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, foxes increase at a rate b0
+ x. 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

Questions Part 1B [5 marks]
When populations are nite and small, stochastic e ects can be signi cant. In the case of the population
model above, a stochastic model can account for the extinction of very small populations.
To study these extinction events,
(a) Formulate and implement a Gillespie algorithm of the model above, and do a similar exploration of the
same parameters, now using the stochastic model. What features are now salient due to stochasticity?
(b) We de ne the system to be sustainable if the probability of extinctions in a time horizon of 10 years is
less than 0:2. Using Montecarlo as well as your Gillespie implementation, determine what is the largest
hunting rate that guarantees a sustainable ecosystem. Explain how you set up your computational
experiments and the reasoning behind your answer.

What should you submit for this question?
You will have to submit
• Description and implementation of the Gillespie algorithm.
• Discussion of the e ects of stochasticity, comparing the deterministic and stochastic models.
• Explanation of how to estimate, using Montecarlo, the probability of extinctions happening in a time
horizon up to t = 10. Reasoning and evidence of your chosen hunting rate to guarantee a sustainable
system as de ned.