Python代写 | FIT3139: Assignment 2


Part 1
This exercise uses the Lotka-Volterra model to explore the sustainability of an ecosystem with rabbits, foxes
and hunting. We start by exploring the e ect of hunting rates rates in a deterministic model.
A stochastic extension further allow us to explore the sustainability of the ecosystem using a combination
of Gillespie and Montecarlo simulations.
Questions Part 1A [2 marks]
The system tracks the number of rabbits x, and foxes y, and the dynamics are given by:
_ x = g(x; y)
_ y = h(x; y)
In the absence of foxes and hunting, rabbits grow exponentially with a growth rate b and a death rate
d (b > d); the number of rabbits eaten by foxes is proportional to the product of the number of rabbits
present and the number of foxes present by a factor . Foxes decline exponentially in isolation, with a
birth and death rate b0
and d0
, respectively (d0
> b0
). A parameter is the proportional constant of foxes
surviving by eating rabbits { thus foxes in the presence of rabbits increase at a rate b0
+ x proportional
to the number of foxes. Rabbits are hunted at a rate f, where as foxes are twice as hard to hunt.
(a) Formulate g(x; y) and h(x; y) re ecting the assumptions above∗.
(b) Using your own RK2 numerical integrator, inspect the dynamics of this system for di erent hunting
rates f. Assume b = 0:8, d = 0:1, b0
= 0:1, d0
= 0:6, = 0:04, and = 0:01. The initial population
consist of 25 rabbits, and 15 foxes. You should study at least 3 di erent hunting rates, and present
the time-evolution of the system as well as a phase plot of the dynamics for each parameter set.
What should you submit for this question?
You will have to submit
• Your own implementation of RK2.
• For each of your chosen f rates, a time-evolution plot, a phase plot and an interpretation of the