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).