3.2 Predator Prey Systems

Question 1

State the prey equation and define all terms

Answer 1

dR/dt = (λ - αF) R
dR/dt - Rabbit growth rate 
λR - Constant rowth rate in absence of predators + infinite grass
αFR - Rate rabbits are eaten 
F - Number of foxes
R - Number of rabbits

Question 2

State the predator equation and define all terms

Answer 2

dF/dt = - (η - βR) F
dF/dt - Growth rate of foxes
-ηF - is the rate of fox competition
βRF - Dependence on number of foxes and rabbits

Question 3

Where do the fixed points exist in the predator prey equations?

Answer 3

At R = F = 0 where there is no solution
- saddle point but meaningless
At R = η/β and F = λ/α
- Centre point

Question 4

Describe the properties of the constant of the motion, C, and how we obtain it

Answer 4

Obtain it by integrating dR/dF from the predator-prey equations.
Each value of C specifies an orbit around the centre
- Know the equation for each orbit C
- Global information

Question 5

Describe the phase plane for the F R population graph

Answer 5

Small circle centre

Then gradually more elongated ellipses towards the right hand side of the graph

Question 6

What is the problem with the phase plane graph?

Answer 6

Small changes in the system (error, fluctuations) can lead to large differences in the population later
- (see spaces between the contours which increases)