2023年11月25日

建模代写 | COM3001 MODELLING AND SIMULATION OF NATURAL SYSTEMS

本次英国代写主要为自然建模与仿真限时测试

1. a) General concepts for modelling natural systems.
(i) Describe what a natural system is and give two distinct examples. Give one
example of a non-natural system that in your view could be modelled as a
natural system.
[10%]
(ii) State and briefly explain four valid reasons for modelling a natural system.
[20%]
(iii) You are presented with two models fitting a set of data. One of these models
has 15 free parameters and error 0. The other has 4 free parameters and 2%
error. Give a definition of what a free parameter is. What are the disadvantages
of a model with too many free parameters? Briefly discuss which of the two
models you would use.
[30%]
b) Numerical methods for solving differential equations (Euler’s method).
(i) Explain Euler’s method. As part of your answer, provide the corresponding
algorithm. How should one choose the discretisation time-step ∆t? Justify
your answer.
[20%]
(ii) Use Euler’s method to solve the differential equation dx
dt
= ax, for initial con-
ditions t0 = 0, x0 = 1 and a = 100. Find the values of x(t) for t=0.1 and
t=0.2. (Hint: choose the discretisation time-step ∆t in a convenient way for
this purpose.) How does the choice of ∆t affect the quality of the solution?
[10%]
(iii) A first order linear differential equation can be solved with Euler’s method,
or analytically (via integration). How do the two solutions differ in terms of
accuracy?
[10%]

a) Figure 1 (last page of this document) shows the population of a species x at time
tn+1 versus the population of the same species at time tn (index n refers to the time
discretisation).
(i) Define mathematically when a system is in steady-state and explain what this
means in the context of modelling populations. Explain what the steady-state
points are.
[10%]
(ii) Use Figure 1 to find the steady-state points of the population. Which are stable
and which are unstable (if any)? Interpret and justify your response. (Hint:
you may want to copy the figure in your answer sheet and use it to justify your
response.)
[15%]
(iii) Which differential equation describes the growth of the population x (shown
in Figure 1)? Justify your response. The plot in Figure 1 can be described
as a line y = βx. Express mathematically the parameter(s) of the differential
equation as a function of β.
[25%]
b) Solve the differential equation αdx
dt
= x(1 − βx). Also, find the stable steady-state
point(s), if any. Assume α > 0 and β > 0 (Hint: you don’t need to solve the equation
to do the stability analysis). [25%]
c) (i) What is a spiking neuron model? Briefly describe the integrate-and-fire model.
[15%]
(ii) Describe the problem of neuronal coding. As part of your reply, give one example
according to which information could be transmitted in brain systems.
[10%