Stage Based Population Projection Matrices and the Power of Linear Algebra

Michael Bruce and Emily Shernock

Leslie Matrix Population Projection Technique Developed in 1945 Each age class represents a stage Fi represents reproductive output

at age class i Pi represents reproductive output

at age class i

Initial Age Distribution Vector

V1

V2

V3

.

.

V i

vt =

The Leslie Matrix

F1 F2 F3 … F i-1 F i P 1 0 0 … 0 0 0 P 2 0 … 0 0 0 0 P 3 … 0 0 . . . … . . 0 0 0 0 P i-1 0

L =

Population Projection

Lvt vt+1

L2vt vt+2

L3vt vt+3

. .

. .

. .

Livt vt+i

=

=

=

=

Lefkovitch Stage Based Population Matrix The Leslie matrix is only a specific

case of a general matrix Lefkovitch matrix does not relate

age and stage Organisms do not always develop

at the same rate Some animals are not easily

identified as to age

New Factors for Lefkovitch Technique Fi represents reproductive output

at age class i Pi represents the probability of not

perishing and not changing stages Gi represents probability of

surviving and moving on to next stage

General Form of the Lefkovitch Matrix

P 1 F2 F3 … F i-1 F i G1 P 2 0 … 0 0 0 G2 P 3 … 0 0 0 0 G3 … 0 0 . . . … P i-1 . 0 0 0 0 G i-1 P i

Carreta-Carreta

Loggerheads in Danger

Listed as “threatened” species in 1978

Conservation focused on protecting nests

Could this focus be bettered?

Constructing Our Stage Based Matrix

Class Number

Survival Rate Average Number of Eggs Laid Per Year

1 0.6747 0 2 0.7857 0 3 0.6758 0 4 0.7425 0 5 0.8091 127 6 0.8091 4 7 0.8091 80

Our Stage Based Population Projection Matrix

0 0 0 0 127 4 80

0.6747 0.7370 0 0 0 0 0

0 0.0486 0.6610 0 0 0 0

0 0 0.0147 0.6907 0 0 0

0 0 0 0.0518 0 0 0

0 0 0 0 0.8091 0 0

0 0 0 0 0 0.8091 0.8089

A =

Growth Rate Calculation

m = er

m = 0.9450 and r = -5.65%

Impact of Various Stages When Their Survival Is 100%

Stage Number Altered Growth Rate

1 -0.04

2 0.033

3 0.11

4 0.04

5 -0.04

6 0.04

7 0.02

Conservation Efforts Misdirected

Data from Lefkovitch matrix shows where to focus conservation efforts

Egg and nest conservation not the best focus

The Main Danger Facing Sea Turtles

The Turtle Excluder Device

Conclusion

TED’s required in most U.S. waters of Gulf of Mexico and South Atlantic

Florida Loggerhead population stable

Florida home to one third of world Loggerhead population