本次澳洲代写主要为python计算建模相关的assignment

Background

This exercise uses the Lotka-Volterra model to explore the sustainability of an ecosystem with rabbits, foxes

and hunting. We start by exploring the eect 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 dierent 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 dierent 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

dynamics.

Questions Part 1B [5 marks]

When populations are nite and small, stochastic eects can be signicant. 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 dene 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.

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