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Bifurcation of Periodic Solutions of Integrodifferential Systems with Applications to Time Delay Models in Population Dynamics

J. M. Cushing
SIAM Journal on Applied Mathematics
Vol. 33, No. 4 (Dec., 1977), pp. 640-654
Stable URL: http://www.jstor.org/stable/2100758
Page Count: 15
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Bifurcation of Periodic Solutions of Integrodifferential Systems with Applications to Time Delay Models in Population Dynamics
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Abstract

A Fredholm alternative is proved for a general linear system of Stieltjes integrodifferential equations. This result is used to derive necessary and sufficient conditions for the bifurcation of nontrivial periodic solutions of a nonlinear perturbation of the system containing n parameters. The results are applied to several models from mathematical ecology which describe the dynamics of various species interactions (included are models of mutualistic, competitive and predator-prey interactions) under the influence of time delays. These applications illustrate how, for such models, the existence of multi-dimensional manifolds of periodic solutions of various periods in the presence of unstable equilibria can occur as a result of the presence of time delays at least for birth rates near certain critical values.

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