## Salmon are special

Salmon life histories are somewhat different than most fish life histories, and a separate scientific jargon has grown up around salmon life histories (fisheries science has its own jargon that is distinctive from ecology although the same problems are studied, and salmon biology has its own jargon that is somewhat distinctive from the rest of fisheries science). Eggs are laid by adults returning from some time in the ocean in nests, called redds, in freshwater. In general (for all Pacific salmon, but not necessarily for steelhead trout or Atlantic salmon) adults die shortly after spawning and how long an adult stays alive on the spawning ground is itself an interesting question (McPhee and Quinn 1998). Eggs are laid in the fall and offspring emerge the following spring, in stages called aelvin, fry, and parr. Parr spend some numbers of years in freshwater and then, in general, migrate to the ocean before maturation. A Pacific salmon that returns to freshwater for reproduction after one sea winter or less is called a jack; an Atlantic salmon that returns early is called a grilse. Salmon life histories are thus described by the notation X • Y meaning X years in freshwater and Y years in seawater.

When individuals die after spawning, we use dynamics that connect the number of spawners in one generation, S(t), with the number of spawners in the next generation, S(t +1). In the simplest case all individuals from a cohort will return at the same time and using the Ricker stock-recruitment relationship we write

In this case (Figure 6.11) the steady state population size at which S(t + 1) = S(t) satisfies 1 = ae-bS (see Exercise 6.11 below) and the stock that can be harvested for a sustainable fishery is the difference S(t +1) — S(t), keeping the stock size at S(t). Thus, the maximum sustainable yield occurs at the stock size at which the difference S(t + 1) — S(t) is maximized (also shown in Figure 6.11).

Salmon fisheries can be managed in a number of different ways. In a fixed harvest fishery, a constant harvest H, is taken thus allowing S(t) — H fish to "escape" up the river for reproduction. The dynamics are then S(t + 1) = a(S(t)— H)e—b(S(t)—H). In a fixed escapement fishery, a fixed number of fish E is allowed to "escape" the fishery and return to spawn. The harvest is then S(t) — E as long as this is positive

and zero otherwise. With a policy based on a constant harvest fraction, a fraction q of the returning spawners are taken, making the spawning stock (1 - q) and the dynamics become S(t + 1) = a(1 - q) S(t)e-b(1-q)S(t). More details about salmon harvesting can be found in Connections.

Exercise 6.11 (M)

This is a long and multi-part exercise. (a) Show that the steady state of Eq. (6.28) satisfies S = (1/b) log (a). For computations that follow, choose a = 6.9 and b = 0.05. (b) Draw the phase plane showing S(t) (x-axis) vs S(t) (y-axis) and use cob-webbing to obtain a graphical characterization of the data. (If you do not recall cob-webbing from your undergraduate days, see Gotelli (2001)). (c) Next, numerically iterate the dynamics, starting at an initial value of your choice, for 20 years, to demonstrate the dynamic behavior of the system. (d) Show that Eq. (6.28) can be converted to a linear regression of recruits per spawner of the form log [S(t + 1)/S(t)] = log(a)- bS(t) so that a plot of S(t) (x-axis) vs log [S(t + 1)/S(t)] (y-axis) allows one to estimate log(a) from the intercept and b from the slope. (e) My colleague John Williams proposed that Eq. (6.28) could be modified for habitat quality by rewriting it as S(t + 1) = ah(t)S(t) exp (-bS(t)/h(t)), where h(t) denotes the relative habitat, with h(t) = 1 corresponding to maximum habitat in year t. What biological reasoning goes into this equation? What are the alternative arguments? (f) You will now conduct a very simple power analysis (Peterman 1989, 1990a, b) for habitat improvement. Assume that habitat has been reduced to 20% of its original value and that habitat restoration occurs at a rate of 3% per year (so that h(t + 1) = 1.03h(t), until h(t) = 1 is reached). Find the steady state population size if habitat is reduced to 20% of its original value. Starting at this lower population size, increase the habitat by 3% each year (without ever letting it exceed 1) and assume that the population is observed with uncertainty, so that the 95% confidence interval for population size is 0.5S(t) to 1.5S(t). Use this plot to determine how long it will be before you can confidently state that the habitat improvement is having the positive effect of increasing the population size of the stock. Interpret your result. See Korman and Higgins (1997) and Ham and Pearsons (2000) for applications similar to these ideas.

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