Confirm Reed S==D theorem, evaluate in Sethi context

[cboettig]

Current results are not supporting S==D.

1. Debug deterministic solution
2. Compare in small-noise limit
3. Compare impact of harvesting cost on S vs D (linear cost, no cost, quadratic costs)
4. Compare in Sethi noise cases

[cboettig]

See Reed vs Clark comparison. Not confirming S==D:

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[cboettig]

This occurs because Reed enforces the population to only increase, absent fishing (regardless of the shock). If we replace f(x) with x+f(x), we should recover the Reed result.

[cboettig]

Changing the growth noise to only allow increasing population fixes this, confirming the Reed D==S.
diff

[cboettig]

So the self-sustaining condition requires that min(Z_t) > x/f(x), which sets the smallest permissible multiplicative shock at x == D, the escapement level. Roughly speaking, are large shock sizes are just over this in the uniform noise case, and clearly log-normal noise cannot satisfy this since x/f(x) > 0 everywhere. More discussion in the lab notebook: http://www.carlboettiger.info/2012/07/30/notes.html