Modeling and simulation
Modeling
Simulation
Analysis
The simulation loop:
In this lecture we focus on the modeling part of the loop
What do we mean by modeling?
Important!
Ex.
•A simplification of a complex problem in nature
•Remember that the model,
in most cases, only give us
a crude picture of reality
•A way of structure data from an experiment or from
other measurements
•Understand the limitations
of the model
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Predator-prey systems
- Ecological examples
Disaster of Lake Victoria
•The lake supported hundreds of small fishing
communities fishing several species
•A new big fish was introduced in the lake
•Result: The new fish wiped out many other
speices….
•More: Diseases, felling of trees for fuel…
Murray J.D., “Mathematical biology”, Springer-Verlag, 1993
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A simple predator-prey system
Our toy system: An island where two species lives, predators (foxes)
and preys (rabbits)
Our first assumptions:
•The foxes eat rabbits and breed
•The rabbits eat grass and breed
•The area is limited, but the grass grows faster than the rabbits can
eat
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Let us build a mathematical model
R = Number density of rabbits
(Lotka-Volterra system)
F = Number density of foxes
1. Only rabbits, inifite food supply
8
x 10
5
Prey model ODE
Rabbits
6
dR
= aR
dt
4
a = growth rate
2
0
0
5
10
15
20
Limited food supply
Prey model ODE
dR ⎛ R ⎞
= a⎜1− ⎟R
dt
⎝ K⎠
K = Carrying capacity
600
200
0
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Rabbits
400
0
5
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15
20
4
R2
dR ⎛ R ⎞
= a⎜1− ⎟R = aR −
dt
K
⎝ K⎠
Steady state solutions:
R1 = 0
Unstable
R2 = K
Stable
Prey model ODE
600
Rabbits
400
200
0
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0
5
10
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5
Predator-Prey model ODE
250
2. Only foxes
dF
= −dF
dt
Foxes
200
150
d = death rate
100
50
0
0
5
10
15
20
3. Rabbits and foxes
Assumptions:
More rabbits => more food for foxes => foxes tends to breed more
More foxes => more rabbits are killed
Fox model:
Rabbit model:
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dF
= −dF + cRF
dt
Lotka-Volterra equations
dR
= aR − bFR
dt
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Solving an ODE-system using Matlab
⎧ dR
⎪⎪ dt = aR − bFR
⎨
⎪ dF = cRF − dF
⎪⎩ dt
a ≈ the rabbits growth rate
bF ≈ the rabbits death rate
cR ≈ the foxes growth rate
d ≈ the foxes death rate
To solve this ODE-system we have to use a numerical method.
In the software Matlab there is a collection of methods that are easy
to use.
Note: Different choices of parameters may take longer/shorter to
solve depending on numerical method
Later in this course you will learn how to choose a numerical method
best fitted to solve a specific problem
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Typical solutions…
Predator-Prey Trajectory a=0.4, b=0.001, c=0.001, d=0.9
1000
Predator-Prey model ODE
1500
Rabbits
Foxes
800
1000
Foxes
600
400
500
200
0
0
20
40
60
80
0
100
0
500
1000
1500
Rabbits
Predator-Prey Trajectory a=0.4, b=0.004, c=0.004, d=0.9
800
Predator-Prey model ODE
1000
Rabbits
Foxes
800
600
Foxes
600
400
200
200
0
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0
20
40
60
80
100
0
0
200
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600
Rabbits
800
1000
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Limitations of our model?
• Infinite grass supply leads to exponential growth of the rabbit
population if no foxes are alive
• The fox population can grow without saturation
• Only two species…
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More realistic predator-prey systems
Trajectory a=0.4, b=0.004, c=0.004, d=0.9, K=800
800
Predator-Prey model ODE
R ⎞
⎛
a → a ⎜1 −
⎟
K ⎠
⎝
Rabbits
Foxes
600
600
Foxes
Finite grass resources
800
400
200
0
400
200
0
20
40
60
80
0
100
0
200
400
600
Rabbits
RF
→
RF
4000
R
1 +
S
3000
S = Fox eating saturation
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Predator-Prey model ODE
Trajectory a=0.4, b=0.004, c=0.004, d=0.9, S=2500
5000
5000
Rabbits
Foxes
4000
Foxes
Finite prey consumption
2000
1000
0
3000
2000
1000
0
20
40
60
80
100
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0
1000
2000
3000
Rabbits
4000
5000
10
Results from a model with both limited grass supply and
limited prey consumption
Trajectory a=0.4, b=0.004, c=0.004, d=0.9, K=800, S=600
600
Predator-Prey model ODE
800
Rabbits
Foxes
500
600
Foxes
400
400
300
200
200
100
0
0
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40
60
80
100
0
0
200
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Rabbits
600
800
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Predator-prey laboration 1
Exercises
1. Implement and solve the standard predator-prey equations
2. Add limited grass supply (see model above)
3. Add limited prey consumption (see model above)
4. Add a third species
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So, what is Matlab???
•A simple programming environment
•Mathematical modules (ODE solvers, statistical tools, algebraic tools…)
•Graphics
•Possibility to connect to external simulations
How do we use Matlab?
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% A Matlab code for solving the Predator-Prey system
clear, clear global
global a b c d
Save as PredatorPrey.m
%Define equation coefficients
a=0.4;
%Rabbit growth rate
b=0.001; %Rabbit death coefficient
c=0.001; %Fox growth coefficient
d=0.9;
%Fox death rate
Save as rabbitfox.m
% Define the variables used as inputs in the ODE-solver
t0=0;
tmax=1000;
Ntot=1000;
p=0.4;
RabbitIni=(1-p)*Ntot;
FoxIni=p*Ntot;
options = odeset('RelTol',1e-4,'AbsTol',[1e-7 1e-7]);
function dy = rabbitfox(t,y);
dy = zeros(2,1);
global a b c d
% Definition of the ODE
% y(1) = Number of rabbits
% y(2) = Number of foxes
% Solve ODE system
[T,Y] = ode45(@rabbitfox,[t0 tmax],[RabbitIni FoxIni],options);
%Plot
figure(1)
plot(T,Y(:,1),'r')
hold on
plot(T,Y(:,2),'--b')
legend('Rabbits','Foxes')
title('Predator-Prey model ODE')
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dy(1) = a*y(1) - b*y(2)*y(1);
dy(2) = c*y(1)*y(2) - d*y(2);
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General remarks about ODE systems (a very short overview…)
y
j
y
1000
We have seen plots like those to the right
Note, for example, the spiral motion into a single point…
Foxes
800
600
400
200
Predator-Prey model ODE
Trajectory a=0.4, b=0.004, c=0.004, d=0.9, K=800
800
800
0
Rabbits
Foxes
0
500
1000
1500
Rabbits
600
Foxes
600
400
800
400
200
0
0
0
20
40
60
80
100
Foxes
600
200
0
200
400
600
400
200
Rabbits
0
Such a point is called a fixed point and represents
a steady state situation:
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4000
3000
2000
1000
0
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400
5000
Foxes
⎧R = F = 0
⎨
⎩ R = d / c, F = a / b
200
Rabbits
(The unmodified case)
dR
dF
= 0,
=0
dt
dt
0
0
1000
2000
3000
Rabbits
4000
5000
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Typical state space trajectories near fixed points
Node
Repellor
Spiral
Spiral
node
repellor
R = F = 0 is a saddle point
R = d / c, F = a / b is a cycle
Cycle
Saddle
point
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Hilborn R.C., “Chaos and nonlinear dynamics”, Oxford University Press Inc., 1994
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State space transitions
•If the parameters in an ODE-system are changed, we can have a
transition from one type of state space behavior to another.
•Such a transition is called a bifurcation
•In order to study bifurcation it is very useful to write the ODE in nondimensional form. In the unmodified case we make the following
substitutions:
u=
Non-dimensional form
bF
d
cR
,v=
, τ = at , α =
d
a
a
⎧ du
⎪⎪ dt = (1 − v )u
⎨
⎪ dv = α (u − 1)v
⎪⎩ dt
Ex.
⎧ dF
d ⎛ av ⎞ a 2 dv
=
⎜ ⎟=
⎪
⎪ dt d (τ / a ) ⎝ b ⎠ b dτ
⎨
2
⎪− dF = − d ⎛⎜ av ⎞⎟ = − ad v = − a αv
⎪⎩
b
b
⎝ b ⎠
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ONLY ONE PARAMETER!!
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Further reading for those who are interested:
1. N. Britton, “Essential Mathematical Biology”, Springer-Verlag,
2003
2. Murray J.D., “Mathematical biology”, Springer-Verlag, 1993
3. Hilborn R.C., “Chaos and nonlinear dynamics”, Oxford University
Press Inc., 1994
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An other way to simulate predator-prey
system…
•The ODE model above describes the evolution of the
population densities
• We do not get any information about dynamics of individual
animals
•Is it possible to construct a simulation of individuals
that give us similar results as the ODE model?
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Cellular Automata and Agent Based Models
Game of Life, John Conway (1970)
-Individuals are distributed on a grid
-Each individual has a well defined neighborhood
-Simple rules stating the birth and death of
individuals
-The whole grid is updated each time step
Ex
Simple rules on microscopic level (individuals)
=> Complex behavior on macroscopic scales
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Predator-Prey simulation using a Cellular Automata (CA)
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Predator-prey laboration 2
Exercises
1. Implement a simulation, based on individuals, with following properties:
-
One predator and one prey
-
The species should be able to move around on a rectangular grid.
Each individual should have limited energy capacity (they need to eat).
OR
You may use non-moving individuals and make a “Game of life”-like
simulation.
-
The behavior of individuals should be controlled by SIMPLE rules
2. Add limited grass supply
3. Add limited prey consumption
4. Add a third species
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Hints to get started
•
Represent the domain by a matrix.
Represent your different species by different numbers.
empty spaces = 0
rabbits = 1
foxes = 2
(It may in some cases be advantageous to use different matrices for different
species, e.g. a separate matrix for foxes and one for rabbits.)
•
How to treat boundaries: let the domain be an island.
•
Random matrix with different species.
•
Count the number of species.
>> island = zeros(5);
>> island(:,1) = 9;
>> island(:,5) = 9;
>> island(1,:) = 9;
>> island(5,:) = 9
island =
>> nrFox = sum(sum(island==2))
>> floor(3*rand(2,3))
ans =
2
0
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1
2
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9
9
9
9
9
0
0
0
9
9
0
0
0
9
9
0
0
0
9
9
9
9
9
9
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Hints to get started (continued)
•
You can use logic in if-cases
if (wall(i,,j)==0) & (Foxes(i,,j)==1)
..........
•
You can use logicals instead of for-loops to find
sites where two matrices overlap
>> F=[0 0 1; 1 1 1; 1 0 0];
>> R=[1 1 1; 0 0 2; 0 0 0];
>> F&R
ans =
•
Example of hunting routine
0
0
0
0
0
0
1
1
0
foxrabbit = F&R;
% Creates a matrix where R and F overlap – here the foxes may hunt
killrabbit=(rand(size(foxrabbit))<=killprobability)&foxrabbit
% Creates a matrix indicating where the hunt
was successful
R(killrabbit)=0;
% The rabbits that were killed are removed.
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Test your code
•
Remove birth → if your population still grows you have a bug?
•
Remove death → if your population decreases you have a bug?
•
Remove grass → does your rabbits still survive?
•
Remove rabbits → does your foxes still survive?
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How to implement graphics (optional)
•
Suppose you have a matrix, Tot, in which 1 represents rabbits and 2
represents foxes.
The following algorithm within the time-loop will display of the evolution in
real time.
figure(1)
set(gcf,'DoubleBuffer','on')
subplot(1,2,1)
plot(main,Popfox(main),'.b')
hold on
plot(main,Poprabbit(main),'.r')
hold on
plot(main,Popgrass(main),'.g')
legend('Foxes','Rabbits','Grass',2)
subplot(1,2,2)
Tot(1,1)=0; Tot(1,2)=2;
pcolor(Tot)
axis equal
colormap(hot)
pause(0.00001)
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% makes the real time update smooth
% this will plot the number of each species
% Popfox is a vector with the fox popnr for each timestep
% main is a vector that keeps track of the time
% this will show the species on a grid in real time
% this is to help fix colorscales
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Example of predator-prey agent based simulation
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