Deviations from S = D

pdg-control

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Exploring causes for deviation from S = D in the Reed model: arises from self-sustaining assumption. Working out the interpretation of the self-sustaining clause, which sounds like it needs to ensure that the stock is non-decreasing in the absence of harvest,

\[ P(X_{t+1} \geq x | X_n = x ) = 1 \]

This sounds like a very awkward condition to enforce for a non-trivial escapement level \( x = S \). Why should the population be guarenteed to increase from it’s escapement population size?

Given the definition

\[ X_{t+1} = Z_t f(X_t) \]

and the condition that we have to have

\[ \operatorname{E}(Z_t f(X_t) ) = f(X_t) \]

that is, $ Z_t $ has to be mean unity, then some shocks must result in

$ X_{t+1} f(X_t) $

for any non-trivial \(Z_t\). So how do we enforce that these decreases do not violate the self-sustaining principle?
It would seem to require at least that \(Z_t\) is a function of \(X\) as well?

Reed seems to imply that this is a much more trivial requirement, such as stating only that \(f(x)\) is compensating density dependence (such as Beverton Holt), and not overcompensating (such as the discrete logistic).

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