Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Population Dynamics
• Deductive modelling: based on physical laws
• Inductive modelling: based on observation + intuition
• Single species:
Birth (in migration) Rate, Death (out migration) Rate
dP
dt= BR−DR
• Rates proportional to population
BR = kBR × P ;DR = kDR × P
dP
dt= (kBR − kDR)P
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 1/43
kBR = 1.4, kDR = 1.2 : Exponential Growth
population
trajectory
0 10 20 30 40 50
1000000
2000000
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 2/43
kBR = 1.4, kDR = 1.2 : log(Exponential Growth)
population
trajectory
0 10 20 30 40 50
1E2
1E3
1E4
1E5
1E6
1E7
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 3/43
kBR = 1.2, kDR = 1.4 : Exponential Decay
population
trajectory
0 10 20 30 40 50
20
40
60
80
100
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 4/43
Logistic Model
• Are kBR and kDR really constant ?
• Energy consumption in a closed system → limits growth
Epc =Etot
P
P ↑→ Epc ↓→ kBR ↓ and kDR ↑ until equilibrium
• “crowding” effect:
ecosystem can support maximum population Pmax
dP
dt= k × (1−
P
Pmax
)× P
• crowding is a quadratic effect
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 5/43
kBR = 1.2, kDR = 1.4, crowding = 0.001
population
trajectory
time0 20 40 60
popu
latio
n
0
100
200
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 6/43
Disadvantages
• NO physical evidence for model structure !
• But, many phenomena can be well fitted by logistic model.
• Pmax can only be estimated once steady-state has been reached.
Not suitable for control, optimisation, . . .
• Many-species system: Pmax, steady-state ?
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 7/43
Multi-species: Predator-Prey
• Individual species behaviour + interactions
• Proportional to species, no interaction when one is extinct:
product interaction Ppred × Pprey
dPpred
dt= −a× Ppred + k × b× Ppred × Pprey
dPprey
dt= c× Pprey − b× Ppred × Pprey
• Excess death rate a > 0, excess birth rate c > 0,
grazing factor b > 0, efficiency factor 0 < k ≤ 1
• Lotka-Volterra equations (1956): periodic steady-state
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 8/43
Predator Prey (population)
predator
prey
trajectories
0 10 20 30
200
400
600
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 9/43
Predator Prey (phase)
predprey
phaseplot
200 400 600
100
200
300
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 10/43
Competition and Cooperation
• Several species competing for the same food source
dP1
dt= a× P1 − b× P1 × P2
dP2
dt= c× P2 − d× P1 × P2
• Cooperation of different species (symbiosis)
dP1
dt= −a× P1 + b× P1 × P2
dP2
dt= −c× P2 + d× P1 × P2
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 11/43
Grouping and general n-species Interaction
• Grouping (opposite of crowding)
dP
dt= −a× P + b× P 2
• n-species interaction
dPi
dt= (ai +
n∑
j=1
bij × Pj)× Pi, ∀i ∈ {1, . . . , n}
• Only binary interactions,
no P1 × P2 × P3 interactions
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 12/43
Forrester System Dynamics
• based on observation + physical insight
• semi-physical, semi-inductive methodology
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 13/43
Methodology
1. levels/stocks and rates/flows
Level Inflow Outflow
population birth rate death rate
inventory shipments sales
money income expenses
2. laundry list: levels, rates, and causal relationships
birth rate → birth → population
3. Influence Diagram (+ and -)
4. Structure Diagram (functional relationships)
dP
dt= BR−DR
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 14/43
Causal Relationships
latent variable
beerconsumption
graduates
standardof
living(SOL)
time
graduates
SOL
beerconsumption
SOL
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 15/43
Archetypes
• Bellinger http://www.outsights.com/systems/
• influence diagrams
• Common combinations of reinforcing and balancing structures
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 16/43
Archetypes: Reinforcing Loop
state1 state2
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 17/43
Archetypes: Balancing Loop
adjustment state
desired state
action
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 18/43
Forrester System Dynamics
Predator Prey
Grazing_efficiency
uptake_predatorloss_prey
predator_surplus_DR
prey_surplus_BR
2−species predator−prey system
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 19/43
Inductive Modelling: World Dynamics
• BR: BirthRate
• P : Population
• POL: Pollution
• MSL: Mean Standard of Living
• . . .
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 20/43
Inductive Modelling: Structure Characterization
BR = f(P, POL,MSL, . . .)
BR = BRN × f (1)(P, POL,MSL, . . .)
BR = BRN × P × f (2)(POL,MSL, . . .)
BR = BRN × P × f (3)(POL)× f (4)(MSL) . . .
• f (3)(POL) inversely proportional
• f (4)(MSL) proportional
• compartmentalize to find correllations
• . . . Structure Characterization !
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 21/43
Structure Characterisation: LSQ fit
X(t) = - gt /2 + v_0 t2
X(t) = A sin (b t )
t
X
LSQ (sin) < LSQ (t ) 2
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 22/43
Feature Extraction
1. Measurement data and model candidates
2. Structure selection and validation
3. Parameter estimation
4. Model use
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 23/43
Feature Rationale
Minimum Sensitivity to Noise
Maximum Discriminating Power
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 24/43
Throwing Stones
Candidate Models
1. x = − 12gt
2 + v0t
2. x = Asin(bt)
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 25/43
Feature 1 (quadratic model)
gi =2xi
t2i−
2xi
ti, i = A,B
F1 = gA/gB
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 26/43
Feature 2 (sin model)
1
btg(bt) =
xi
xi
solve numerically for b
F2 = 200|bA − bB|
bA + bB
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 27/43
Feature Space Classification
Feature t2
feature sin
F1 = gA/gBgi = 2xi/ti^2 - 2xi_der/ti
F2 = 200 |bA -bB|/(bA + bB)1/b(tg(bt)) = xi/xi_der
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 28/43
Forrester’s World Dynamics model
• “Club of Rome” World Dynamics model
• Few “levels”, note the depletion of natural resources
• implemented in Vensim PLE (www.vensim.com)
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 29/43
Population
birth rate normalbirth rate normal 1
population initial
births crowdingmultiplier
births food multiplier
births deathsbirths material multiplier
births pollution multiplier
switch time 1<Time>
death rate normaldeath rate normal 1
deathscrowdingmultiplier
deaths food multiplier
deaths material multiplier
deaths pollution multiplier
switch time 3
crowding
land areapopulation
density normal
Population & Food
birthscrowdingmult tab
deathscrowdingmult tab
food ratio
food coefficientfood coefficient 1
food crowding multiplier
food percapita normal
food per capita potential
food pollutionmultiplier
switch time 7
<Time>food
pollutionmult tab
<pollution ratio>
<capital ratio agriculture>
food per capitapotential tab
<foodcrowdingmult tab>
<Time>
capital agriculture fraction indicated
births foodmult tab
deaths foodmult tab
capital agriculturefraction indicated tab
<material standard ofliving>
deathsmaterialmult tab
birthspollutionmult tab
births materialmult tab
<materialstandard of
living>
deaths pollutionmult tab
<pollution ratio>
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 30/43
Capital
capital initial
capital ratio
capital depreciation
capitalinvestment
rate normal 1
capital depreciationnormal
capital depreciationnormal 1
switch time 5
capital ratioagriculture
capitalagriculture
fraction normal
capitalinvestmentmultiplier
effective capital ratio
capitalinvestment
mult tab
capital investment
capitalinvestmentrate normal
<Population>
switch time 4<Time>
Capital & Quality of Life
CapitalAgriculture
Fraction
capital agriculturefraction adjustment
time
<capitalagriculture fraction
indicated>
capitalagriculture
fraction initial
capital investmentfrom quality ratio
<natural resource extraction multiplier>
quality material multiplier
qualitymaterialmult tab
material standard of living
effectivecapital ratio
normal
capitalinvestment
quality ratio tab
<quality foodmultiplier>
<natural resourceextraction
multiplier> quality of life
qualitycrowdingmultiplier quality food
multiplier
quality of lifenormal
quality pollutionmultiplier
<crowding>quality
crowdingmult tab
<food ratio>qualityfood
mult tab
<pollution ratio>quality
pollutionmult tab
<Time>
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 31/43
NaturalResources
pollutionabsorption
time tab
naturalresources
initial
pollution ratio
pollution standard
nat res matl multiplier natural resourceutilization normal
natural resourceutilization normal 1
<capital ratio>
switch time 2
pollutioncapital
mult tab
natural resourceutilization
Pollution & Natural Resources
Pollution
pollutioninitial
pollution capitalmultiplier
pollution percapita normal
pollution percapita normal 1
<Population>
switch time 6<Time>
pollutionabsorption time
pollutionabsorption
pollutiongeneration
<material standard of living> natural resource material mult tab
natural resourceextractionmultiplier
natural resourceextraction mult tab
natural resourcefraction remaining <Time>
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 32/43
World Model Results
6 B Person10 B Capital units40 B Pollution units
1e+012 Resource units2 Satisfaction units
0 Person0 Capital units0 Pollution units0 Resource units
0.4 Satisfaction units1900 1929 1957 1986 2014 2043 2071 2100
Time (Year)
Population : run1 PersonCapital : run1 Capital unitsPollution : run1 Pollution unitsNatural Resources : run1 Resource unitsquality of life : run1 Satisfaction units
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 33/43
The Process influences Productivity
“Adding manpower to a late software project makes it later”
Fred Brooks. The Mythical Man-Month.
(www.ercb.com/feature/feature.0001.html)
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 34/43
Why Brooks’ Law ? Team Size.
Model in Forrester System Dynamics
using Vensim PLE (www.vensim.com)
development rate =
nominal_productivity* (1-C_overhead*(N*(N-1)))*N
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 35/43
Team Size N = 5
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 36/43
Team Size N = 3 . . . 9
Optimal Team Size between 7 and 8
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 37/43
The Effect of Adding New Personnel (FSD model)
development rate = nominal_productivity*
(1-C_overhead*(N*(N-1)))* (1.2*num_exp_working + 0.8*num_new)
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 38/43
5 New Programmers after 100 days
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 39/43
5 New Programmers after 100 days
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 40/43
0 . . . 6 New Programmers after 100 days
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 41/43
Individual-based (discrete-event) model:Productivity.
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 42/43
Individual-based (discrete-event) model:Remaining work.
Hans Vangheluwe [email protected] Modelling and Simulation: Forrester System Dynamics 43/43