Extra Information on the Graphical Analysis Near the End of Chapter 14


Chapter 14 explains that traditional fishery studies often use graphical analysis to find the MSY, the Maximum Sustainable Yield. Chapter 14 also explains how the salmon model may be used to find the maximum yield using multiple simulations. Therefore, we don't need a graphical analysis to find the MSY. But it is common in fishery studies, so it is useful to illustrate the graphical approach. The illustration will serve to reinforce the findings from the dynamic model.

The graphical approach begins with the juvenile survival curve shown in Figure 14.11 of the book. Juvenile loss is the only loss in the model that depends on density. All other losses may be combined in a single line whose slope varies with changes in the harvesting fraction. Figure 14.11demonstrates with four lines corresponding to harvest fractions of 0%, 50%, 75% and 90%. The fact that the straight lines intersect the survival curve is a sign that the harvest fraction is sustainable.

Figure 14.11 (in book) Graphical analysis of the number of smolts and fry
in a sustainable system with different values of the harvest fraction.

To demonstrate, algebraically, why this is true, it is helpful to define the following terms:
 h = harvest fraction x1 = 25% adult migration loss fraction
A = adults arrive at Columbia x2 = 50% egg loss fraction
H = harvest x3 = 90% smolt loss fraction
 F = emergent fry x4 = 35% ocean loss fraction in 1st year
 S = smolts ready to migrate x5 = 10% ocean loss fraction in 2nd year
 ff = 50% fraction female epr = 3.9 thousand eggs per redd

In a sustainable situation, we should expect to see:
F = A (1-h) (1-x1) (ff) (epr) (1-x2) or
F = A (1-h) (.73)

This gives the fry, F, as a function of A, the number of adults. If we happen to know S,
the number of smolts to begin the spring migration, we could find A as follows:
A = S (1-x3) (1-x4) (1-x5) = S (.0585)

Now, insert the expression for A into the previous expression for F to obtain:
F = S (.0585) (1-h) (.73) or
F = S (.0427) (1-h)

At this point, we have F as a linear function of S. But you can see from Figure 14.11, that we need to express S as a linear function of F. This may be done as follows:
S = 23 F/(1-h)

For any value of h, we can express S as a linear function of F, as shown by the four lines in Figure 14.11. Recall that the purpose is to find where the four lines intersect the nonlinear curve. These intersection points reveal the number of fry and the number of smolts that are consistent with all the assumptions of the model andare consistent with the assumption that the harvest is sustainable. With 50% harvesting, for example, the intersection point shows around 8 million fry and around 360,000 smolts. (These values agree with the values shown in chapter 14.) Now, to find the annual harvest, we set
H = hA,

and we express A as proportional to S using the previous expression:
A = S (.0585)

Now suppose we find S from the intersection points in Figure 14.11. We could calculate the annual harvest as:
H = .0585*h*S

Taking the three lines in Figure 14.11 as illustrative, we could eyeball the value of S to obtain rough approximations to the annual harvest as follows:
 h S H = .0585*h*S
50% 380,000 11,000
75% 330,000 14,000
90% 220,000 11,500

These values for H, the annual harvest, confirm the two findings from Figure 14.10. That is:

1) the maximum sustainable yield is around 14,000 salmon/yr, and

2) it is possible to obtain yields of over 10,000 salmon/yr with harvest fractions ranging anywhere from 50% to 90%.