On the Dynamic Dichotomy between Positive Equilibria and Synchronous 2-cycles in Matrix Population Models

Persistent Link:
http://hdl.handle.net/10150/613591
Title:
On the Dynamic Dichotomy between Positive Equilibria and Synchronous 2-cycles in Matrix Population Models
Author:
Veprauskas, Amy
Issue Date:
2016
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
For matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death. Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties. We show that this dynamic dichotomy is a general feature of synchrony models which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).
Type:
text; Electronic Dissertation
Keywords:
Applied Mathematics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Applied Mathematics
Degree Grantor:
University of Arizona
Advisor:
Cushing, Jim

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleOn the Dynamic Dichotomy between Positive Equilibria and Synchronous 2-cycles in Matrix Population Modelsen_US
dc.creatorVeprauskas, Amyen
dc.contributor.authorVeprauskas, Amyen
dc.date.issued2016-
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractFor matrix population models with nonnegative, irreducible and primitive inherent projection matrices, the stability of the branch of positive equilibria that bifurcates from the extinction equilibrium as the dominant eigenvalue of the inherent projection matrix increases through one is determined by the direction of bifurcation. However, if the inherent projection matrix is imprimitive this bifurcation becomes more complicated. This is the result of the simultaneous departure of multiple eigenvalues from the unit complex circle. Matrix models with imprimitive projection matrices commonly appear in models of semelparous species, which are characterized by one reproductive event that is often followed by death. Due to the imprimitivity of the projection matrix, semelparous Leslie models exhibit two contrasting dynamics, either equilibria in which all age classes are present or synchronized cycles in which age classes are separated temporally. The two-stage semelparous Leslie model has index of imprimitivity two, meaning that two eigenvalues simultaneously leave the unit circle when the dominant eigenvalue increases past one. This model exhibits a dynamic dichotomy in which the two steady states have opposite stability properties. We show that this dynamic dichotomy is a general feature of synchrony models which are characterized by the simultaneous creation of a branch of positive equilibria and a branch of synchronous 2-cycles when the extinction equilibrium destabilizes (Chapter 3). A synchrony model must, necessarily, have index of imprimitivity two but is not limited to models of semelparous species. We provide a specific example of a synchrony model for an iteroparous species which is motivated by observations of a cannibalistic gull population (Chapter 2). We also extend the study of the synchrony model to a Darwinian model which couples population dynamics with the dynamics of a suite of evolving phenotypic traits (Chapter 4). For the evolutionary synchrony model, we show that the dynamic dichotomy occurs provided that fitness, as measured by the spectral radius, is maximized. In addition, we examine the dynamic dichotomy for semelparous species in a continuous-time setting (Chapter 5).en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectApplied Mathematicsen
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineApplied Mathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorCushing, Jimen
dc.contributor.committeememberChesson, Peteren
dc.contributor.committeememberTabor, Michaelen
dc.contributor.committeememberWatkins, Joeen
dc.contributor.committeememberCushing, Jimen
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