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Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals†

Published online by Cambridge University Press:  14 February 2012

In-Ding Hsü  and
Nicholas D. Kazarinoff
In-Ding Hsü
Affiliation:
State University of New York at Buffalo
Nicholas D. Kazarinoff
Affiliation:
State University of New York at Buffalo
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Synopsis

A 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 1-2 , 1977 , pp. 163 - 175
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Hopf, E.Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems. Ber. Verh. Sachs. Akad. Wiss. Leipzig. Math.-Nat. 94 (1942), 3–22.Google Scholar
2Hsü, I.-D. and Kazarinoff, N. D.An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J. Math. Anal. Appl., 55 (1976), 61–89.CrossRefGoogle Scholar
3Pimbley, G. H. Jr.Periodic solutions of third order predator-prey equations simulating an immune response. Arch. Rat. Mech. Anal. 55 (1974), 93–123.CrossRefGoogle Scholar
4Wan, Yieh-Hei A remark on the Hopf bifurcation formula, to appear.Google Scholar