Introduction

Suppose fishery \(i\) follow the Pella-Tomlinson growth model: \[ \dot{B_i} = \frac{\phi+1}{\phi}g_iB_i\left(1-\left(\frac{B_i}{K_i}\right)^\phi\right) \] Now supose there are \(N\) fisheries. Notice that \(\phi\) is a common parameter across all \(N\) fisheries, but \(g_i\) and \(K_i\) are specific to the fishery. We often want to aggregate fisheries into a single growth model. These notes explain one way to do that.

The concept

First, suppose that each fishery \(i\) is at \(\alpha\) fraction of its carrying capacity, so \(B_i=\alpha K_i\). Then the growth for fishery \(i\) becomes: \[ \dot{B_i} = \frac{\phi+1}{\phi}g_i\alpha K_i\left(1-\alpha^\phi\right) = \frac{\phi+1}{\phi}\alpha\left(1-\alpha^\phi\right)(g_iK_i) \] Now suppose we want the growth rate for the sum of all fisheries. Let \(B=\sum_iB_i\). The growth rate of the aggregate population is: \[ \dot{B} = \frac{\phi+1}{\phi}\alpha\left(1-\alpha^\phi\right)\left(\sum_ig_iK_i\right) \]

Parameters for the Aggregated Fishery

You can see that these take the same form. We need to ensure that: \[ \sum_i g_iK_i = gK \] where \(g\) and \(K\) are the parameters for the aggregated fishery. If we make the assumption that \(K=\sum_i K_i\), then we can pin down \(g\) as follows: \[ g = \frac{\sum_i g_i K_i}{\sum_i K_i} \]

Then the reference points for the aggregated fishery just follow from this:

  1. \(B_{MSY}=\frac{K}{(\phi+1)^{1/\phi}}\)
  2. \(MSY=\frac{gK}{(\phi+1)^{1/\phi}}\)
  3. \(F_{MSY}=g\)

Example

Take a two-fishery example where \(\phi=.188\), \(g_1=.2\), \(g_2=.3\), \(K_1=10\), and \(K_2=40\). Those growth curves are plotted below:

Using the formula above, we get: \(K =\) 50 and \(g =\) 0.28. The aggregated growth curve is shown below in green.