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

Transcription

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 Rates proportional to population BR DR BR kbr P;DR kdr P dp dt k BR k DR P McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 1/33

2 k BR 1 4 kdr 1 2 : Exponential Growth trajectory population McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 2/33

3 k BR 1 4 kdr 1 2 : log(exponential Growth) 1E7 trajectory population 1E6 1E5 1E4 1E3 1E McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 3/33

4 k BR 1 2 kdr 1 4 : Exponential Decay 100 trajectory population McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 4/33

5 and kdr Are k BR and k DR really constant? Logistic Model Energy consumption in a closed system limits growth P Epc kbr E pc E tot P until equilibrium crowding effect: ecosystem can support maximum population P max dp dt k 1 P P max P crowding is a quadratic effect McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 5/33

6 k BR 1 2 kdr 1 4 crowding trajectory 200 population population time McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 6/33

7 Disadvantages NO physical evidence for model structure! But, many phenomena can be well fitted by logistic model. P max can only be estimated once steady-state has been reached. Not suitable for control, optimisation,... Many-species system: P max, steady-state? McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 7/33

8 b Ppred 0, efficiency factor 0 Multi-species: Predator-Prey Individual species behaviour + interactions Proportional to species, no interaction when one is extinct: product interaction P pred Pprey dp pred dt a Ppred k P pred Pprey dp prey dt c Pprey b P prey Excess death rate a grazing factor b k 0, excess birth rate c 1 0, Lotka-Volterra equations (1956): periodic steady-state McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 8/33

9 Predator Prey (population) trajectories 600 predator prey McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 9/33

10 Predator Prey (phase) phaseplot 300 predprey McGill, December, 2001 CS A Modelling and Simulation 10/33

11 P1 P1 P1 P1 Competition and Cooperation Several species competing for the same food source dp 1 dt a P1 b P 2 dp 2 dt c P2 d P 2 Cooperation of different species (symbiosis) dp 1 dt a P1 b P 2 dp 2 dt c P2 d P 2 McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 11/33

12 n Grouping and general n-species Interaction Grouping (opposite of crowding) dp dt a P b P 2 n-species interaction dp i dt ai n b i j 1 j Pj Pi i 1 Only binary interactions, no P 1 P2 P 3 interactions McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 12/33

13 Forrester System Dynamics based on observation + physical insight semi-physical, semi-inductive methodology McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 13/33

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 population 3. Influence Diagram (+ and -) 4. Structure Diagram dp dt BR DR McGill, December, 2001 CS A Modelling and Simulation 14/33

15 Causal Relationships graduates standard of living (SOL) SOL graduates beer consumption latent variable time beer consumption SOL McGill, December, 2001 CS A Modelling and Simulation 15/33

16 Archetypes Bellinger influence diagrams Common combinations of reinforcing and balancing structures McGill, December, 2001 CS A Modelling and Simulation 16/33

17 Archetypes: Reinforcing Loop state1 state2 McGill, December, 2001 CS A Modelling and Simulation 17/33

18 Archetypes: Balancing Loop action adjustment state desired state McGill, December, 2001 CS A Modelling and Simulation 18/33

19 Forrester System Dynamics uptake_predator loss_prey Grazing_efficiency prey_surplus_br predator_surplus_dr Predator Prey 2 species predator prey system McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 19/33

20 Inductive Modelling: World Dynamics BR: BirthRate P: Population POL: Pollution MSL: Mean Standard of Living... McGill, December, 2001 CS A Modelling and Simulation 20/33

21 POL POL MSL 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 f 4 MSL f 3 inversely proportional f 4 proportional compartmentalize to find correllations... Structure Characterization! McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 21/33

22 Structure Characterisation: LSQ fit X 2 X(t) = - gt /2 + v_0 t X(t) = A sin (b t ) t 2 LSQ (sin) < LSQ (t ) McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 22/33

23 Feature Extraction 1. Measurement data and model candidates 2. Structure selection and validation 3. Parameter estimation 4. Model use McGill, December, 2001 CS A Modelling and Simulation 23/33

24 Feature Rationale Minimum Sensitivity to Noise Maximum Discriminating Power McGill, December, 2001 CS A Modelling and Simulation 24/33

25 Throwing Stones Candidate Models 1. x 2. x 1 2 gt2 v0t Asin bt McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 25/33

26 Feature 1 (quadratic model) g i 2x i t 2 i 2ẋ i t i i A B F1 ga g B McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 26/33

27 Feature 2 (sin model) 1 b tg bt x i ẋ i solve numerically for b F2 200 b A bb b bb A McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 27/33

28 Feature Space Classification feature sin F2 = 200 ba -bb /(ba + bb) 1/b(tg(bt)) = xi/xi_der Feature t 2 F1 = ga/gb gi = 2xi/ti^2-2xi_der/ti McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 28/33

29 Forrester s World Dynamics model Club of Rome World Dynamics model Few levels, note the depletion of natural resources implemented in Vensim PLE ( McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 29/33

30 births pollution mult tab births material mult tab <material standard of living> birth rate normal birth rate normal 1 switch time 1 <Time> births material multiplier births pollution multiplier <pollution ratio> births food multiplier births food mult tab births births crowding multiplier births crowding mult tab Population & Food Population crowding land area population density normal population initial <food crowding mult tab> deaths deaths crowding multiplier deaths crowding mult tab deaths pollution death rate normal mult tab death rate normal 1 <pollution ratio> switch time 3 <Time> deaths pollution multiplier deaths material multiplier deaths food multiplier deaths food mult tab deaths material mult tab <material standard of living> food pollution mult tab food pollution multiplier <Time> switch time 7 food coefficient food coefficient 1 food ratio food per capita normal food crowding multiplier food per capita potential capital agriculture fraction indicated <capital ratio agriculture> food per capita potential tab capital agriculture fraction indicated tab McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 30/33

31 capital investment rate normal capital investment rate normal 1 capital investment mult tab capital investment multiplier effective capital ratio normal Capital & Quality of Life switch time 4 <Time> capital investment quality material multiplier capital investment from quality ratio capital initial <Population> material standard of living capital <quality food investment multiplier> quality ratio tab Capital effective capital ratio <capital agriculture fraction indicated> capital depreciation capital ratio switch time 5 <Time> capital agriculture fraction normal <natural quality resource extraction multiplier> <natural resource material extraction mult tab multiplier> capital depreciation normal capital ratio agriculture Capital Agriculture Fraction capital agriculture fraction adjustment time capital depreciation normal 1 <crowding> quality crowding mult tab capital agriculture fraction initial quality crowding multiplier quality of life <food ratio> quality food mult tab quality food multiplier quality pollution multiplier <pollution ratio> quality pollution mult tab quality of life normal McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 31/33

32 switch time 6 <Time> pollution per capita normal 1 pollution per capita normal pollution capital mult tab <capital ratio> pollution capital multiplier Pollution & Natural Resources pollution generation <Population> pollution initial Pollution pollution ratio pollution absorption pollution absorption time pollution standard pollution absorption time tab natural resource fraction remaining natural resource extraction multiplier natural resource extraction mult tab natural resources initial Natural Resources <material standard of living> natural resource utilization nat res matl multiplier switch time 2 <Time> natural resource utilization normal 1 natural resource utilization normal natural resource material mult tab McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 32/33

33 World Model Results 6 B Person 10 B Capital units 40 B Pollution units 1e+012 Resource units 2 Satisfaction units 0 Person 0 Capital units 0 Pollution units 0 Resource units 0.4 Satisfaction units Time (Year) Population : run1 Capital : run1 Pollution : run1 Natural Resources : run1 quality of life : run1 Person Capital units Pollution units Resource units Satisfaction units McGill, December, 2001 hv@cs.mcgill.ca CS A Modelling and Simulation 33/33