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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling

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Parameterizing the growth-decline boundary for uncertain population projection models.

Joan Lubben¹, Derek Boeckner, Richard Rebarber

¹Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA.

Theoretical Population Biology

|December 25, 2008

Summary

This study provides methods to analyze population model uncertainty by identifying leading eigenvalues. These methods help determine parameter combinations leading to population growth or decline, even with uncertainty.

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Area of Science:

Ecology
Mathematical Biology
Population Dynamics

Background:

Population models use matrices or operators to project population size over time.
The leading eigenvalue of these matrices/operators dictates population growth or decay rates.
Parameter uncertainty in population models complicates eigenvalue analysis.

Purpose of the Study:

To develop methods for assessing the impact of parameter uncertainty on the leading eigenvalue of population models.
To provide sufficient conditions for identifying the leading eigenvalue in matrices and integral operators.
To enable clear identification of parameter combinations causing population growth versus decay under uncertainty.

Main Methods:

Considered discrete time linear population models of the form n(t+1)=An(t).
Developed criteria to identify the leading eigenvalue for a broad class of matrices and integral operators.
Utilized preselection of the leading eigenvalue to 1 to analyze parameter uncertainty effects.

Main Results:

Sufficient conditions were established for an eigenvalue to be the leading eigenvalue.
The preselection method effectively distinguishes parameter combinations leading to asymptotic population growth or decay.
Applied to thistle population models (matrix and integral), demonstrating practical utility.

Conclusions:

The developed methods offer a robust way to handle parameter uncertainty in population dynamics.
These techniques can be generalized for any preselected leading eigenvalue.
Facilitates more accurate predictions of population persistence and decline under varying conditions.