The various relationships between the spawning stock and recruits may be: exploitation K. Limburg, lecture notes, Fisheries Science

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1 S-R II: More about stockrecruitment relationships & exploitation K. Limburg, lecture notes, Fisheries Science Last time, we discussed the following concepts about exploited populations of fish. We identified the stock (S) as the exploited portion of a population, and also as the portion wherein reproduction is generally carried out. By convention, entrants to the stock are called recruits (R), and their entry is called recruitment. The various relationships between the spawning stock and recruits may be: (a) density-independent: no relationship exists between the two (b) density-dependent: - compensatory (per capita recruitment declines with increasing size of the spawning stock) - overcompensatory (really declines with high S, e.g. when cannibalism occurs, or disease outbreaks due to overcrowding) - depensatory (per capita recruitment drops when S goes below a certain level Allee effect) There are two very famous fisheries models that are used to describe the relationships between stocks and recruits. These were both developed in the 1950s and hailed the age of industrial fishing. Both of these models are density-dependent. Neither of them is capable of showing depensation. Both of them caused a great deal of debate! (1) The Beverton-Holt model (from Beverton, R.J.H. and S.J. Holt On the dynamics of exploited fish populations. Fisheries Investment Series 2, Volume 19. U.K. Ministry of Agriculture and Fisheries, London) R α S = β + S Beverton As S gets very large, R α

2 R Recruits, R α S R = β + S Beverton-Holt stock recruitment relationship X β alpha = 1000 beta = Spawning stock, S S (2) The Ricker model (from Ricker, W.E Handbook of Computations for Biological Statistics of Fish Populations. Fisheries Research Board of Canada, Bulletin No. 119) R S = α Se β α is the slope of the curve near 0. As S gets large, e -βs gets very small! By varying the parameters, you can get the Ricker model to look more or less like a Beverton-Holt S-R relationship alpha = 10 beta = R ecruits, R Ricker stock recruitment relationship Spawning stock, S alpha = 10 beta =

3 Alternative models: DeRiso model ( one style fits all ) R = as( 1 bcs) 1 c R = as( 1 bcs) By varying the parameter c, you get different forms of the model: c R = as = Dens. independent c 1 as R = (1 + bs) c = 0 bs R = ase c = 1 R = as(1 bs) = Beverton-Holt 1 c Ricker Logistic Depensatory recruitment: How well do these models fit the real data??? R = as δ S 1+ b δ R ecruits, R Depensatory Bev-Holt model δ = Spawners, S (thousands) Figure from Jennings et al Marine Fisheries Ecology. Blackwell.

4 Figure from Ross Fisheries Conservation & Management. Prentice-Hall. Figure from Ross Fisheries Conservation & Management. Prentice-Hall. Data have been used to fit curves as imaginative as the one in Figure Except for a single, influential, right-hand point in this graph, there is in fact little evidence that recruitment decreases at high stock levels as the curve suggests (p. 161). Figure from King Fisheries Biology, Assessment & Management. Fishing News Books. Figure from Hilborn and Walters Quantitative Fisheries Stock Assessment. Chapman and Hall.

5 One problem is measurement error. This can creep in, for example, when measuring stock size. Results of simulation study Walters (right) and Ludwig (below) showed in a simple model how error in S could affect the S-R relationship Sˆ = S e 2 ν [ ν σ / 2] Error term with mean = 0 and standard deviation σ ν Photos: Univ. of British Columbia Hilborn and Walters 1992 Ransom A. Myers (yet another Canadian fisheries scientist, died in 2006) w/colleagues examined many fish stocks that have been seriously overfished. Myers et al Population dynamics of exploited fish stocks at low population levels. Science 269: Also, Myers published a sensational, but controversial article in Nature about declines in marine predatory fishes These researchers took data from 128 over-fished stocks, all kinds of fish, and fit the depensatory model to it. If δ is over 1, depensation is occurring. Myers et al. (1995) found depensatory dynamics in 3 stocks (2 salmonids, 1 herring) and not in the rest.

6 Their conclusion: most stocks will recover if overfishing is ceased. Unfortunately, this is not always true even some of the supposedly resilient stocks have not come back from serious crashes! Why? Insufficient data? Wrong predictive models? A fairly recent example of depensation. Rougier et al. (2012) Collapse of allis shad, Alosa alosa, in the Gironde system (southwest France): environmental change, fishing mortality, or Allee effect? ICES J. Mar. Sci. 69(10): The allis shad, largest herring in Europe Gironde-Garonne- Dordogne Native range of Allis shad Rougier et al. (2012)

7 Hilborn and Walters issue warnings: Warning 1: do not blindly believe the average behavior predicted by the model A variant on Beverton-Holt, but with depensation (parameter d ) Warning 2: do not ignore the variability in the real data use it to generate estimates of uncertainty (variance) in the predictive models Rougier et al. (2012) Summary: 1. A number of stock-recruitment relationships have been formalized as models 2. The most frequently used ones (even today) are the Beverton-Holt and Ricker models, both density-dependent 3. Alternative models exist (e.g., depensatory) 4. Few really fit the data well! Next time: density dependence and the logistic model