive Nicholson’s blowflies equation, (Chen, W.; Hu, S., Proceedings of the International Conference on Dynamical Systems and Differential Equations, Springfield, MO U.S.A. May 29-June 1, 1996, Volume 11 (1998)) · Zbl 0923.35195
[4] So, J. W.-H.; Yang, Y., Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. Diff. Eqns., 150, 317-348 (1998) · Zbl 0923.35195
[5] J.W.-H. So and X. Zou, Travelling waves for the diffusive Nicholson’s blowflies equation (Preprint).; J.W.-H. So and X. Zou, Travelling waves for the diffusive Nicholson’s blowflies equation (Preprint). · Zbl 1027.35051
[6] Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqns., 31, 53-98 (1979) · Zbl 0476.34034
[7] Akveld, M. E.; Hulshof, J., Travelling wave solutions of a fourth-order semilinear diffusion equation, Appl. Math. Lett., 11, 3, 115-120 (1998) · Zbl 0932.35017
[8] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer: Springer New York · Zbl 0487.47039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.