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Introduction to Simulation WS01/02 - L 02 1/39 Graham Horton
Modelling with ODEs
24.10.01
Introduction to Simulation WS01/02 - L 02 2/39 Graham Horton
Motivation and Goals
• Motivation:– Ordinary Differential Equations (ODEs) are very
important in all branches of Science and Engineering– ODEs form the basis for the simulation of almost all
continuous phenomena
• Goals:– To introduce ODEs– To give some examples of how to build simple ODE
models
Introduction to Simulation WS01/02 - L 02 3/39 Graham Horton
Continuous Processes
• Continuous processes occur everywhere.• Here, we are interested in cases with discrete variables• Some examples:
– An object falling to the ground– The motion of the planets orbiting the sun– The current and voltage in an electrical circuit– The level of alcohol in my blood on January 1st– The populations of a predator and its prey (!)
• In almost all cases, the relationships between the variablesare defined by an ODE
Introduction to Simulation WS01/02 - L 02 4/39 Graham Horton
Ordinary Differential Equations
• An Ordinary Differential Equation (ODE) describes the rateof change of a quantity y as a function of its current valueand the time t :
• In addition, we usually know the value of y(0):
),( tyfdtdy =
0)0( yy =
Introduction to Simulation WS01/02 - L 02 5/39 Graham Horton
Initial Value Problems
• Given an ordinary differential equation ...
• ... and an initial condition ...
• ... find
),( tyfdtdy =
0)0( yy =
)(ty
Introduction to Simulation WS01/02 - L 02 6/39 Graham Horton
Example for an IVP
• Differential equation and initial condition:
• Solution:
1)0(
2
=
=
y
tdtdy
1)( 2 += tty
Introduction to Simulation WS01/02 - L 02 7/39 Graham Horton
Example for an IVP
• Differential equation and initial condition:
• Solution:
0)0(
)1(
=
−=
y
ydtdy λ
tety λ−−=1)(
Introduction to Simulation WS01/02 - L 02 8/39 Graham Horton
Example for an IVP
• Differential equation and initial condition:
• Solution:
0)0(
)cos(
=
−= −
y
ytedtdy t
)sin()( tety t−=
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Example for an IVP
• A system of two differential equations and initial conditions:
• Solution:
)cos()( ),sin()( ttvttu ==
1)0( ,0)0(
,
==
−==
vu
udtdvv
dtdu
Introduction to Simulation WS01/02 - L 02 10/39 Graham Horton
Solving ODEs
• We were able to solve all these examples analytically
• These were therefore special cases
• Usually, there is no analytic solution for systems of ODEs
• We are forced to use numerical methods to perform the
integration
• The numerical integration of ODEs forms the most
important part of continuous simulation
• We will look at this next week
Introduction to Simulation WS01/02 - L 02 11/39 Graham Horton
Balance Equations
• Most ODEs are balance equations• A balance equation basically says:
Change = Increase - Decrease
• Example: ODE for the amount of water x in a tank:
O l/s
I l/sdx/dt = I - O
Introduction to Simulation WS01/02 - L 02 12/39 Graham Horton
A Falling Object
• Consider dropping an object from a certain height• We will develop a model for the object's location and speed
• Variables:– p(t) : distance fallen at time t. (Units [m])– v(t) : velocity of object at time t. (Units [m/s])
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Effect of Gravity
• The Earth exerts a force F on the object. (Units [N])• Let the mass of the object be m. (Units [kg])
• Newton's law (F = m · a) states that Force = Mass · Acceleration
• Therefore, a = F /m
• Parameters:– a : acceleration due to gravity. (Units [m/s/s])
Introduction to Simulation WS01/02 - L 02 14/39 Graham Horton
A Falling Object
• The situation at time t:
p meters
v meters per secondF Newtons
Mass m
Introduction to Simulation WS01/02 - L 02 15/39 Graham Horton
The Model
• Acceleration is the rate of change of velocity: dv/dt = g = 9,81 [m/s/s]
• Velocity is the rate of change of position: dp/dt = v
• At time t =0 we define: p(0) = 0 v(0) = 0
Introduction to Simulation WS01/02 - L 02 16/39 Graham Horton
Simulation Results
• Simulation results:
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Wind Resistance
• We have assumed that the object is falling in a vacuum• Now let us consider wind resistance• The standard physical model states:
– The force due to wind resistance is proportional to thesquare of the velocity
• We obtain
dv/dt = g - a · v 2
for some constant a with units [1/m]
Introduction to Simulation WS01/02 - L 02 18/39 Graham Horton
Wind Resistance
• Result for a = 0.0033 (skydiver):
Terminal velocity≈ 56 m/s
Introduction to Simulation WS01/02 - L 02 19/39 Graham Horton
Population Biology
• Consider the population x of some animal species
• Assume for the moment...– No deaths– Infinite space and food supply
• The birth rate is proportional to the current population:
dx/dt = a · x
• The solution of this ODE is
x = eax
• We have exponential population growth0
5
1 0
1 5
20
25
30
0 0. 5 1 1 . 5
Introduction to Simulation WS01/02 - L 02 20/39 Graham Horton
Population Biology
• Now let us add deaths to the model• The death rate is also proportional to the current population• This gives
dx/dt = a · x - b · x = (a - b) · x• If a > b, we still have exponential population growth• We can just set c = a - b to obtain dx/dt = c · x
Introduction to Simulation WS01/02 - L 02 21/39 Graham Horton
The Logistic Equation
• Now assume a limited food supply• As the population increases, food per capita goes down• This leads to decreasing birth and increasing death rates• This is called crowding or competition
• One solution is to use the logistic equation
dx/dt = c · x - d · x2
• We will achieve steady state when x = c / d• This is an assumption which works well in practice
Introduction to Simulation WS01/02 - L 02 22/39 Graham Horton
The Logistic Equation
• Simulated solutions for c = 0.5, 1.0 and 2.0:
Introduction to Simulation WS01/02 - L 02 23/39 Graham Horton
Predator-Prey Models
• Consider an area in which hares and foxes live
• Denote the population of hares by h and of foxes by f
• Observations show:
– Foxes must eat hares in order to survive
– Hares have an unlimited supply of food
• Without foxes, hares multiply according to dh/dt = a · h
• Without hares to eat, foxes die according to df /dt = -b · f
Introduction to Simulation WS01/02 - L 02 24/39 Graham Horton
Lotka-Volterra Equations
• The probability of a meeting is proportional to h · f
• The increase in foxes is d · h · f
• The rate at which hares are eaten is c · h · f
• The resulting equations are:
• These are the famous Lotka-Volterra equations
fhchadtdh ⋅⋅−⋅= fhdfb
dtdf ⋅⋅+⋅−=
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Lotka-Volterra Equations
• Simulation results forparameters
– h (0) = 400
– f (0) = 37
– a = 0.175
– b = 0.125
– c = d = 0.001
– 0 < t < 150
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Lotka-Volterra Equations
• State space representation: Plot h (t ) against f (t )
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Lynxes and Hares
• From the records of the Hudson Bay Company• Number of lynx and hare skins purchased 1845-1935
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Another Example
• Petri-dish populations of the bacteria
– Paramecium Aurelia (predator)
– Saccharomices Exiguns (prey)
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The Gypsy Moth
• The Gypsy moth is a pest
• Its larvae eats the leavesof the larch
• This defoliates and oftenkills the tree
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The Gypsy Moth
• Population of Gypsy moths in the Engadin 1954 - 1976:
0
50
100
150
200
250
300
350
1954 1959 1964 1969 1974
Year
Popu
latio
n
Introduction to Simulation WS01/02 - L 02 31/39 Graham Horton
The Gypsy Moth
• This looks very much like half of the Lotka-Volterrasimulation result:
• Have we proved that the Gypsy moth is part of apredator-prey relationship?
0
50
100
150
200
250
300
350
1954 1959 1964 1969 1974
Year
Popu
latio
n
Introduction to Simulation WS01/02 - L 02 32/39 Graham Horton
Chaos
• Chaotic behaviour is a (relatively) new discovery in ODEs• Discovered by Edward Lorenz in the early 60s• Three equations from meteorology• Discovered by accident• "Chaos" in physics means...
"Sensitive dependence oninitial conditions"
• The "Butterfly Effect" zcyxdtdz
yxbzxdtdy
xyadtdx
⋅−⋅=
−⋅+⋅−=
−⋅= )(
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Types of Behaviour
• What can the long-term behaviour of a system look like?
• Explosion:
– One or more values becomes infinite
• Steady-state / stationary
– Nothing changes
• Periodic
– Values repeat indefinitely
• Chaotic
– Finite, non-stationary, non-periodic
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Lorenz's Equations
• Solution in the time domain:
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The Lorenz Attractor
• Simulation result ("Lorenz attractor"):
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The Lorenz Attractor
• Simulation result in the x-y, x-z and y-z planes:
Introduction to Simulation WS01/02 - L 02 37/39 Graham Horton
Three Species
• Now let us build a new three-species predator-prey model:– Species z preys on species x and y– There is crowding between x and y
yzxzzdtdz
yzyxyydtdy
xzxyxxdtdx
0005.0005.0
001.0001.00015.0
01.0001.0001.0
2
2
++−=
−−−=
−−−=
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Three Species
• The three-species result in a non-periodic 3-D state space:
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The Last Slide
• An ODE is an equation of the form dy/dt = f (y,t)• Almost all continuous processes are described by ODEs• These are thus very important in simulation• There is a 2+2 SWS course "Kontinuierliche Simulation"• ODEs are usually a type of balance equation:
– Change = Increase - Decrease
• Next week:– How to solve ODEs