Although the Beverton-Holt and Ricker models are most widely used, other density-dependent models have been proposed, such as the Deriso-Schnute, Shepherd, and gamma models. Many are generalizations of the Beverton-Holt or Ricker models or both. The following generalization of the Beverton-Holt model allows for depensation (or Allee effect: a decline in spawning success (see section titled 'Spawning success and density dependence') as S becomes small):
Here, depensation occurs for 7 >1.
In the recruitment models described above, recruitment and spawning success ^ are considered a function purely of S. More detailed models add explanatory factors such as water temperature, water mixing, prey density, or effects of contaminants.
Catch-curve analysis is a heuristic method of estimating Z. Consider a cohort exploited at constant Fa for all ages a > ar When the logarithm of relative abundance at age is graphed against age, the right-hand portion of the graph, representing a > a„ should be a straight line with slope — Z. Due to stochasticity and sampling error, the decline will not be precisely linear, but a straight line can be fitted to estimate Z. A major assumption of this method is that ar is chosen correctly. To make the analysis easier, a 'synthetic cohort' is often used, that is, relative abundance at age in a single year is graphed; this entails the additional strong assumption of constant recruitment. The method has fallen into relative disuse because estimates are quite sensitive to violations of assumptions, and those violations are difficult to detect without additional information.
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