Schaefer model through the Fish Banks role playing

Schaefer model through the Fish. Banks role playing game

Fish. Banks • Fish Banks – is a computer-assisted interactive, role-playing simulation in which groups manage a fishing company – Participants try to maximize their assets in a world with renewable natural resources and economic competition. – the game has been created in 1986 by Dennis Meadows • • A diversity of strategies Management of incomes or resources ? Free Access => negociations Discussion about the type of measurements – => need of an agreement about the diagnostic • Discussion <=> institution – Rules - Control - Sanctions 2 Espaces et Ressources

Fish. Banks A message Privatization of the profits, socialization of losses Social cost extern to the market A standard model Schaefer 3 Espaces et Ressources

Fish. Banks Typical evolution of three key indicators Boats Captures Fishes 12 10 8 6 4 2 0 1 2 3 4 5 6 4 Time Espaces et Ressources

The Schaeffer model Carrying capacity K and equilibrium • Based on Verhulst model (Logistic equation) : dx/dt = rx (1 - x/K) + K/2 K x 5 Espaces et Ressources

Renewable resource with harvest h(t) • Constant harvest h(t) • dx/dt = rx (1 - x/K) - h(t) • MSH = Maximum sustainable harvest, XK/2 h. RMS + h - K X 1 =unstable equilibrium x X 2 =stable equilibrium 6 Espaces et Ressources

Harvest h(t) with effort E and capturability q • Harvest: h(t) = q. E. x • Fishing effort E (number of boats), q capturability • dx/dt = rx (1 - x/K) - h(t) + . x E. q h + -- h’ x q. E’. K X =Equilibrium =Equilibre x 7 Espaces et Ressources

Renewable resource Harvest h(t) with effort E and capturability q • dx/dt = rx (1 - x/K) - h(t) • h(t) = q. E. x At the biological equilibrium, one harvests what is produced . x E. q x* 8 Espaces et Ressources

Yield as a function of effort in situations of equilibrium MSY = Maximum Sustainable Yield Production-population curve MSY + x r. K/4 Yield-effort curve q. E. Y E r/2 q r/q x*=K/2 9 Espaces et Ressources

Renewable resource Estimation of the parameters • h=Y = q. E. x* = q. E. K. (1 - q. E/r) • Y/E = -(q 2. K/r). E +q. K Catches per ship Y/E Capture per effort – effort curve The Schaefer line q. K Y/EMSY=q. K/2 -q 2. K/ r EMSY=r/2 q r/q E Number of ships 10 Espaces et Ressources

Schaeffer “Regression line” Fish. Banks game 30 S 1 Catches per ship 25 S 4 S 2 S 3 20 ? ? 15 10 5 S 5 0 0 10 20 30 40 Number of ships 50 60 70 11 Espaces et Ressources

The Schaeffer model Basic assumptions • Isolated Resource • Constant environment • Dynamics at the equilibrium • Free access • Rational and optimal behaviour • Fixed natural mortality • Fixed Prices • Homogenous efforts. Schaefer, M. B. , 1957. Some considerations of population dynamics and economics in relation to the management of the marine fisheries. Jal. of the Fisheries Research Board of Canada, 14 (5) : 669 -681. 12 Espaces et Ressources

Discrete Systems Dynamics Espaces et Ressources

Definitions • The state of the system at time t Xt = (at, bt, …) • The state at time t+1 only depends on the state at time t Xt+1= F(Xt) • X 0 and the function F are sufficient to define all possible values No need to run simulation to solve the system 14 Espaces et Ressources

Discrete dynamics • Arithmetic growth, xt+1= r + xt => xt = x 0+r. t • Geometric growth, xt+1= r. xt => xt = x 0. rt • Logistic equation, 15 Espaces et Ressources

Linear equation Xt+1 = f(Xt) Xt = g(Xo, t) 16 Espaces et Ressources

Example Phases space : Temporal equation : 17 Espaces et Ressources

Example Phases space : Temporal equation : 18 Espaces et Ressources

Example Phases space : Temporal equation : Þ Converge whatever the initial value of Xo X∞ = 10/0. 6=16, 66666 19 Espaces et Ressources

Example Phases space : Temporal equation : 20 Espaces et Ressources

Limits of linear equation 21 Espaces et Ressources

Attractors • An attractor is a point or a set of points • Fixed point xt+1=xt • Cyclic point xt+n=xt 22 Espaces et Ressources

Logistic equation • Populations dynamics • bilge mathematician Pierre-François Verhulst. 1838 • Non-linear dynamic model 23 Espaces et Ressources

Populations dynamics Growth factor R • R =b –m – b: natality rate – m: mortality rate • Whose solution is : • => Exponential growth – Malthus model 24 Espaces et Ressources

Populations dynamics Reproduction factor r • Growth is limited. • R decreasing function of x. R(x) • Example: logistic equation (Verhulst. 1838) growth factor Reproduction rate Carrying capacity 25 Espaces et Ressources

Solution: 26 Espaces et Ressources

Logistic equation Dynamics and carrying capacity K dx/dt = r. x. (1 - x/K) 27 Espaces et Ressources

carrying capacity and equilibrium + K/2 - K x 28 Espaces et Ressources

From continuous to discret model: Deterministic chaos and by using the approximation growth factor => Reproduction rate Carrying capacity Feigenbaum (1970), the discret model is richer than the continuous 29 one. Espaces et Ressources