Let time t be measured in discrete units (such as years). Let B(t) denote the biomass of a population at time t (beginning of time interval [t,t + 1[). We consider the so called Schaefer model
| (1) |
where Biol is the population dynamics and h(t) is the harvesting. Notice that, in the time interval [t,t + 1[, growth of the stock occurs first, followed by harvesting1 .
The sustainable yield he = Sust(Be) solves Be = Biol(Be) − he, which gives:
| (2) |
The carrying capacity of the habitat is the level K > 0 of positive biomass such that Biol(K) = K, that is Sust(K) = 0.
The maximum sustainable yield hmse and the corresponding maximum sustainable equilibrium Bmse are
| (3) |
From [1, p. 258] and numerical simulations, we shall consider the Pacific yellowfin tuna example as in Table 1.
Pacific yellowfin tuna | |
yearly intrinsic growth | R = 2.25 |
carrying capacity | K = 250 000 metric tons |
catchability | q = 0.0 000 385 per SFD |
price | p = 600 $ per metric ton |
cost | c = 2 500 $ per SFD |
The Beverton-Holt model is characterized by the discrete time dynamics mapping
| (4) |
We have
| (5) |
Question 2 Select one biomass level Be between the maximum sustainable biomass Bmse and the carrying capacity K. Compute the corresponding sustainable yield he.
Draw the corresponding steady trajectory of the Schaefer model (1) with the Beverton-Holt dynamics (4) and h(t) = he. Pick up two different initial conditions in the neighborhood of the equilibrium biomass Be. Draw the corresponding trajectories. Does the figure confirm or not the fact that the equilibrium biomass Be is attractive?
Recall that, for an equilibrium, being stable or attractive are unrelated notions.
Question 3 Does the figure confirm or not the fact that the equilibrium biomass Be is stable? Be specific in your justifications. What can you say about asymptotic stability of the equilibrium biomass Be?
Question 4 Find an equilibrium state Be which is not attractive. Illustrate that Be is not attractive with some trajectories. What can you say about asymptotic stability of the equilibrium biomass Be?
With price p, catchability coefficient q and harvesting unitary costs c, the private property equilibrium (ppe) is the equilibrium solution (Bppe,hppe) = (Bppe,Sust(Bppe)) which maximizes the rent as follows:
| (6) |
The common property equilibrium Bcpe makes the rent null and is given by
| (7) |
The logistic model is characterized by the discrete time dynamics mapping
| (9) |
where R ≥ 1 and r = R − 1 ≥ 0 is the per capita rate of growth (for small populations), and κ is related to the carrying capacity K (which solves Biol(K) = K) by:
| (10) |
We have
| (11) |
Question 6 Adapt the previous Scilab code to the logistic model, and compare the results.
The Ricker model is characterized by the discrete time dynamics mapping
| (12) |
Question 7 Adapt the previous Scilab code to the Ricker model, and compare the results. Try numerical procedures: type help fsolve to obtain information about Scilab solver.