Schaefer model through the FishBanks role playing game

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Schaefer modelthroughthe FishBanks role playing
game
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FishBanks

• 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 gt negociations
• Discussion about the type of measurements
• gt need of an agreement about the diagnostic
• Discussion ltgt institution
• Rules - Control - Sanctions

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FishBanks
A message
Privatization of the profits, socialization of
losses Social cost extern to the market
A standard model
Schaefer
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FishBanks Typical evolution of three key
indicators
Boats
Captures
Fishes
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10
8
6
4
2
0
Time
1
2
3
4
5
6
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The Schaeffer model Carrying capacity K and
equilibrium

• Based on Verhulst model (Logistic equation)
  dx/dt rx (1 - x/K)

K
K/2
x
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Renewable resource with harvest h(t)

• Constant harvest h(t)
• dx/dt rx (1 - x/K) - h(t)
• MSH Maximum sustainable harvest, XK/2

hRMS

h
-
K
x
X2 stable equilibrium
X1 unstable equilibrium
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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)

K
x
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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
q.E.x
x
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Yield as a function of effort in situations of
equilibrium
MSY Maximum Sustainable Yield
Y Yield-effort curve
Production-population curve
MSY
rK/4
q.E.x
E
xK/2
r/q
r/2q
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Renewable resource Estimation of the parameters

• hY q.E.x q.E.K.(1 - q.E/r)
• Y/E -(q2.K/r).E qK

Catches per ship
Y/E
Capture per effort effort curve The Schaefer
line
qK
-q2.K/r
Y/EMSYqK/2
r/q
E
EMSYr/2q
Number of ships
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Schaeffer Regression line FishBanks game
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S4
S1
S2
25
S3
20
15
Catches per ship
10
5
S5
0
0
10
20
30
40
50
60
70
Number of ships
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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.
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DiscreteSystems Dynamics
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Definitions

• The state of the system at time t
  Xt (at, bt, )
• The state at time t1 only depends on the state
  at time t Xt1 F(Xt)
• X0 and the function F are sufficient to define
  all possible values

No need to run simulation to solve the system
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Discrete dynamics

• Arithmetic growth, xt1 r xt gt xt
  x0r.t
• Geometric growth, xt1 r.xt gt xt x0.rt
• Logistic equation,

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Linear equation
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Example
Phases space
Temporal equation
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Example
Phases space
Temporal equation
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Example
Phases space
Temporal equation

• Converge whatever the initial value of Xo X8
  10/0.616,66666

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Example
Phases space
Temporal equation
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Limits of linear equation
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Attractors

• An attractor is a point or a set of points
• Fixed point xt1xt
• Cyclic point xtnxt

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Logistic equation

• Populations dynamics
• bilge mathematician Pierre-François Verhulst.
  1838
• Non-linear dynamic model

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Populations dynamics Growth factor R

• R b m
• b natality rate
• m mortality rate
• Whose solution is
• gt Exponential growth
• Malthus model

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Populations dynamics Reproduction factor r

• Growth is limited.
• R decreasing function of x. R(x)
• Example logistic equation (Verhulst. 1838)

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Solution
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Logistic equation Dynamics and carrying capacity
K
dx/dt r.x.(1 - x/K)
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carrying capacity and equilibrium
x
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From continuous to discret model Deterministic
chaos
and by using the
approximation
gt
Feigenbaum (1970), the discret model is richer
than the continuous one.
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Evolution and attractors of logistic equation
0 lt a lt 1
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Evolution and attractors of logistic equation
1 lt a lt 3
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Evolution and attractors of logistic equation
3 lt a lt 4
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Initial conditions sensibility
Dynamics of Xt for a 4 and X00.01 or
X00.01001 ? 1
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a4 Xo0.2 and Xo0.2002
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Attractors of logistic equation
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Bifurcation diagram Feigenbaum diagram
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Bifurcation diagram Feigenbaum diagram (zoom)
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Strange attractors and fractal dimensions
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dimension 2
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Henon attractor
Attractor shape You cant predict x,y at t The
system is organized fractal properties
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Lorenz attractor
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References

• Discrete system dynamics Pavé, Tu,
• Deterministic chaos Casti, Cvitanovic
• Discrete control Clark