Computational modeling techniques
Lecture 3: Modeling change (2) Modeling using proportionality
Modeling using geometric similarity
Ion Petre Department of IT, Abo Akademi
http://www.users.abo.fi/ipetre/compmod/
September 8, 2014 1 http://users.abo.fi/ipetre/compmod/
Content of the lecture
• Modeling change with systems of difference equations: three examples • Modeling using proportionality • Modeling using geometric similarity
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 2
MODELING CHANGE WITH SYSTEMS OF DIFFERENCE EQUATIONS September 8, 2014 http://users.abo.fi/ipetre/compmod/
3
Modeling with systems of difference equations
• Example: car rental company; one office in Tampa, the other in Orlando o Travelers may rent a car from either office and drop it at either one
– Data shows that 40% of those who rent in Orlando, return it in Tampa, 30% of those who rent in Tampa, return it in Orlando
– The question is whether the car distribution will eventually become unbalanced and cars will have to be moved (empty) from one place to the other – extra cost
o Problem: build a model for how the number of cars in the two offices varies
– On=number of cars in Orlando after n time units – Tn=number of cars in Tampa after n time units – On+1=0.6On+0.3Tn – Tn+1=0.4On+0.7Tn
o Equilibrium values (O,T): – O=0.6O+0.3T – T=0.4O+0.7T – Solution: O=0.75 T
– Note that On+Tn= On+1+Tn+1, for all n – So, O+T=O0+T0 – Final solution: (3/7(O0+T0), 4/7(O0+T0))
4
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 1.22, page 36
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Example (continued)
Example • On+1=0.6On+0.3Tn
• Tn+1=0.4On+0.7Tn
• Total number of cars: 7000 • (3000,4000) equilibrium
• Start with various initial
distributions of the cars between Orlando and Tampa
• Test the numerical evolution
• Conclusion: after about 7 time units, we approach the equilibrium value (3000,4000)
– It suggests an asymptotically stable equilibrium
5 September 8, 2014
010002000300040005000600070008000
0 5 10 15
Cars
Time units
O(n)
T(n)
0
1000
2000
3000
4000
5000
6000
0 5 10 15
Cars
Time units
O(n)
T(n)
0
1000
2000
3000
4000
5000
6000
0 5 10 15
Cars
Time units
O(n)
T(n)
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 1.23, page 37
Example: battle of Trafalgar
Trafalgar, 1805: French and Spanish naval forces under Napoleon against British naval forces under Admiral Nelson
• French and Spanish: 33 ships • British: 27 ships
• Model for battle outcome:
during an encounter, each side has a loss equal to 5% of the number of ships of the enemy
Scenario I: Straight-on series of confrontations
• Bn+1=Bn-0.05Fn • Fn+1=Fn-0.05Bn
6 September 8, 2014
0
5
10
15
20
25
30
35
0 5 10 15 20 25
Ships
Number of encounters
Scenario I: straight-on battle
B(n)
F(n)
http://users.abo.fi/ipetre/compmod/
Example (continued)
Recall the model: • French and Spanish: 33 ships • British: 27 ships • During an encounter, each side has
a loss equal to 5% of the number of ships of the enemy
Scenario II: divide-and-conquer
• Napoleon’s ships were arranged in three groups
• Strategy – Engage force A with 13 British ships – Engage force B with all available
ships – Engage force C with all available
ships
7
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 1.25, page 39
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Example (continued)
Battle A: engage French force A (3 ships) with 13 British ships
Outcome of battle A:
• French force A reduced from 3 to 0.4385
• French force B, C left intact: 17, 13, resp.
– did not engage
• British force: – from 13 ships engaged
in the battle, 12.5935 remaining
– 14 did not engage
September 8, 2014 8
0
2
4
6
8
10
12
14
0 1 2 3 4 5
Ships
Number of encounters
Scenario II: divide and conquer Battle A
B(n)
F(n)
http://users.abo.fi/ipetre/compmod/
Example (continued)
Battle B: engage French force B (17 ships) with all available British ships (26.5935)
Outcome of battle B:
• French force A: 0.4385 – did not engage
• French force B, reduced
from 17 to 0.1260
• French force C intact: 13 – did not engage
• British force:
– From 26.5935, reduced to 20.0704
September 8, 2014 9
0
5
10
15
20
25
30
0 5 10 15 20
Ships
Number of encounters
Scenario II: divide and conquer Battle B
B(n)
F(n)
http://users.abo.fi/ipetre/compmod/
Example (continued)
Battle C: engage French force B (13 ships) with all available British ships (20.0704)
Outcome of battle C:
• French force A: 0.4385 – did not engage
• French force B: 0.126
– did not engage
• French force C: from 13, reduced to 0.3111
• British force: – From 20.0704, reduced
to 15.0101
September 8, 2014 10
0
5
10
15
20
25
0 5 10 15 20
Ships
Number of encounters
Scenario II: divide and conquer Battle C
B(n)
F(n)
http://users.abo.fi/ipetre/compmod/
Modeling exercise
• Another strategy for defeating a superior force is through better technology
o Formulate a model where the British forces have better weaponry and as a result, each encounter results in
– A loss for the French-Spanish fleet of 10% of the number of British ships – A loss for the British ships of 5% of the number of French-Spanish ships – Assume that French-Spanish start with 33 ships, British start with 27 ships
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 11
Modeling exercise
• Another strategy for defeating a superior force is through better technology
o Formulate a model where the British forces have better weaponry and as a result, each encounter results in
– A loss for the French-Spanish fleet of 15% of the number of British ships – A loss for the British ships of 5% of the number of French-Spanish ships – Assume that French-Spanish start with 33 ships, British start with 27 ships
• Result o Bn+1=Bn-0.05Fn
o Fn+1=Fn-0.15Bn
o B0=27, F0=33
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 12
Example: competitive hunter model
• Two species, say owls and hawks co-exist in a habitat: O, H
• In the absence of the other species, each species exhibits unconstrained growth, proportional to its current level:
o On+1-On=k1On, Hn+1-Hn=k2Hn, where k1,k2>0
• The effect of each species is to diminish the growth of the other o Model it here as proportional to the number of possible encounters
between the two species: o On+1-On=k1On- k3OnHn; Hn+1-Hn=k2Hn- k4OnHn; k1,k2,k3,k4>0
• Model:
o On+1=(1+k1)On- k3OnHn; Hn+1=(1+k2)Hn- k4OnHn
September 8, 2014 13 http://users.abo.fi/ipetre/compmod/
Example (continued)
• Model: o On+1=(1+k1)On- k3OnHn; Hn+1=(1+k2)Hn- k4OnHn
• Question: what are the equilibrium points (O,H)?
o O=(1+k1)O- k3OH; H=(1+k2)H- k4OH o i.e. O(k1 – k3H)=0; H(k2- k4O)=0 o i.e., (O,H)=(0,0) or (O,H)=(k2/k4, k1/k3)
• Numerical simulation for k1=0.2, k2=0.3, k3=0.001, k4=0.002
o Equilibrium points: (0,0) and (150, 200)
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 14
September 8, 2014 15
0
50
100
150
200
250
0 5 10 15
Individuals
Time units
Initial value: (150,200)
O(n)
H(n)
0
100
200
300
400
500
0 10 20 30
Individuals
Time units
Initial value: (151,199)
O(n)
H(n)
0
200
400
600
800
1000
0 10 20 30
Individuals
Time units
Initial value: (149,201)
O(n)
H(n)
This suggests an unstable equilibrium point; sensitivity to initial conditions
http://users.abo.fi/ipetre/compmod/
Modeling exercise
• Spotted owl’s primary food source is mice • Here is an ecological model for the populations of owls and mice:
o Mn+1=1.2 Mn-0.001 OnMn o On+1=0.7 On+0.002 OnMn
• Explain the significance of every term and constant in the model
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 16
MODELING USING PROPORTIONALITY AND GEOMETRIC SIMILARITY
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 17
Modeling using proportionality
• y is proportional to x, denoted y~x, if y=kx, for some k>0
• y~x2 if y=k1x2, for some constant k1>0
o In this case, x~y1/2
• y~ln(x) if y=k2ln(x), for some constant k2>0 • y~ex if y=k3ex, for some constant k3>0
• Note: If z~y and y~x, then z~x
18
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 2.8, 2.9, page 66
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Modeling using geometric similarity
• Two objects are said to be geometrically similar if there is a one-to-one correspondence between the points of the objects such that the ratio of distances between corresponding points is the same for all pairs of points
• Example: two boxes o l/l’=w/w’=h/h’=k, with k>0 o Ratio of their volumes:
V/V’=(lwh)/(l’w’h’)=k3
o Ratio of their surface areas: S/S’=(2lh+2wh+2wl)/(2l’h’+2w’h’+2w’l’) =k2
o In other words: – S/S’=l2/l’2, V/V’=l3/l’3
19
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 2.18, page 76
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Example
• Example: Bass fishing derby o Problem: predict the weight of a
fish in terms of some easily measurable dimensions
o Solution: assume geometric similarity
– V~l3; W~V – Conclusion: W~l3 – Calculate the proportionality
constant based on just one fish
20
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 2.19-2.21, pages 79-81
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Verify the model:
Example (continued) • The rule above does not reward catching a fat fish • Changes in the model
o Assume that the majority of the weight comes from the main body, without head and tail
o Length of the main body (effective length): leff
o The main body has a varying cross-sectional area. Consider the average cross-sectional area: Aavg
– Model: W~V~ Aavgleff o How do we measure/estimate the effective
length and the average cross-sectional area – Answer: Model it through geometric similarity! – Assume leff~l – Measure the circumference of the fish at its
widest point: the girth g – Assume Aavg~g2
– Final model: W~lg2 – Test the model: W=0.0187 lg2
21
Giordano et al. A first course in mathematical modeling. (3rd edition) Fig. 2.22, pages 82
September 8, 2014 http://users.abo.fi/ipetre/compmod/
Learning objectives for lectures 2-3
• Understand the concept of modeling the change in a discrete dynamical systems • Able to write a linear and an affine model with difference equations for a simple real-life phenomenona • Understand the diverse behavior that a linear dynamical system model can have • Understand the notion of equilibrium point • Understand the different types of stability of equilibrium points
• Ability to write multi-variable models as systems of difference equations • Understand the basic paradigm of modeling proportionality • Understand the basic paradigm of modeling geometric similarity
September 8, 2014 http://users.abo.fi/ipetre/compmod/ 22