To derive the Beverton-Holt stock-recruitment relationship, let us follow the fate of a cohort of offspring from the time of spawning until they are considered recruits to the population at time T and let us denote the size of the cohort by N(t), so that N(0) = N0 is the initial number of offspring. If survival were density independent, we would write dN/dt = — mNfor which we know the solution at t = T is N(T) = N0e—mT. This is perhaps the simplest form of a stock-recruitment relationship once we specify the connection between S and N0 (e.g. if we set N0 = fS, where f is per-capita egg production, and a = fe—mT, we then conclude R = aS).

We can incorporate density dependent survival by assuming that m = m(N) = mi + m2N for which we then have the dynamics of N

and which needs to be solved with the initial condition N(0) = N0.

Use the method of partial fractions (that is, write \/(m\N + m2N2) = (A/N) + [B/(m1 + m2N)] to solve Eq. (6.3) and show that g—m\T Nq

Now set N0 = fS, make clear identifications of a and b from Eq. (6.2), and interpret them.

Figure 6.3. The Ricker and Beverton-Holt stock-recruitment relationships are similar when stock size is small but their behavior at large stock sizes differs considerably. I have also shown the 1:1 line, corresponding to R = S (and thus a steady state for a semelparous species).

At this point, we can get a sense of how a fishery model might be formulated. Although in most of this chapter we will use discrete time formulations, let us use a continuous time formulation here with the assumptions of (1) a Beverton-Holt stock-recruitment relationship, and (2) a natural mortality rate M and a fishing mortality rate F on spawning stock biomass (we will shortly explore the difference between M and F, but for now simply think of F as mortality that is anthropogenically generated). The dynamics of the stock are dN aN(t - T) / N

This is a nonlinear differential-difference equation (owing to the lag between spawning and recruitment) and in general will be difficult to solve (which we shall not try to do). However, some simple explorations are worthwhile.

The steady state population size satisfies aN/ (b +N)— MN — FN — 0. Show that N — a/(M + F) — b and interpret this result. Also, show that the steady state yield (or catch, or harvest; all will be used interchangeably) from the fishery, defined as fishing mortality times population size will be Y(F) — FN — F(a/(M + F)-b) and sketch this function.

There are other stock-recruitment relationships. For example, one due to John Shepherd (Shepherd 1982) introduces a third parameter, which leads to a single function that can transition between Ricker and Beverton-Holt shapes aS

Here there is a third parameter c; note that I used the parameter b that characterizes density dependence in yet a different manner. I do this intentionally: you will find all sorts of functional relationships between stock and recruitment in the literature, with all kinds of different para-metrizations. Upon encountering a new stock-recruitment relationship (or any other function for that matter), be certain that you fully understand the biological meaning of the parameters. A good starting point is always to begin with the units of the parameters and variables, to make certain that everything matches.

Each of Eqs. (6.1), (6.2), and (6.6) have the property that when S is small R ~ aS, so that when S — 0, R — 0. We say that this corresponds to a closed population, because if spawning stock size is 0, recruitment is 0. All populations are closed on the correct spatial scale (which might be global in the case of a highly pelagic species). However, on smaller spatial scales, populations might be open to immigration and emigration so that R > 0 when S = 0. In the late 1990s, it became fashionable in some quarters of marine ecology to assert that problems of fishery management were the result of the use of models that assume closed populations. Let us think about the difference between a model for a closed population model and a model for an open population:

The equation on the left side is the standard logistic equation, for which dN/dt = 0 when N = 0 or N = K. The equation on the right side is a simple model for an open population that experiences an externally determined recruitment R0 and a natural mortality rate M.

Sketch N(t) vs t for an open population and think about how it compares to the logistic model.

For the open population model, dN/dt is maximum when N is small. Keep this in mind as we proceed through the rest of the chapter; it will not be hard to convince yourself that the assumption of a closed population is more conservative for management than that of an open population.

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