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MXB261 - Modelling and Simulation Science

Assignment 2 - A Simulation Project

Due Date: end of Tuesday 22 October 2019 (Week 13)

This is a Group Project with 4 students per group, and is worth 40% of your final grade.

There are four individual tasks available, and each student must choose a different task.

If circumstances dictate that your group has only 3 students, then you may omit Task 3. Note:

Tasks 1, 2 and 4 are still to be completed.

The final mark for each student is made up of the score for their individual component (out of

20 marks), plus a contribution from the Group Report (out of 20 marks) - this contribution will

depend on the percentage the student contributed to the report. For example, if all students

contributed equally (25% each), then all will receive the same score for the report, otherwise the

mark will be reduced according to the percentage effort.

The individual component includes an oral discussion (5 of the 20 marks) of your contribution,

with your tutor. This discussion will cover your understanding of the programming (and any

issues) associated with the individual task you have selected. You should find an opportunity to

complete this oral discussion during the workshop in Week 11, 12 or 13.

The theme of this project is how parameter values can determine the system dynamics of a

parasite model, in both a temporal and spatial setting.

The group is to submit a zip archive containing all the m-files, movie files (.avi), a pdf of the

report, and a Statement of Contribution showing who has worked on which individual task,

together with the percentage contributed to the Group Report; these percentages must total

100.

The Parasite Model

Consider the following differential equation system

dX1 dt = k1X1X2 ∈ k2X1 dX2 dt = k3 3 k4X2 ∈ k5X1

where X1 represents a population of parasites that feeds off a population X2. Parameter k1

represents the birth rate of X1, k2 represents the death rate of X1, k3 represents the rate of food

growth, k4 represents food decay, and k5 represents food consumption by the parasites. All these

constants are positive. You will solve the differential equation system using MATLAB’s ODE45.

1

Task 1: Delay Dynamics and a Parameter Sweep

Consider first the one-dimensional logistic equation of parasite growth

dX(t)

dt = r X(t) 1 1X(t) K

where K is the carrying capacity and r is the growth rate. We can simulate this using the

difference scheme

Xn+1 = Xn (1 + h r r h rXn/K)

with h being the stepsize.

Now introduce a delay term that could, for example, represent a delay in the food source

for the parasites:

dX(t)

dt = r X(t) 1 1 X(t t τ ) K .

The initial value is X0, defined on [[τ, 0], where τ represents the delay. The associated

difference scheme is (with τ = s h) Xn+1 = Xn (1 + h r r h rXn s/K).

For example, if s = 1, we will have an initial value for the system defined on [[h, 0] (with

a delay of h), while s = 10 corresponds to the initial values defined on [[10 ∗ h, 0] with a

delay of 10h.

(a) For K = 100, X0 = 50, r = 2, h = 0.01, investigate the dynamics of the numerical

solution for delays s = 50, 100, 150, 200, 250, 300, 350, 400 over 2000 steps. Repeat this

investigation for r = 12

, h = 0.05, with the same delays. You should see an oscillatory

solution in the delay case. What is the relationship between the oscillatory peaks, their

period, and the delay term? What do you observe about the parasite population?

Now consider the 2D representation of the parasite/food system given on page 1 of the

assignment specification.

Suppose the following parameters have values as indicated:

k1 = 1, k2 = 2, k5 = 3

and that (after some scaling), the initial values are X1(0) = 1, X2(0) = 1. The time span

for the investigation is [0, T] where T = 20 units.

(b) In addition to these values, consider first the case that k4 = 4. You are to perform

a parameter sweep on k3 ∈ [0, 50] to find k3-values that result in X1 → 0 + Tol or

X2 → 2 ± Tol, for a specified tolerance Tol (reasonable values for Tol are 10 1 or

10 2

). It is also a requirement that both populations X1 and X2 do not drop below

0; if this happens, you must discard those parameter choices. Plot your successful

k3-values, according to the characteristics you observe in the system dynamics. You

should determine the boundary of the regions that characterise k3 with respect to the

observed dynamics.

(c) As above, but now implement a parameter sweep on both k3 and k4 (over [0, 50]). Plot

successful parameter pairs (k3, k4), showing the region(s) corresponding to different

system dynamics.

2

(d) Now fix k3 = 10 and perform a similar parameter sweep on both k4 and k5 over

[0, 50], plotting successful parameter pairs (k4, k5). Are your results consistent with

the previous parameter sweep from (b)? Why/why not?

(e) Write a concise paragraph to explain your results.

Task 2: Latin Hypercube Sampling in 3D

In this task, we fix two of the parameters: k1 = 1, k2 = 2. You are to implement Latin

Hypercube Sampling on the 3D space of parameters k3, k4, k5 (each in the range [0,50]). For

an appropriate mesh size and number of trials, build a population of successful parameter

3-tuples, that reflect the system of equations exhibiting the characteristics that either

X1 → 0 + Tol or X2 → 2 ± Tol, with the constraint that neither X1 nor X2 drops below

zero in [0, T]; if this happens, you must discard those parameter choices. Take T = 20 time

units, and have X1(0) = 1, X2(0) = 1.

You are to write your own code to carry out the Latin Hypercube Sampling. You can

check that your code is working by comparing results with the built-in MATLAB command

lhsdesign.

In your exploration of the 3D parameter space, you are to group your successful parameter

3-tuples so that a region in the 3D space corresponds to particular characteristics of the

system dynamics. In addition to this, you should find the boundary between these regions.

Demonstrate your results in a 3D visualisation that is appropriately labelled (axis labels

and title). Write a concise paragraph to explain your results.

Task 3: Latin Hypercube Sampling in 4D

This task requires a Latin Hypercube Sampling to be implemented as for Task 2, but over

a 4D parameter space. Here k1 is fixed at k1 = 1. Again, the parameter space is explored

to determine regions that correspond to certain system dynamics.

Visualisation of the successful parameter 4-tuples is required, with appropriate labelling.

Discuss the approaches you take to represent the 4th dimension in your plots. Write a

concise paragraph to explain your results.

The Latin Hypercube Sampling requirements of Task 3 are very similar to the coding required

for Task 2. It is expected that the students involved in Tasks 2 and 3 may like to discuss

their programming approach for the sampling, but then to develop their own visualisations.

3

Task 4: Spatial agent-based implementation

The spatial implementation allows exploration of the system dynamics when interactions

take place at an individual level rather than at population level.

Construct a grid of cells (200 x 200) and populate that grid with F agents for food and

P agents for parasites, according to various densities (see below). The parasites will be

positioned randomly, while the positions of the food will be either random or localised

according to the food-placement strategy being investigated. There is to be one agent only

per cell. The rules for the simulation are as follows:

• Each parasite in turn will attempt to move to a neighbouring cell that is N, S, E or

W of its current position; if the new cell is empty, the parasite moves; if the new cell

is already occupied by a parasite, the move does not take place; if the new cell is

occupied by food, the food is consumed (the food is replaced by the parasite), and a

birth event takes place with the new parasite being placed in the original cell.

• A parasite dies after f1 iterations and is removed from the grid.

• For all food agents, each agent dies if a uniform random sample (u ∼ U(0, 1)) u < f2,

and it is removed from the grid. Note u has to be different for each food agent.

• After all the parasites and food agents have been processed for the current step, create

f3 new food agents and position them either randomly in empty cells, or localised (in

empty cells). Localised means in a region such as a quadrant of the grid. Note that

when choosing a cell at random, if it is already occupied, place the new food agent in

a nearby unoccupied cell.

For a selection of parameter values, and for various initial F and P population densities of

10%, 20%, 30% and 40% of the grid (assume F and P have the same population density),

simulate the evolution of the system and hence describe the relationship between initial

population density, food-placement strategies, and the observed system dynamics. For the

scale of this spatial simulation, it will be useful to consider a value of f1 ∈ [0, 15], a value

of f2 ∈ [0, 0.1], and f3 values such as 100, 200, 300 or 400.

To demonstrate your simulation, create a movie of the spatial stochastic model above,

capturing the output at regular time points as the simulation evolves.

4

The Group Report

The purpose of the Group Report is to combine the results from the individual tasks, and to

compare and contrast the results obtained from the different approaches and strategies.

The Report (which should be approximately 10 pages) should have the following structure:

• an Introduction, setting the scene for the content;

• a Methods (by Task) section, where you describe the approaches taken, for each Task;

include here an analysis of the equilibrium solutions of the parasite model;

• a Results and Discussion section, where you combine, compare and contrast the results

for parameter regions and the characteristics of the system dynamics from Tasks 1, 2 and

3; include here plots that justify your answers; also include results from Task 4 (the spatial

agent-based simulation) and the interpretation of the effect of initial population densities

and food-placement strategies; include a screenshot taken from the movie;

• a Conclusions section, where you summarise the main results in no more than a few

sentences.

Guide to the Marking Schedule

5

Marks Breakdown

Individual component

Structure 2 2 Code is well structured and well documented.

0 Code is not structured and/or not documented.

Solution 8

8 Solution is correct, all aspects of the task have

been considered.

4 Parts of the solution may be incorrect, most aspects of the task have been considered.

0 Little/no solution.

Methods (by task) 3

3 Methods for the task have been clearly and correctly outlined for reproducibility.

1 Methods for the task have been outlined, but

may not be correct, clear or reproducible.

0 Little/no methods.

Results 2 2 Relevant and concise figures that clearly demonstrate the solution have been produced. Figures

are clear, correctly and completely labelled.

0 Irrelevant, excessive or no figures that do not

clearly demonstrate the solution.

Oral component 5 5 Conveys a depth of understanding of the programming and issues for the Individual Task.

2.5 Conveys some understanding of the programming and issues for the Individual Task.

0 No oral discussion, or poor level of understanding conveyed.

Subtotal 20

Group component

Structure and presentation 3 3 Report has a clear and logical structure. Presentation is of a professional standard. No major

spelling/grammatical errors.

0 Report does not have a clear or logical structure.

Presentation is not professional. Major spelling

or grammatical errors.

Introduction 3 3 Introduction provides a short relevant background for the exercises performed in the report.

Motivation for each task and an outline of the

structure of the report is provided.

1.5 Introduction missing key components, but outlines the main features of the report.

0 Little/no introduction.

Discussion and comparison 10 10 Discussion shows concisely presented, but extensive, detailed and relevant insight. Results from

individual tasks have effectively been compared.

Plots, figures and tables are clear, concise and

relevant.

5 Discussion shows some relevant insight. Results from individual tasks have been compared.

Plots, figures and tables may not be clear, concise and relevant.

0 Little/no discussion or comparison.

Conclusions 4 4 Conclusions accurately and concisely summarise

main findings of the project.

2 Conclusions summarise some main findings of

the project.

0 Little/no conclusions.

Subtotal 20

Total 40

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