1. Modelling and Simulating Social Systems with MATLAB
Modelling and Simulating Social Systems with MATLAB
Lesson 3 – 3D plots and population growth
A. Johansson & M. Moussaid
© ETH Zürich |

2. Lesson 3 - Contents Repetition Additional plotting commands
Lesson 3 - Contents
Repetition
Additional plotting commands
3-dimensions plots
Case studies: exponential and logistic growth

3. Repetition Plotting functions >> x=[0:0.2:10]; >> a=3;
Repetition
Plotting functions
>> x=[0:0.2:10];
>> a=3;
>> y=a*exp(-x);
>> plot(x,y,’b+-’)

4. Repetition Plotting functions
Repetition
Plotting functions
>> xlabel(“X”) >> ylabel(“Y=3*exp(-x)”) >> title(“Plotting a simple function”) 4 4

5. Repetition Plotting functions >> x=randn(10^4 , 1);
Repetition
Plotting functions
>> x=randn(10^4 , 1);
>> hist(x); 5 5

6. Repetition Statistics >> mean(x) >> std(x) >> min(x)
Repetition
Statistics
>> mean(x)
0.021
>> std(x)
1.005
>> min(x)
-3.797
>> max(x)
4.198 6 6

7. Plotting Two useful commands >> x=[-5:0.1:5];a=2;b=3;
Plotting
Two useful commands
Hold on|off
Grid on|off
>> x=[-5:0.1:5];a=2;b=3;
>> y1=exp(-x.^2);
>> y2=a*exp(-x.^2);
>> y3=exp(-(x.^2)/b); 7 7

8. Plotting Two additional useful commands >> plot(x,y1);
Plotting
Two additional useful commands
>> plot(x,y1);
>> hold on
>> plot(x,y2,’r’);
>> plot(x,y3,’g’); 8 8

9. Plotting Two additional useful commands >> plot(x,y1);
Plotting
Two additional useful commands
>> plot(x,y1);
>> hold on
>> plot(x,y2,’r’);
>> plot(x,y3,’g’);
>> grid on 9 9

10. Plotting Saving a plot >> hgsave(‘fileName.fig’) 07.05.2018 10

11. 3 dimensions plots Plot a surface surf(X,Y,Z) with: x2 x1 x3 y1 y2 y3
3 dimensions plots
Plot a surface
surf(X,Y,Z) with:
X a vector containing the x-axis
Y a vector containing the y-axis
Z a matrix with Z(i,j) is the corresponding value for Y(i) and X(j)
Number of lines and columns in Z must respectively equal length(Y) and length(X). x2 x1 x3 y1 y2 y3
z11
z12
z13
z21
z22
z23
z31
z32
z33 11 11

12. 3 dimensions plots Plot a surface >> X=-2:0.1:2;
3 dimensions plots
Plot a surface
>> X=-2:0.1:2;
>> Y=0:0.1:1;
>> Z = Y(2)*exp(-X.^2)
Z =
>> for i=1:length(Y)
Z(i,:)= Y(i)*exp(-X.^2);
end 12 12

13. 3 dimensions plots Plot a surface >> surf(X,Y,Z) 07.05.2018 13

14. 3 dimensions plots Plot a surface >> surf(X,Y,Z) 07.05.2018 14

15. 3 dimensions plots Plot a surface >> surf(X,Y,Z) 07.05.2018 15

16. 3 dimensions plots Plot a surface
3 dimensions plots
Plot a surface
>> contourf(X,Y,Z) >> contour(X,Y,Z) 16 16

17. Case 1 Exponential growth
Case 1
Exponential growth
Consider a population that has a constant birth rate through time and is never limited by food or disease.
At each new generation the population increases with a given birth rate r
This is called an exponential growth: the birth rate alone controls how fast the population grows.
Generation 1
Generation 2
Generation 3
Generation 4

18. Case 1 Exponential growth
Case 1
Exponential growth
An exponential growth can be described as follow:
Xt+1 = xt * (1 + r)
XT : Size of population at time T
r : birth rate 18 18

19.
Case 2
Logistic growth
In most real populations, there is an upper limit to the number of individuals the environment can support.
This limit is called the "carrying capacity" of the environment : K
Populations in this kind of environment show what is known as logistic growth.
Generation 1
Generation 2
Generation 3
Generation 4 19 19

20. Case 2 Logistic growth The logistic growth can be described as follow:
Case 2
Logistic growth
The logistic growth can be described as follow:
Xt+1 = xt exp(r (1-xt/K) )
XT : Size of population at time T
r : birth rate
K: Carrying capacity 20 20

21. Application Logistic growth Food NEST
Application
Logistic growth
Experimental data: Ants recruitment dynamic
Each ant recruit N others
Limited space around the food source
Data in: antsData.m
Food
NEST 21 21

22. Exercise 1 Exponential Growth
Exercise 1
Exponential Growth
Write a function expGrowth.m that computes the population size xT, after T generations. The initial population x0, the birth rate r and the final generation T must be given as function parameters.
For an initial population of 10 individuals and a birth rate of 0.1, use the function to compute the population size after T=30, 40 and 50 generations.
Modify the function so that it returns a vector ST that contains the sequence x0, x1, x2, … xT. Plot S30 (with the same parameters).
On the same figure plot S30 for r = -0.05, 0, 0.05, Set title, axis names and save the file.
Download and launch file exercise1.m . Read the code and understand how it works. Add some comments. 22 22

23. Exercise 2 Logistic Growth
Exercise 2
Logistic Growth
Write a function logGrowth.m that compute the evolution of population size ST, after T generations. The initial population x0, the birth rate r, the capacity K and the final generation T must be given as function parameters.
Consider x0=10 , r=0.3 , and K = 500 , plot evolution of population after T=30.
On a new plot, draw the evolution of population size for r=0.3, 0.5, 1, 1.5, and 2. What can you say about the overall dynamic? How do you interpret it?
Use the functions surf or contourf to determine the critical value of r beyond which the stationary dynamic switch to an oscillatory dynamic. In order to do so, build and plot a matrix that contains the evolution of population during 30 generations with r varying from 0.3 to 2.5 with a step of 0.01 23 23

24. Exercise 3 Ants recruitment
Exercise 3
Ants recruitment
Download the data file antsData.m and read the data description written into the file.
On the same plot, represent the data from the 5 experiments: number of ants as a function of time. Use the option ‘.’ to plot dots instead of lines.
Build a vector called avgAnts of length 15 that contains the average value of the 5 experiments at each time step.
Always on the same plot, draw the vector avgAnts in red.
Try to estimate the recruitment rate r and the capacity K. Choose x0=6 and T=15. 24 24

25. Exercise 4 (optional) Ants recruitment : least-square fitting
Exercise 4 (optional)
Ants recruitment : least-square fitting
The ‘trial and error' approach is obviously not the best way to fit a curve. The least squares method assumes that the best-fit curve is the one that has the minimal sum of the squared deviations from a given set of data.
1. Build a function lsDeviation that computes the least-square error for a given set of parameters, as follow:
ls = lsDeviation (observedData, r,K,x0,T) where
observedData is the observed mean values vector (use the vector avgAnts from exercise3)
r,K,x0,T is a given set of parameters
ls is the sum of the squared deviations between observed data and the theoretical data at each time step:
ls = ( avgAnts(i) - S(i) ) with S = logGrowth(x0,r,K,T) 25 25

26. Exercise 4 (optional) Ants recruitment : least-square fitting
Exercise 4 (optional)
Ants recruitment : least-square fitting
For r = 0.1:0.01:1 and K=50: compute and plot the matrix LS , defined as :
LS(i,j) = lsDeviation(avgAnts, r(i), K(j) , x0 , T) Choose x0=6 and T=15;
What is the minimum value of LS?
What are the corresponding optimal parameters r and K ? 26 26