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

✄ � ✆ � ✂ � ☎ ✁ ✄ ✁ ✁ � ✂ ✁ 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 BR DR dt Rates proportional to population BR k BR P ; DR k DR P dP k BR k DR P dt Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 1/33

✝ ✞ ✟ ✝ ✞ k BR 1 4 k DR 1 2 : Exponential Growth trajectory population 2000000 1000000 0 10 20 30 40 50 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 2/33

✝ ✞ ✟ ✝ ✞ k BR 1 4 k DR 1 2 : log(Exponential Growth) trajectory 1E7 population 1E6 1E5 1E4 1E3 1E2 0 10 20 30 40 50 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 3/33

✞ ✟ ✝ ✞ ✝ k BR 1 2 k DR 1 4 : Exponential Decay trajectory 100 population 80 60 40 20 0 10 20 30 40 50 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 4/33

� ✄ ✠ ☛ � ✠ ✡ ✁ ☎ ☛ ✁ ✂ ✆ ✠ ✄ � � ✡ Logistic Model Are k BR and k DR really constant ? Energy consumption in a closed system limits growth E tot E pc P P E pc k BR and k DR until equilibrium “crowding” effect: ecosystem can support maximum population P max dP P k 1 P dt P max crowding is a quadratic effect Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 5/33

✝ ✞ ✟ ✝ ✞ ✟ ✝ ✞ k BR 1 2 k DR 1 4 crowding 0 001 trajectory population 200 population 100 0 0 20 40 60 time Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 6/33

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

✄ � � ✄ ✂ ✄ ✄ ☞ ✄ ✄ ✂ ✁ ✁ ✌ ✌ ✌ ✄ ✍ ✎ � � ✄ Multi-species: Predator-Prey Individual species behaviour + interactions Proportional to species, no interaction when one is extinct: product interaction P pred P prey dP pred a P pred k b P pred P prey dt dP prey c P prey b P pred P prey dt 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

Predator Prey (population) trajectories predator prey 600 400 200 0 10 20 30 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 9/33

Predator Prey (phase) phaseplot predprey 300 200 100 200 400 600 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 10/33

✄ ✄ � ✂ ✁ ✄ � ✁ ✄ ✁ ✂ ✄ ✄ ✄ ✄ ✂ ☞ ✄ ☞ ✁ ✄ ✄ ✂ ✄ Competition and Cooperation Several species competing for the same food source dP 1 a P 1 b P 1 P 2 dt dP 2 c P 2 d P 1 P 2 dt Cooperation of different species (symbiosis) dP 1 a P 1 b P 1 P 2 dt dP 2 c P 2 d P 1 P 2 dt Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 11/33

✔ ✄ ✖ ✖ ✑ ✄ ✆ ✘ ✄ � � ✏ ✄ ☞ ☎ ✕ ✁ ☞ ✄ � ✁ ✂ ✄ Grouping and general n -species Interaction Grouping (opposite of crowding) dP P 2 a P b dt n -species interaction n dP i ∑ a i b ij P j P i i 1 n dt ✑✗✖ ✑✓✒ j 1 Only binary interactions, no P 1 P 2 P 3 interactions Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 12/33

� � 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

✠ ✠ ✂ ✁ 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 BR DR dt Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 14/33

Causal Relationships graduates standard of graduates SOL living (SOL) beer consumption time beer consumption latent variable SOL Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 15/33

� � � 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

Archetypes: Reinforcing Loop state1 state2 Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 17/33

Archetypes: Balancing Loop action state adjustment desired state Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 18/33

Forrester System Dynamics uptake_predator loss_prey Grazing_efficiency prey_surplus_BR predator_surplus_DR Predator Prey 2−species predator−prey system Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 19/33

� � � � � 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

✆ ☎ ✆ ✖ ✁ ✖ ✄ ✖ ✄ ✆ ✙ ☎ ✚ ✚ ✖ ✑ ✙ ✑ ✖ ✖ ✖ ✄ ✆ ✁ ☎ ✄ ✚ ✖ ✖ ✙ ✙ ✁ � ☎ � ✑ ✆ ✑ ☎ ✚ ✖ ✖ ✆ ✁ ✑ � ✄ ✆ ✙ ☎ ✚ ☎ ✚ ✑ ✙ ✑ � ✄ Inductive Modelling: Structure Characterization BR f P POL MSL ✑✗✖ 1 BR BRN f P POL MSL 2 BR BRN P f POL MSL 3 4 BR BRN P f POL f MSL 3 f POL inversely proportional 4 f MSL proportional compartmentalize to find correllations . . . Structure Characterization ! Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 21/33

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

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

Feature Rationale Minimum Sensitivity to Noise Maximum Discriminating Power Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 24/33

☎ ✁ ✂ ☞ ✁ ✆ Throwing Stones Candidate Models 1 2 gt 2 1. x v 0 t 2. x Asin bt Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 25/33

✁ ✑ ✁ ✛ ✁ ✑ ✂ Feature 1 (quadratic model) 2 x i 2˙ x i g i i A B t 2 t i i F 1 g A g B Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 26/33

✜ ☞ ✂ ☎ ✜ ✆ ✁ ✁ Feature 2 (sin model) 1 x i btg bt x i ˙ solve numerically for b b A b B F 2 200 b A b B Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 27/33

Feature Space Classification F2 = 200 |bA -bB|/(bA + bB) feature sin 1/b(tg(bt)) = xi/xi_der 2 Feature t F1 = gA/gB gi = 2xi/ti^2 - 2xi_der/ti Hans Vangheluwe hv@cs.mcgill.ca Modelling and Simulation: Forrester System Dynamics 28/33

� � � 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