The theory of marine reserves

No-take marine reserves (or marine protected areas), in which all forms of catch are prohibited, are gaining increasing attention as conservation and management tools. Rather than provide a comprehensive review, I point you to recent issues of the £w//etzn of Marine Science (66(3), 2000) and Eco/ogica/Applications (13(1) (Supplement), 2003). A summary of these is that there is general agreement that no-take marine reserves are likely to be effective tools for conservation, but it is still not clear if they will enhance fishery catches, either in the short-term or the long-term (Mangel 1998, 2000a, b, c, Botsford et a/. 2001, Lockwood et a/. 2002).

In this section, we analyze a relatively simple model for reserves, because it allows us to use a variety of our tools and to see things in a new way. Other modeling approaches are discussed in Connections.

Figure 6.12. A model for marine reserves involves a habitat that is divided into a reserve zone and a harvest zone. In the harvest zone, a fraction u of the stock is taken by the fishery. Following fishing, the stocks in the two zones merge for reproduction

Figure 6.12. A model for marine reserves involves a habitat that is divided into a reserve zone and a harvest zone. In the harvest zone, a fraction u of the stock is taken by the fishery. Following fishing, the stocks in the two zones merge for reproduction

Envision a stock that grows logistically, again in discrete time, in a known habitat. Rather than fishing in the entire habitat, we set aside a fraction a of it as a reserve in which there is no fishing. We then allow a fraction u of the stock in the non-reserve area to be taken by the fishery (Figure 6.12). If N(t) is the size of the stock at the start of fishing season t, then after fishing the stock size in the reserve is aN(t) and in the fishing region is (1 - a)(1 - u)N(t). Hence the total stock after fishing but before reproduction is aN(t) + (1 - a)(1 - u)N(t) — [1 - u(1 - a)]N(t). With logistic dynamics, we have

+ r[1 - «(1 - a)]N(,)(1 -[1 ' u(1 - a)]N(t)) (6.31)

To begin, as always, we ask about the steady state.

Exercise 6.14 (E)

Show that the steady state of Eq. (6.31) is given by u( 1 — a)

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