Understanding Recruitment
Recruitment
Recruitment refers to the act of young fish transitioning between two stages of life. What is important here is what happens before and after this transition:
In the “before” stage, fish begin life as a fertilized egg that hatches into a larval fish. Mortality for egg and larval stages are considered density-independent (i.e., density-independent mortality). This means that natural mortality does not depend on the density of fish and the survival of an individual fish is not influenced by the number of other fish within the population. It is thought that mortality at this stage is density-independent because environmentally-driven variability in mortality tends to be strongest in this stage.
Eventually, they settle into habitat or aggregrate into schools. This settlement phase is where most scientists think mortality begins to be density-dependent (Walters and Juanes, 1993). During this transitional period (i.e., the recruitment period), the natural mortality does depend on the density of fish (i.e., density-dependent mortality). This means that mortaltiy will decrease (and survival will increase) with decreasing density of fish. This is based on the idea that resources during this period are limited and decreased densities reduce competition for food, space, and refuge (Walters and Juanes 1993). Fish that survive the density-dependent mortality recruitment stage are referred to as recruits.
As fish grow in size through the recruitment period, they become less vulnerable to predators and grow large enough that density-dependent mortality ceases. This is the “after” stage.
In the context of stock assessment, we need to estimate recruitment that reproduce the observed catch and match survey trends and age composition. We also need to forecast recruitment to provide advice on future stock trajectory and catch advice.
Deep dive into origin of stock-recruitment relationship
Stock-recruitment functions describe the production of new recruits to a fish population and the dependence of that production on the spawning component of the population. Stock-recruitment functions pass through the origin and exhibit some form of density dependence.
Beverton and Holt (1957) found that spawning stock is generally a poor predictor of recruitment (i.e., recruitment being independent of abundance) except at relatively low stock sizes. However, this does not mean there is no stock-recruit relationship. There is still a link between spawning biomass and recruits; the number of recruits spawned depends on the mature component of the stock. The average survival rate from eggs to recruitment decreases with increasing spawning stock size. When there is low amount of recruits, the survival rate is greater. This is all as a result of density dependence and competition.
This theory refers to one of the most commonly assumed forms of the stock-recruitment function: the Beverton-Holt (Beverton and Holt 1957). In the Beverton-Holt function, recruitment is a function of spawning biomass \(S\) that increases towards an asymptotic value with increased spawning biomass, and density dependence pertains to coexisting pre-recruits (e.g., through competition).
Another common stock-recruitment function is the Ricker (1954). For the Ricker function, recruitment is an asymmetric dome-shaped function of spawning biomass \(S\). Density dependence pertains to spawning biomass (e.g., through cannibalism, competition, predation).
Derivations
Beverton-Holt
In the original derivations, the units of spawning stock was eggs. The Beverton-Holt is derived as:
\[ R_{t + a_r} = \dfrac{E_t e^{-M_I a_r}}{1 + \dfrac{M_D}{M_I} (1-e^{-M_I a_r}) E_t} \tag{1}\]
- \(a_r\): age of recruitment
- \(E_t\): number of eggs spawned at time t
- \(M_D\): instantaneous density-dependent mortality rate
- \(M_I\): instantaneous density-independent mortality rate
It is more commeon to reparameterize these functions in terms of spawning biomass because the number of eggs produced is usually unknown. This is how it’s done:
\[ E_t = \sum_a f_{t,a} m_{t,a} w_{t,a} N_{t,a} \tag{2}\]
- \(f_{t,a}\): relative feduncity (eggs per unit mass)
- \(m_{t,a}\): maturity at age
- \(w_{t,a}\): weight at age
- \(N_{t,a}\): abundance at age
Relative fecundity is assumed to be invariant to mass or age, so that:
\[ E_t = f \sum_a m_{t,a} w_{t,a} N_{t,a} = f S_t \tag{3}\]
This allows total egg production to be proportional to spawning biomass \(S_t\).
This transform the Beverton-Holt function to:
\[ R_{t+a_r} = \dfrac{\alpha S_t}{1 + \beta S_t} \tag{4}\]
where
\[ \alpha = f e^{-M_I a_r} \tag{5}\]
and
\[ \beta = f \dfrac{M_D}{M_I} (1-e^{-M_I a_r}) = \dfrac{M_D}{M_I}(f-\alpha) \tag{6}\]
\(\alpha\) is proportional to the fraction surviving the pre-recruit stage from density-independent mortality, and it is the rate of recruitment when \(S_t=0\). In other words, \(\alpha\) is the maximum average survival rate absent of density effects. \(\beta\) is a “scaling” parameter that includes density-dependent and -independent mortality components. Note that \(\beta\) is a function of \(\alpha\).
Ricker
For Ricker, the mature stock is expressed in terms of the initial number of eggs laid (Ricker 1954). Thus, the function is:
\[ R_{t+a_r} = E_t e^{-(M_I + M_E E_t)a_r} \tag{7}\]
- \(a_r\): age of recruitment
- \(E_t\): number of eggs spawned at time t
- \(M_I\): instantaneous density-independent mortality rate
- \(M_E\): represent density-dependent mortality that is proportional to the initial number of eggs \(E_t\)
The specification for spawning biomass is the same in the Beverton-Holt and Ricker functions (Eq 2 and 3). The Ricker function of spawning biomass is defined as:
\[ R_{t+a_r} = \alpha S_t e^{-\beta_E S_t} \tag{8}\]
where
\[ \alpha = f e^{-M_I a_r} \tag{9}\]
and
\[ \beta_E = f M_E a_r \tag{10}\]
The \(\alpha\) and \(\beta_E\) terms are independent, unlike in the Beverton-Holt (aside from them being a function of the age at recruitment and relative fecundity).
References
Beverton, R. J. H., & Holt, S. J. (1957). On the dynamics of exploited fish populations. Chapman and Hall, London, Fish and Fisheries Series No. 11, fascimile reprint 1993.
Camp, E., Collins, A. B., Ahrens, R. N., & Lorenzen, K. (2020). Fish population recruitment: what recruitment means and why it matters: FA222, 3/2020. EDIS, 2020(2), 6-6.
Miller, T.J. and Brooks, E.N., 2021. Steepness is a slippery slope. Fish and Fisheries, 22(3), pp 634-645.
Myers, R. A., & Barrowman, N. J. (1996). Is fish recruitment related to spawner abundance? Fishery Bulletin, 94(4), 707–724.
Ricker, W. E. (1975). Computation and interpretation of biological statistics of fish populations. Bulletin of the Fisheries Research Board of Canada, Number 119, 382 p.
Walters, C. J., & Juanes, F. (1993). Recruitment limitation as a consequence of natural selection for use of restricted feeding habitats and predation risk taking by juvenile fishes. Canadian Journal of Fisheries and Aquatic Sciences, 50(10), 2058-2070.