Population Dynamics Deductive modelling: based on physical laws - - PowerPoint PPT Presentation

▶
population dynamics
SMART_READER_LITE
LIVE PREVIEW

Population Dynamics Deductive modelling: based on physical laws - - PowerPoint PPT Presentation

Oct 07, 2022 360 likes •709 views

Population Dynamics Deductive modelling: based on physical laws Inductive modelling: based on observation + intuition Single species: Birth (in migration) Rate, Death (out

slide-1
SLIDE 1

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 1/33

slide-2
SLIDE 2

kBR

✝

1

✞

4

✟

kDR

✝

1

✞

2 : Exponential Growth

population

trajectory

10 20 30 40 50 1000000 2000000

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 2/33

slide-3
SLIDE 3

kBR

✝

1

✞

4

✟

kDR

✝

1

✞

2 : log(Exponential Growth)

population

trajectory

10 20 30 40 50 1E2 1E3 1E4 1E5 1E6 1E7

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 3/33

slide-4
SLIDE 4

kBR

✝

1

✞

2

✟

kDR

✝

1

✞

4 : Exponential Decay

population

trajectory

10 20 30 40 50 20 40 60 80 100

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 4/33

slide-5
SLIDE 5

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 5/33

slide-6
SLIDE 6

kBR

✝

1

✞

2

✟

kDR

✝

1

✞

4

✟

crowding

✝ ✞

001

population

trajectory time

20 40 60

population

100 200

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 6/33

slide-7
SLIDE 7

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 7/33

slide-8
SLIDE 8

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 8/33

slide-9
SLIDE 9

Predator Prey (population)

predator prey

trajectories

10 20 30 200 400 600

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 9/33

slide-10
SLIDE 10

Predator Prey (phase)

predprey

phaseplot

200 400 600 100 200 300

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 10/33

slide-11
SLIDE 11

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 11/33

slide-12
SLIDE 12

Grouping and general n-species Interaction

  • Grouping (opposite of crowding)

dP dt

✁ ✂

a

✄

P

☞

b

✄

P2

  • 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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 12/33

slide-13
SLIDE 13

Forrester System Dynamics

  • based on observation + physical insight
  • semi-physical, semi-inductive methodology

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 13/33

slide-14
SLIDE 14

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 14/33

slide-15
SLIDE 15

Causal Relationships

latent variable

beer consumption graduates standard

  • f

living (SOL) time graduates SOL beer consumption SOL

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 15/33

slide-16
SLIDE 16

Archetypes

  • Bellinger http://www.outsights.com/systems/
  • influence diagrams
  • Common combinations of reinforcing and balancing structures

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 16/33

slide-17
SLIDE 17

Archetypes: Reinforcing Loop

state1 state2

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 17/33

slide-18
SLIDE 18

Archetypes: Balancing Loop

adjustment state desired state action

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 18/33

slide-19
SLIDE 19

Forrester System Dynamics

Predator Prey Grazing_efficiency uptake_predator loss_prey predator_surplus_DR prey_surplus_BR

2−species predator−prey system

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 19/33

slide-20
SLIDE 20

Inductive Modelling: World Dynamics

  • BR: BirthRate
  • P: Population
  • POL: Pollution
  • MSL: Mean Standard of Living
  • . . .

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 20/33

slide-21
SLIDE 21

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 21/33

slide-22
SLIDE 22

Structure Characterisation: LSQ fit

X(t) = - gt /2 + v_0 t 2 X(t) = A sin (b t ) t X LSQ (sin) < LSQ (t ) 2

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 22/33

slide-23
SLIDE 23

Feature Extraction

  • 1. Measurement data and model candidates
  • 2. Structure selection and validation
  • 3. Parameter estimation
  • 4. Model use

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 23/33

slide-24
SLIDE 24

Feature Rationale

Minimum Sensitivity to Noise Maximum Discriminating Power

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 24/33

slide-25
SLIDE 25

Throwing Stones

Candidate Models

  • 1. x
✁ ✂

1 2gt2

☞

v0t

  • 2. x
✁

Asin

☎

bt

✆

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 25/33

slide-26
SLIDE 26

Feature 1 (quadratic model)

gi

✁

2xi t2

i

✂

2˙ xi ti

✑

i

✁

A

✑

B F1

✁

gA

✛

gB

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 26/33

slide-27
SLIDE 27

Feature 2 (sin model)

1 btg

☎

bt

✆ ✁

xi ˙ xi

solve numerically for b

F2

✁

200

✜

bA

✂

bB

✜

bA

☞

bB

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 27/33

slide-28
SLIDE 28

Feature Space Classification

Feature t 2 feature sin

F1 = gA/gB gi = 2xi/ti^2 - 2xi_der/ti F2 = 200 |bA -bB|/(bA + bB) 1/b(tg(bt)) = xi/xi_der

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 28/33

slide-29
SLIDE 29

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 hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 29/33

slide-30
SLIDE 30

Population

birth rate normal birth rate normal 1 population initial births crowding multiplier births food multiplier births deaths births material multiplier births pollution multiplier switch time 1 <Time> death rate normal death rate normal 1 deaths crowding multiplier deaths food multiplier deaths material multiplier deaths pollution multiplier switch time 3 crowding land area population density normal

Population & Food

births crowding mult tab deaths crowding mult tab food ratio food coefficient food coefficient 1 food crowding multiplier food per capita normal food per capita potential food pollution multiplier switch time 7 <Time> food pollution mult tab <pollution ratio> <capital ratio agriculture> food per capita potential tab <food crowding mult tab> <Time> capital agriculture fraction indicated births food mult tab deaths food mult tab capital agriculture fraction indicated tab <material standard of living> deaths material mult tab births pollution mult tab births material mult tab <material standard of living> deaths pollution mult tab <pollution ratio> Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 30/33

slide-31
SLIDE 31

Capital

capital initial capital ratio capital depreciation capital investment rate normal 1 capital depreciation normal capital depreciation normal 1 switch time 5 capital ratio agriculture capital agriculture fraction normal capital investment multiplier effective capital ratio capital investment mult tab capital investment capital investment rate normal <Population> switch time 4 <Time>

Capital & Quality of Life

Capital Agriculture Fraction

capital agriculture fraction adjustment time <capital agriculture fraction indicated> capital agriculture fraction initial capital investment from quality ratio <natural resource extraction multiplier> quality material multiplier quality material mult tab material standard of living effective capital ratio normal capital investment quality ratio tab <quality food multiplier> <natural resource extraction multiplier>

quality of life

quality crowding multiplier quality food multiplier quality of life normal quality pollution multiplier <crowding> quality crowding mult tab <food ratio> quality food mult tab <pollution ratio> quality pollution mult tab <Time> Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 31/33

slide-32
SLIDE 32

Natural Resources

pollution absorption time tab natural resources initial pollution ratio pollution standard nat res matl multiplier natural resource utilization normal natural resource utilization normal 1 <capital ratio> switch time 2 pollution capital mult tab natural resource utilization

Pollution & Natural Resources

Pollution

pollution initial pollution capital multiplier pollution per capita normal pollution per capita normal 1 <Population> switch time 6 <Time> pollution absorption time pollution absorption pollution generation <material standard of living> natural resource material mult tab natural resource extraction multiplier natural resource extraction mult tab natural resource fraction remaining <Time>

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 32/33

slide-33
SLIDE 33

World Model Results

6 B Person 10 B Capital units 40 B Pollution units 1e+012 Resource units 2 Satisfaction units Person Capital units Pollution units Resource units 0.4 Satisfaction units 1900 1929 1957 1986 2014 2043 2071 2100 Time (Year)

Population : run1 Person Capital : run1 Capital units Pollution : run1 Pollution units Natural Resources : run1 Resource units quality of life : run1 Satisfaction units

Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 33/33