Y2 Computational
Physics Laboratory
Exercise 4
Modelling physical systemsa predator-prey
simulation
Aims and objectives:



To gain exposure to using computers to model complex, physical systems
To make use of ‘expertise’ acquired in other parts of the degree program to
make predictions of the behaviour of systems, based in inspection and
manipulation of equations; to understand results of the code in the light of the
predictions
To be comfortable using functions, and passing variables between them
Duration: 3 weeks
Introduction
This exercise is intended to introduce you to the use of computers as invaluable tools
for the study of complex systems characterised by interdependent variables. The
system we have chosen to scrutinise is the simplest form of the so-called ‘predatorprey’ class of models. As the name suggests, these models have important application
in a biological and ecological context. However, they are also of use to economists,
and physicists. Examples of physical systems for which predator-prey-like models
find useful application are in the complex dynamics of turbulence, and the
competition between particles for water in rain clouds.
The system we will study is the Lotka-Volterra model, which is described by two
linked differential equations, and was developed, independently, by two
mathematicians. Vito Volterra was an Italian mathematician who was asked by his
son-in-law (a biologist) to try and explain the varying population numbers of fish in
the Adriatic sea, on the basis of available data on growth, death and interactions rates
of predator and prey species. Alfred Lotka developed the same set of equations for
describing a hypothetical chemical reaction. The most famous example of the use of
this simple model in a real biological system has been for explaining numbers of
Canadian lynx (predator) and snowshoe hare (prey) [of which more later].
In this exercise you will first make some predictions of the expected behaviour of the
Lotka-Volterra system, from inspection and manipulation of the mathematical form of
the equations. You will then write a C++ program to solve, numerically, the equations
to show the behaviour over time. You will be making use of numerical techniques
very similar to those introduced in Exercise 2.
The Lotka-Volterra Model
To introduce the model we will use the language of biology, and talk in terms of
species of predator and prey, death, food etc. The system describes the interaction
between one predator species and one prey species. Several, key assumptions are
implicit in the model (assumptions which would be unrealistic in most real biological
scenarios). First, we assume the prey has an unlimited supply of food. In the absence
of any form of predator, the numbers of prey, x, would grow exponentially over time,
t, i.e.,
dx
 ax .
[1]
dt
In the above, the constant, a, provides suitable normalization and reflects, implicitly,
natural birth and death rates of the population. Next, let us introduce the effect of the
single predator species, y. We assume the rate at which predators ‘find’ prey is
proportional to the product of the total numbers of predators and prey; and that some
fraction of these encounters leads to death of prey. In mathematical form, we
therefore now have:
dx
 ax  bxy .
[2]
dt
Here, the constant b fixes the death rate.
This is all well and good, but it says nothing about the predator population. In the
absence of any source of prey, the number of predators would fall exponentially (at a
rate fixed by some constant, c):
dy
 cy .
[3]
dt
Let us assume predators rely solely on the prey species for their food. The growth rate
of the predator population is assumed to go in proportion to the product of the two
population totals. We might choose to normalize this with the same constant, b, which
was used to fix the death rate of prey. However, to keep things as general as possible
we use a fourth constant, p, so that:
dy
 cy  pxy .
[4]
dt
Notice the analogy between Equations [2] and [4], and the importance of the signs of
the terms. These two equations give us the Lotka-Volterra model:
dx

 ax  bxy 

dt

dy
 cy  pxy

dt
[5]
Problem 1: Getting to grips with the modelsome predictions
For the first part of the exercise, we begin by making some predictions of the likely
behaviour of the model, based on inspection (and some manipulation) of Equations
[5].
The problemwhat is required? Your write-upwhat is needed?
1. Look carefully at the equations. What are the expected equilibrium values for x
and y? Include a clear explanation in your notebook. [1 mark]
2. Combine the equations to eliminate dt; separate the variables and integrate the
equations. You should get the following result, which describes a family of
curves:
a ln y  by  c ln x  px  constant .
[6]
Include a neatly laid out derivation in your notebook. [1 mark]
3. What form do these curves take? How might the values from question (1) fit in?
What do the forms of the curves imply for how x and y might vary over time?
[1 mark]
You should use your ‘curve sketching’ skills to help you here. It may also help to
test some ideas, and check some numbers, with MathCAD or Excel. (Hint: think
about the case where all the constants are unity.) Answer the questions, and
include clear, concise explanations of your logic in your notebook, together with a
rough sketch of the expected forms for the curves. (Very rough, but neat, will do!)
Problem 2: A numerical solution of the Lotka-Volterra model
Equations [5] can be solved numerically, and estimates of the population numbers, x
and y, made as a function of time using techniques like Euler’s method, which we met
in Exercise 2. Let us recap briefly.
Here, the small increment h used in Euler’s method is a small increment in time (say
t). The Euler formulas for the Lotka-Volterra model are then:
x(t  t )  x(t )  [ax(t )  bx(t ) y (t )]t 

y (t  t )  y (t )  [cy (t )  px(t ) y (t )]t 
[7]
Provided we have initial values for x and y, and chosen values for the constants a, b, c
and p, Equations [7] can be used to estimate x and y versus time. A plot of pairs of x
and y can then also be made (the ‘trajectories’ of the system; cf. question (2) above).
Exercise 2 also introduced us to some of the shortcomings of Euler’s method.
Accuracy can be lost, even for very simple differential equations. Here, you will use a
slightly more sophisticated version of Euler’s method.
In the basic method, the slope of the function is determined only at the starting-point
of the step. We know this can cause problems if the slope changes significantly over
the interval. To help retain accuracy in the solution a small increment, h, is therefore
required. A desirable improvement would be to return an estimate of the slope
averaged across the entire range of the increment. But this would need an estimate of
the slope at the as yet unknown end-point of the interval!
There is a way forward. We can do things in two steps. First, a predictor stepwe use
the basic Euler method to get an estimate of the function (say ŷ) at the end of the
increment. We then use ŷ in place of the actual value of y (t+t) to get an estimate of
the slope at the end of the incrementthis is the second, corrector step. We can then
get our prediction for the next value of y using an average value of the estimated
slopes from the start and end points of the increment.
This is one example of the family of Runge-Kutta methods. In fact, it is a secondorder Runge-Kutta method, i.e., we use an evaluation of the function at two points
along each step to get the slope.
In this exercise we are going to use a more accurate fourth-order approach, where the
mean slope is instead determined from four points along the step. Derivations of the
fourth-order formulae (which make use of the Taylor expansion) can be found in
numerical textbooks. The formulae you will need are as follows:
x(t  t )  x(t )  [k1  2k 2  2k 3  k 4 ] / 6
y (t  t )  y (t )  [m1  2m2  2m3  m4 ] / 6
Where:
k1  t f ( x, y )  t [ax(t )  bx(t ) y (t )]
m1  t g ( x, y )  t [cx(t )  px(t ) y (t )]
k 2  t f ( x  0.5k1 , y  0.5m1 )
m2  t g ( x  0.5k1 , y  0.5m1 )
k 3  t f ( x  0.5k 2 , y  0.5m2 )
m3  t g ( x  0.5k 2 , y  0.5m2 )
k 4  t f ( x  k 3 , y  m3 )
m4  t g ( x  k 3 , y  m3 )
To clarify: f (x, y) is the right-hand side of the first Equation in [7], assessed at time t.
The more complicated f (x + 0.5k1, y + 0.5m1) means x(t) should be substituted by
x(t)+0.5k1 before the right-hand side is computed; and, likewise, y(t) should be
substituted by y(t)+0.5m1 prior to calculation. And so on through the progression.
The problemwhat is required?
1. Write a C++ program to generate estimates of the predator and prey populations, x
and y, over time, using the fourth-order Runge-Kutta method.
2. Code should be written to allow the user to alter the values of the constants, and
the step increment t. It should output the results to file [i.e., t, x(t), y(t)] to
facilitate graphical output of the results, using MathCAD or Excel.
3. Start by investigating the case where the constants a = b = c = p = 1. Plot, on the
same graph, estimates of x(t) and y(t) versus time, covering t = 0 up to about 30
seconds. [You will need to find a suitable value for t for the computations.] Also
produce a plot of y(t) versus x(t) for this time period. Is the phase relation between
the curves roughly what you might expect? How do these results compare with
your predictions from Problem 1?
Your write-upwhat is needed?
1. You should include a printout of your code, and a description of it. Detail should
be sufficient to allow the marker to easily follow, and understand, the logic and
function of the code. The code itself should be well commented. [2 marks]
2. You should include clearly labelled plots of the population numbers versus time,
and versus each other, for the case where a = b = c = p = 1. [0.5 marks]
3. You should comment on the relation of the two curves: is this what you might
naively expect (think in ‘predator’ and ‘prey’ terms). Also discuss the results in
the light of your predictions from Problem 1. How do your predictions of the
equilibrium numbers fit in? Does your prediction for the shapes of the curves
match what is seen? [1.5 marks]
Getting started: Passing variables between functions
To keep things tidy, you ought to be thinking in terms of dividing your code into
functions to do the job. Your main program could contain a loop, over the increments
in time; inside the loop functions would then be called to calculate the Runge-Kutta
estimate for that particular time step. Think carefully about what is required (you may
even want to have a function call another function).
If you have forgotten some of the procedures and syntax pertaining to functions, you
will need to look back at the appropriate pages of the Y1 C++ course.
‘Introduction to User-defined functions in C++’
There are different types of functions. Those that have a few variables passed to them,
and which then use these values to calculate a value, which is passed back by the
function:
‘Functions that return values’
Then there are functions which have parameter values passed to them, but which do
not return a function value:
‘Functions with parameters and no return value’
You might ask: how might this second set of functions be useful? They might simply
do some final calculations in the routine, or output the data etc. However, they can
also modify the parameter values passed to them, and return the modified values to
the main program, so that other routines can work on the modified values. This sort of
function will be of use to you here. The population numbers, x and y are variables that
change over time. Initial values might be passed to a Runge-Kutta subroutine,
modified to get the new values (at the next time step), and then returned to the main
program for use as initial values for the next loop.
You must, however, be careful. Unless you tell the program that the function can alter
the actual parameter values, it will only work on copies of them, and the values will
not be altered in the rest of the program. You therefore need to specify the parameters
to be altered are of the ‘call-by-reference’ type.
‘Call-by-reference parameters’
An example of such usage can be found in:
‘Passing parameters between functions’ [From the module pages]
Problem 3: Lynx and hare
Over the course of the second half of the nineteenth, and the first part of the twentieth
centuries, the Hudson Bay Company kept records of numbers of pelts of Canadian
lynx (predator) and snowshoe hare (prey). These fur data, some of which are
reproduced in the table below, may be regarded as a reasonable proxy for the actual
populations (after multiplication by a suitable correction factor).
The problemwhat is required?
1. On the same plot, make a graph of the numbers of predator and prey pelts as a
function of time.
2. See if you can alter the constants in your computer model to match the pelt
records. You should take initial values from the table, i.e., x = 30 and y = 4.
Year Hares (103) Lynx (103) Year Hares (103) Lynx (103)
1900
30
4
1911
40.3
8
1901
47.2
6.1
1912
57
12.3
1902
70.2
9.8
1913
76.6
19.5
1903
77.4
35.2
1914
52.3
45.7
1904
36.3
59.4
1915
19.5
51.1
1905
20.6
41.7
1916
11.2
29.7
1906
18.1
19
1917
7.6
15.8
1907
21.4
13
1918
14.6
9.7
1908
22
8.3
1919
16.2
10.1
1909
25.4
9.1
1920
24.7
8.6
1910
27.1
7.4



A couple of hints:

An average, over time, of the pelt records will give an estimate of each
equilibrium value. Remember how these relate to the constants (see
Problem 1).

Approximate estimates for a, and c, can be made from the steepest rising
and falling parts of the lynx and hare curves. Assume exponential growth
(or decay) dominates; simplified equations then describe the evolution of
the hare or lynx over time. Manipulate these into a form where you can use
data on the ‘steepest’ part to estimate the constants.
Your write-upwhat is needed?
1. You should include a plot showing the observed time variation of the fur pelts,
and your best prediction from your computer model. State the values of the
constants, which best matched the records. You should include a brief discussion
of how you were led to the quoted values for the constants (cf. ‘hints’ above).
[2 marks]