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Spatial structures in predator-prey communities with hereditary effects and diffusion

Cohen, Donald S. and Hagan, Patrick S. and Simpson, Henry C. (1979) Spatial structures in predator-prey communities with hereditary effects and diffusion. Mathematical Biosciences, 44 (3-4). pp. 167-177. ISSN 0025-5564. doi:10.1016/0025-5564(79)90079-8.

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We study a continuous time delay interaction model for predator-prey communities in which the species can also diffuse. Oscillatory traveling wave solutions are obtained. The amplitudes (and also the wavenumbers and frequencies) are seen to evolve slowly in time and to vary significantly over large regions of space due to the effects of small nonlinear terms in the model. These modulations (amplitude and dispersion relation) are found in terms of the parameters of the problem. Other authors have conjectured that these waves (which are nonlinear perturbations of linear waves) are stable, based on various restricted stability analyses. Based on a full stability analysis, we show that all such waves (both steadily progressing and steady solutions) are unstable. We conclude that periodic bifurcations from an equilibrium state are unstable, but periodic structure bifurcating from a finite amplitude spatially homogenous limit cycle can be stable.

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Additional Information:© 1979 Published by Elsevier B.V. Received 27 July 1978, Revised 30 September 1978. Supported in part by the U.S. Army Research Office (Durham) under Contract DAAG29-78-C-0011 and by the National Science Foundation under Grant MCS 78-03036. Supported by the Fannie and John Hertz Foundation.
Funding AgencyGrant Number
Army Research Office (ARO)DAAG29-78-C-0011
NSFMCS 78-03036
Fannie and John Hertz FoundationUNSPECIFIED
Issue or Number:3-4
Record Number:CaltechAUTHORS:20200113-140555733
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Official Citation:Donald S. Cohen, Patrick S. Hagan, Henry C. Simpson, Spatial structures in predator-prey communities with hereditary effects and diffusion, Mathematical Biosciences, Volume 44, Issues 3–4, 1979, Pages 167-177, ISSN 0025-5564, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100689
Deposited By: Tony Diaz
Deposited On:13 Jan 2020 23:32
Last Modified:16 Nov 2021 17:56

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