Consider a renewable ressource whose stock, measured through its biomass B, is considered
globally, as one single unit, without any consideration for the structure population. Its growth is
materialized through the linear Schaefer model
(1)
where h(t) is the harvesting at time t and R(t) = 1 + r(t) is the productivity of the
resource.
Adapted from [1, p. 258], we shall illustrate the results with the Antartic baleen whale example
as in Table 1.
Antartic baleen whale
intrinsic growth
r = 5%
carrying capacity
K = 400000 BWH
catchability
q = 0.0016 WCY
price
p = 7000$ per BWU
cost
c = 600000$ per WCY
Table 1: Antartic baleen whale data for a discrete time logistic model (adapted from [1,
p. 258]). BWH: blue whale unit. WCY: whale-catcher year.
2 Deterministic growth
In this case, the productivity is stationary and deterministic:
(2)
2.1 Try your own strategy
Question 1You have to manage the whale population whose dynamics parameters are givenin Table1. Design your own catch strategyCatch(t,B) (following the model of the randomcatch effort strategy in the scilab code below). Give the scilab code corresponding to yourown catch strategy. Draw simulations of biomass and catch trajectories.
R_whale=1.05;// per capita productivity R=R_whale; K_whale=400000;// carrying capacity K=K_whale; // LINEAR DYNAMICS function y=linear(B) y=R*B,endfunction; // RANDOM CATCH EFFORT function h=rand_effort(t,B) h=rand()*B,endfunction; Horizon=10; years=1:Horizon; yearss=1:(Horizon+1); yearsss=1:(Horizon+2); Binit=K/2; // initial condition trajectory_whale=zeros(yearss); // vector will contain the trajectory B(1),...,B(Horizon+1) catch_whale=zeros(years); // vector will contain the catches h(1),...,h(Horizon) trajectory_whale(1)=Binit; // initialization of vector B(1),...,B(Horizon+1) for t=yearsdo catch_whale(t)=rand_effort(t,trajectory_whale(t)); trajectory_whale(t+1)=linear(trajectory_whale(t)-catch_whale(t)); end; // Graphics xset("window",20+1);xbasc(20+1); plot2d2(yearss,[trajectory_whale;[catch_whale,0]]'); xtitle('Trajectory under linear growth with R='+string(R_whale)+' and random effort', ... 'year (t)','biomass (t)') legends(['Biomass trajectory';'Catch trajectory'],[1,2],'ur');
2.2 Intertemporal discounted utility maximization
We now assume that some planner aims at maximizing the sum of discounted utilities of the
catches plus a final stock utility. The maximization problem is
(3)
where ρ = ∈ [0, 1] is a discount factor and Util(⋅) is a utility function, concave and
strictly increasing on ℝ+. Notice that the final term UtilB(T) corresponds to an
existence or inheritance value of the stock. We shall use a discount factor ρ with typical
values ranging between 0.9 and 1. We restrict the study to the isoelastic utility case
where
(4)
Analytic resolution by dynamic programming
The dynamic programming equation is:
(5)
By backward induction, it can be proved that
(6)
where βd(t) is given by the backward equation
(7)
It can be deduced that, along the optimal path given by
(8)
we have
(9)
scilab simulations
Question 2Simulate optimal catches and biomass trajectories (B⋆(t),h⋆(t)). Change thevalue of the discount factor ρ to modify the location of ρR with respect to 1. Do at least threecases with ρR > 1, ρR = 1 and ρR < 1. What do you observe?
Question 3What is the influence of η on the optimal paths (B⋆(t),h⋆(t))? In particular,what happens when η ≈ 1? Comment knowing that 1 − η may be interpreted as the constantrelative risk aversion of the decision-maker.
Question 4Write a program which evaluates the criterion
(10)
for any catch decision ruleCatch : (t,B)Catch(t,B) (admissible in the sense that
0 ≤Catch(t,B) ≤ B), and where
2.3 Minimal utility maximization or Rawls criterion
We now assume that the planner aims at sustainability and intergenerational equity, and attempts
at maximizing the utility of the “poorest” generation. The maximization problem is
then
(14)
Analytic resolution by dynamic programming
The dynamic programming equation is:
(15)
By backward induction, it can be proved that
(16)
where γd(t) is given by the backward equation
(17)
Consequently, along the maximin optimal path given by
(18)
we have
(19)
scilab simulations
Question 6Adapt the previous scilab code to obtain the maximin path (B⋆(t),h⋆(t)).
Question 7Compare the trajectories of the maximin criterion, of the stock biomasses and of thecatches between
Question 12Show that, as for the corresponding deterministic case in subsection2.2,
(6)–(11) hold true with R replaced by R♭.
scilab simulations
Question 13Generate trajectories with robust optimal decision rule. Adapt the code insubsection2.2to draw trajectories. Compare the criterion values given by the random effortstrategy, by your test strategy in Question1, and by the optimal strategy.
3.2 Worst minimal utility maximization
The maximization problem is
(29)
Analytic resolution by dynamic programming
The dynamic programming equation is:
(30)
Question 14Show that, as for the corresponding deterministic case in subsection2.3,
(16)–(17) hold true with R replaced by R♭.
scilab simulations
Question 15Generate trajectories with robust viable decision rules. Adapt the code insubsection2.3to draw trajectories. Compare the criterion values given by the random effortstrategy, by your test strategy in Question1, and by the optimal strategy.
3.3 Tolerable window and guaranteed harvesting (robust viability)
We assume that the policy is to constrain the biomass level within an ecological window, namely
between conservation and maximal safety values, whatever the scenario:
(31)
A minimal catch is also required:
(32)
The dynamic programming equation for the robust viability kernels is:
(33)
Question 16Define by backward induction
Show that, whenever
the viability kernels are intervals:
Show that the viable decision rulesCatch(t,B) are those which belong to
References
[1]M. Kot. Elements of Mathematical Ecology. Cambridge University Press,
Cambridge, 2001.