Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction–Diffusion Systems
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The distribution of the roots of a second order transcendental polynomial is analyzed, and it is used for solving the purely imaginary eigenvalue of a transcendental characteristic equation with two transcendental terms. The results are applied to the stability and associated Hopf bifurcation of a constant equilibrium of a general reaction–diffusion system or a system of ordinary differential equations with delay effects. Examples from biochemical reaction and predator–prey models are analyzed using the new techniques.
KeywordsSecond order transcendental polynomial Characteristic equation Reaction–diffusion Stability Hopf bifurcation
Mathematics Subject Classification (2010)34K08 34K18 34K20 35R10 92E20
The authors thank two anonymous referees for very helpful comments which greatly improved the manuscript. Parts of this work was done when SSC visited College of William and Mary in 2010–2011, and she would like to thank CWM for warm hospitality.
Partially supported by a grant from China Scholarship Council (Chen), NSF grant DMS-1022648 and Shanxi 100 talent program (Shi), China-NNSF grants 11031002 (Wei).
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