Citation
Simpson, Henry C. (1979) Periodic solutions of integrodifferential equations which arise in population dynamics. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03212013102207354
Abstract
The problem of the existence and stability of periodic solutions of infinitelag integradifferential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range ( ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.
The first chapter is devoted to linear integrodifferential equations with constant coefficients utilizing the method of semigroups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the twotiming perturbation procedure is applied to construct the periodic solutions. The third chapter uses twotiming to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integrodifferential equations with periodic coefficients again using the semigroup approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Applied Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied Mathematics 
Thesis Availability:  Restricted to Caltech community only 
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Thesis Committee: 

Defense Date:  12 June 1978 
Record Number:  CaltechTHESIS:03212013102207354 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:03212013102207354 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7541 
Collection:  CaltechTHESIS 
Deposited By:  Dan Anguka 
Deposited On:  21 Mar 2013 17:44 
Last Modified:  21 Mar 2013 17:44 
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