Lotka-Volterra Modifications for

Endangered Species

Stephanie ConstantinoScott McCalla

Kalab Yohannes

Topic Overview

The kiwi bird is an endangered species indigenous to New Zealand

We used a modified Lotka-Voltera model to show the decrease in population of the kiwi bird

Using differential equations, the graphs show that the kiwi is going extinct in the wild

There are a few ways in which humans are attempting to help the kiwi bird

The KiwiSize of a

chicken

Lays about 2 eggs, 2 to 3 times a year

Can live for up to 40 years in wild

The ProblemEuropeans

introduced dogs and other predators into New Zealand

Only about 5% of kiwi chicks survive to be one year old

Kiwis are going extinct

Logistic Model

Our Model

Coefficients α=4

β=1/5

γ=24

δ=0

M=500 000

N=12 000 000

Our Model

Linearization

(0,0), (12 000 000,0), (0,500 000), and

( -1.8 x 1014,500 000)

.

Equilibrium Pointsequilibrium point (0,0) - source

Equilibrium Pointsequilibrium point (0, 500 000) - sink

Equilibrium Points equilibrium point – (12 000 000,0) -saddle

Equilibrium Points equilibrium point (-1.8x10^14, 500 000) – saddle

Saving the KiwiRemove dogs from New Zealand

Remove them from designated area where kiwis are

Move equilibrium point to positive quadrant

)( ,

• Need to make K coordinate positive• Occurs when

α >

• Increase the reproductive rate of the kiwi, α, which is currently 4,

> 0, aka when

• Lower threshold number of dogs, M, currently 500,000,

• Lower the loss to the kiwi population per encounter with a dog, γ, currently 24,

Conclusion Our model agrees with the situation that kiwis will be extinct

unless humans make more of an effort to conserve them, and validates the existence of Kiwi Preserves in New Zealand.