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Further model development ppt - optimal control theory group - Jim Sanchirico
EBM: Control Freaks
Model Zero: Two interacting species and a regulator who is informationally challenged
“Real world�: two-species competition
Optimal control problem of the informationally challenged regulator
Regulator is choosing effort levels over time to max. profits from fishing each species
Assumption is that the regulator does not know the interactions between the species
g is the (social) discount rate
What do I do…
I solve for the optimal effort level at the steady-state, that is, the solution to optimal control problem at the equilibrium.
Let, this effort level be Eiss
Take Eiss, which is a constant, and plug it into the real world two-species model
“Real world�: two-species competition
This generates a path of fish stocks
Assumption: Manager does not change the policy over time
Comparison to case with no-interaction
I compare these population trajectories to the ones I get from the following:
Experiment 1
Only fish x1, f1>0, f2=0
Look at different levels of the competition term that is assumed equal to each other (ai=a)
Low: a/4
Medium: a/2
High: a
where a is equal to 70% of ri=r
Measure stock in terms of density (with all adjustments to parameter levels)
Experiment 2
Everything the same, but now we are fishing in both patches f1>0, f2>0 (as defined previously)
Look at different levels of the competition term that is assumed equal to each other (ai=a)
Low: a/4
Medium: a/2
High: a
where a is equal to 70% of ri=r
Preliminary conclusions w/ competition
Fishing both species seems to lower the absolute interaction strength (even though the alpha term is the same)
Absolute interaction strength = ax1x2
Results are, of course, specific to the parameter set
We should determine if we could derive these results analytically
Now, looking at predator-prey
Experiment 1
Fishing the prey (f1>0) not the predatory
All other parameters the same as with competition
Natural mortality of predator=.01*r1
Experiment 2
Fishing the predator (f1>0) not the prey
All other parameters the same as with competition
Natural mortality of predator=.01*r1
Experiment 3
Fishing the predator and the prey (fi>0)
All other parameters the same as with competition
Natural mortality of predator=.01*r1
To do…
Analytics of this very special setting
More sensitivity analysis
Red line is what
the regulator is believing to occur,
Blue line is fished population,
Green is unfished
Red line is what
the regulator is believing to occur
Fished population is the prey,
Predator is unfished and red is what the regulator “sees�
Fishing the predator and not the prey
Fishing both predator (x2) and prey (x1)
Created with pptHtml
Model Zero: Two interacting species and a regulator who is informationally challenged
“Real world�: two-species competition
Optimal control problem of the informationally challenged regulator
Regulator is choosing effort levels over time to max. profits from fishing each species
Assumption is that the regulator does not know the interactions between the species
g is the (social) discount rate
What do I do…
I solve for the optimal effort level at the steady-state, that is, the solution to optimal control problem at the equilibrium.
Let, this effort level be Eiss
Take Eiss, which is a constant, and plug it into the real world two-species model
“Real world�: two-species competition
This generates a path of fish stocks
Assumption: Manager does not change the policy over time
Comparison to case with no-interaction
I compare these population trajectories to the ones I get from the following:
Experiment 1
Only fish x1, f1>0, f2=0
Look at different levels of the competition term that is assumed equal to each other (ai=a)
Low: a/4
Medium: a/2
High: a
where a is equal to 70% of ri=r
Measure stock in terms of density (with all adjustments to parameter levels)
Experiment 2
Everything the same, but now we are fishing in both patches f1>0, f2>0 (as defined previously)
Look at different levels of the competition term that is assumed equal to each other (ai=a)
Low: a/4
Medium: a/2
High: a
where a is equal to 70% of ri=r
Preliminary conclusions w/ competition
Fishing both species seems to lower the absolute interaction strength (even though the alpha term is the same)
Absolute interaction strength = ax1x2
Results are, of course, specific to the parameter set
We should determine if we could derive these results analytically
Now, looking at predator-prey
Experiment 1
Fishing the prey (f1>0) not the predatory
All other parameters the same as with competition
Natural mortality of predator=.01*r1
Experiment 2
Fishing the predator (f1>0) not the prey
All other parameters the same as with competition
Natural mortality of predator=.01*r1
Experiment 3
Fishing the predator and the prey (fi>0)
All other parameters the same as with competition
Natural mortality of predator=.01*r1
To do…
Analytics of this very special setting
More sensitivity analysis
Red line is what
the regulator is believing to occur,
Blue line is fished population,
Green is unfished
Red line is what
the regulator is believing to occur
Fished population is the prey,
Predator is unfished and red is what the regulator “sees�
Fishing the predator and not the prey
Fishing both predator (x2) and prey (x1)
Created with pptHtml