Growth Models, Part 5.5

Population Growth Models

Part 5.5: Lobstering Effort

The simplest way to measure lobstering effort would be to use the number of traps as a proxy for effort. However, when large numbers of traps are in use, there is a diminished return per trap -- as well as a probable reduction in the average number of days per year each trap is used. For these reasons, we model the effort as a function of the number of traps, but with a modeling function

E = Effort (Traps)

such that

zero traps yields zero effort (i.e., Effort (0) = 0),
close to 0, effort increases at the same rate as the number of traps (i.e., Effort' (0) = 1), and
because of diminished return per trap, the graph of the function bends downward (i.e., Effort'' (Traps) < 0).

These three conditions assure that the graph of Effort (Traps) starts out tangent to the graph of Effort = Traps, but then bends down and away from that line. There are many such functions -- the one we choose to use is

Here is the graph of this function:

Does this function have all the desired properties? Explain.

When we apply the Effort function to the data on number of traps in use, we get the following scatter plot. (Note that Effort is expressed in arbitrary units.)

What -- if anything -- does this graph tell you about lobstering effort over the period from 1940 to 1976?

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