Demographics of the Hawaiian Green
Sea Turtle

Modeling Population Dynamics using a
Linear Deterministic Matrix Model:
The Leslie Matrix

Introduction

Previously, we examined how the Leslie matrix is used to model population dynamics. Once a model is created, how do you go about analysing the model? In the case of the Leslie matrix model we proceed to find the eigenvalues of the matrix.

The five stages of the Honu or Hawaiian Green Sea turtle along with estimated annual surviorship and eggs laid are given in the table below.

Five Stage Model for Hawaiian Green Sea Turtle Demographics


Stage Description(age) Annual survivorship eggs laid
1 Eggs, Hatchlings (<1) 0.23 0
2 Juveniles (1-16) 0.68 0
3 Sub adults (17-24) 0.75 0
4 Novice breeders (25) 0.89 280
5 Mature breeders (26-50) 0.92 70

The Leslie Matrix

The Leslie matrix for the five-stage model is found by:

The entries in the matrix are computed using

Stage Survivor Probabilities

Stage Duration in years

Eggs laid per female turtle

where is the annual survivor rate for the ith stage, is the duration in years of that stage and is the number of eggs laid by turtles in that stage.

For

let

and

represents the proportion remaining in stage i the following year and represents the proportion that will survive and move into stage i+1.

For our data the Leslie Matrix L is given by:

Given the initial population at each stage were:

The future population after

The current estimated sea
turtle population by stages

years is estimated to be:

Although the accurate population and survivorship data for green sea turtles is not available, this interactive matrix easily allows us to see the effects of different management options that impact stage-specific survival. The population numbers and survival rates used are based upon research from NMFS.

Eigenvalue Analysis

The eigenvalues or characteristic values of the Leslie matrix are found by calculating

the characteristic polynomial for L, P(l) = Det( L - l I), then solving for l.

The roots of this polynomial are the eigenvalues.

The Mathcad Commands for calculating the eigenvalues and eigenvectors of L.

generates a matrix whose columns are the eigenvectors associated with the eigenvalues of L.

Let

Here is the key to the eigenvalue analysis. By changing bases L , we find L is similar to a diagonal matrix D.

The diagonal entries are the eigenvalues of L.

In theory L = U^-1 D U

n times

which simplifies to:

if

Since the largest eigenvalue is less than 1,

The zero matrix.

That is the population does not reach an equillibrium, but becomes extinct.

Graphically, if we plot the novice breeders and the mature breeders on a graph we see the results.

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Lecture 14 Notes