Lotka-Volterra
Modifications for
Endangered Species
Stephanie Constantino
Scott 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 Kiwi
 Size of a
chicken
 Lays about 2
eggs, 2 to 3
times a year
 Can live for
up to 40 years
in wild
The Problem
 Europeans
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 Points
 equilibrium point (0,0) - source
Equilibrium Points
 equilibrium 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 Kiwi
 Remove 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
> 0, aka when
• Increase the reproductive rate of the kiwi, α,
which is currently 4,
α>
• Lower the loss to the kiwi population per
encounter with a dog, γ, currently 24,
• Lower threshold number of dogs, M, currently 500,000,
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.