Problem sheet 1

1. Early warfare

Originally¹, warfare was conducted through hand-to-hand combat. Thus, a single person had the ability to despatch only one other person at a time. Namely, the combat was one on one until one person was unable to fight any longer. Consider two opposing armies of size \(A(t)\) and \(B(t)\) and suppose \(A(0)=A_0\), \(B(0)=B_0\), where \(A_0>B_0\).

1. Suppose the armies are equally matched such that when members of the \(A\) and \(B\) armies meet they are both equally likely to win the fight. Further, assume that the interaction rate constant between the two armies is \(r\). Write the above combat interaction down as two interaction equations.

2. Show that the Law of Mass Action can be used to convert the early warfare interactions into the following ODEs,

   \[\begin{align} \dot{A}=-rAB,\quad A(0)=A_0,\tag{1}\\ \dot{B}=-rAB,\quad B(0)=B_0. \tag{2} \end{align}\]

3. Derive the steady states of equations (1) and (2). What happens when you try to characterise their stability?

4. Show that

   \[A(t)=B(t)+\textrm{Constant}, \tag{3}\]

   where you should define the constant explicitly.

5. In question 3 you should have found that there was an infinite number of steady states. Show how equation (3) collapses all possibilities to just one.

6. Construct the \((A,B)\) phase plane and use it to characterise the steady states’ stability.

7. Show that equations (1) and (2) can be rewritten as

   \[\begin{align} \dot{A}=-rA(A-A_0+B_0),\tag{4}\\ \dot{B}=-rB(B+A_0-B_0). \tag{5} \end{align}\]

8. Derive the steady states of equations (4) and (5).

9. Considering equations (4) and (5) as separate scalar ODEs characterise the stability of the previously derived steady states analytically and graphically through constructing the phase planes \(\left(A,\dot{A} \right)\) and \(\left(B,\dot{B}\right)\).

10. Consider equations (4) and (5) as a system and characterise the stability of the steady states algebraically and graphically by plotting the \((A,B)\) phase plane.

11. Do the results from questions (9) and (10) correspond to the results from question 6?

12. Solve equations (4) and (5) explicitly and sketch the resulting \((t,A(t))\) and \((t,B(t))\) curves on the same axes.

13. You have now derived the same result in three different ways. Without referring to the mathematics, interpret your result.

2. Non-dimensionalisation and interpretation

Consider the following set of interaction equations for two animal populations \(N\) and \(P\).

where \(b, e, K, r\) and \(m\) are positive constants and the initial conditions are \(N_0\) and \(P_0\), respectively.

1. What type of interaction is occurring between \(N\) and \(P\)?

2. Write down a set of interaction equations that could give rise to equations (6) and (7).

3. Non-dimensionalise equations (6) and (7) to get the form

   \[\begin{align} &\frac{\text{ d}u}{\text{ d}\tau} =c_1u\left(1- u\right)-uv,\tag{8}\\ &\frac{\text{ d}v}{\text{ d}\tau} =c_2uv-v, \tag{9} \end{align}\]

   where \(c_1\) and \(c_2\) should be defined explicitly in terms of \(K, b, e, r\) and \(m\).

4. Provide an interpretation of \(c_1\) and \(c_2\).

5. Suppose the predator is a pest and we want to wipe the predator out, whilst maintaining a positive prey population (e.g. consider a fox and hen scenario). What strategy best achieves this, in terms of altering the sizes of \(c_1\) and \(c_2\)? Explicitly, is it best to:

   • increase \(c_1\)?

   • increase \(c_2\)?

   • increase both?

   • decrease \(c_1\)?

   • decrease \(c_2\)?

   • decrease both?

   • increase \(c_1\) and decrease \(c_2\)?

   • decrease \(c_1\) and increase \(c_2\)?

   Explain your choice.

6. Having chosen a strategy, consider how best to enact the changes of \(c_1\) and/or \(c_2\) in terms of altering \(b, e, K, r\) and \(m\). What are the pros and cons? What would your advice be on the best strategy?

3. Using ode45

As in MA0232 I will be offering you the chance to have a go at understanding Matlab. It is free to download; all you need is to follow this (clickable) link https://uk.mathworks.com/academia/tah-support-program/eligibility.html and sign up using your Cardiff email address.

Throughout this course we will be simulating ODEs of many types and, thus, it is good to be able to check your answers. The more you simulate, the better your intuition. In order to simulate ODEs we are going to use Matlab’s inbuilt command ode45. Matlab has several ODE solvers, but ode45 is the most general “catch all” code. Essentially, you try it first and if it does not work you try something more specific. However, for our purposes it will be adequate.

For a high-level description of ode45 type help ode45, or doc ode45 into Matlab and it will bring up a brief description and a full description, respectively. Here, I will provide a brief overview of a general code that will work on most systems throughout this course.

The code below should be available as General_ode45_code.m and it solves problem 1. Either download it directly, or copy and paste it into Matlab. The green text contains comments to help you understand what is going on. Essentially, for different ODE systems you should only really have to change the description in lines 49-50. Note you may also need to change the length of time over which the equations are solved (Tend).

%% Ensure we start from a blank slate
clear all
close all
clc

%% Initialise variables
fs=15; % Set fontsize

Tend=0.3; % Set final time
N=100; % Set number of steps
t=linspace(0,Tend,N); % Set number of time points at which the solution is to be evaluated
IC=[200;
    190]; %Initial conditions as a column vector

%% Solve ODE

% Each component is a row in the matrix y.
[t,y] = ode45(@ODE_function,t,IC);

%% Plot Solution

%%% (time,solution plots)
hold on % Allows us to plot multiple graphs at once
plot(t,y(:,1)) % Plots the first solution
plot(t,y(:,2)) % Plots the second solution
legend('u','v') % Adds a legend and defines the lines for clarity

% Label axes for clarity
xlabel('t') 
ylabel('Solutions')

set(gca,'fontsize',fs) % Changes the axis fontsize

%%% Phasespace plots
figure % Starts a new figure without overwriting the previous one
plot(y(:,1),y(:,2))
xlabel('u') 
ylabel('v')
set(gca,'fontsize',fs)




%% ODE description
function dydt=ODE_function(t,y)
r=1; %Interaction parameter

% Defining the ODEs as a column vector. Each row is one of the ODEs.
dydt=[-r*y(1)*y(2) %Line 49
      -r*y(1)*y(2)]; %Line 50

end

Once you have had a play with the above code try and adapt it to solve equations (8) and (9).

Exam Revision

Problem sheets 3-5 of course MA0232 contain many examples and solutions that will provide adequate practice at: deriving steady states, non-dimensionalisation, characterising stability and sketching graphs.