From Beverton and Holt and Back Again. ICES CM 2011/D:02

Not to be cited without Prior Reference to the Author ICES ASC Gdansk 2011 ICES CM 2011/D:02

Movement Modeling in Stock Assessment: From Beverton and Holt and Back Again

Daniel R. Goethel Steven X. Cadrin

In the computer age of modern fisheries science, it is difficult to imagine working on today‟s models by hand. This is especially true in stock assessment where a laptop computer has become the only tool most assessment scientists ever use. However, Beverton and Holt developed the foundations for much of fisheries science in their 1957 „fisheries bible,‟ using little more than slide rules and adding machines. As models grow in complexity, it becomes easy to forget their origins and basic assumptions. On the forefront of assessment science is the use of tag-integrated models, which allow fish movement between sub-populations by including tagging data within the objective function. Although current tag-integrated models are more complex than could ever be imagined 50 years ago, it is important to remember that the basis for these models were first developed by Beverton and Holt in their seminal work. A review of Beverton and Holt‟s original movement models shows that the assumptions are essential for the successful application of their models. Although these models were largely forgotten in stock assessment, advancements in computation power, data collection techniques, and cooperative research have led to a resurgence of movement models and the successful application of tag- integrated models. In today‟s world of instant computing it is often easy to take for granted what one‟s predecessors have accomplished, thereby undervaluing their work but simultaneously overlooking important lessons learned along the way.

Keywords: Beverton and Holt, movement models, tagging models, tag-integrated models

Contact Author: Daniel Goethel UMASS-Dartmouth School for Marine Science and Technology 200 Mill Rd., Suite 325 Fairhaven, MA 02719 [email protected]

1. Introduction

In his foreword to the 1993 reprint of Beverton and Holt‟s On the Dynamics of Exploited Fish Populations (1957), Daniel Pauly (1993) muses: “I wonder what example will be used for illustrating Beverton and Holt‟s anticipation of ideas when, in a few years or decades, another reprint…is presented to a new generation of fishery scientists?” Although the possibilities are almost infinite, it appears that over the last two decades the honor should be given to Beverton and Holt‟s views on incorporating the effects of fish movement on population dynamics. Section 10 of the „fisheries bible‟, titled „Spatial Variation in the Values of Parameters: Movement of Fish within the Exploited Area,‟ takes up less than 30 of the over 500 pages in Beverton and Holt‟s (1957) book. However, the work presented in this section has been the basis for the recent development of spatially explicit stock assessment models. The most advanced of which, termed tag-integrated models due to the ability to incorporate a tagging component directly in the objective function, represent one of the biggest advancements in stock assessment since the development of statistical catch-at-age (SCAA) models. Even though these models are much more complex than even Beverton and Holt envisioned, the basic tenets and simplifying assumptions used in these models all can be traced back to their 1957 work (Schwarz, 2005). A review of their ideas on incorporating movement along with a brief historical trace of how these simple models have developed into today‟s complex statistical stock assessment models is presented. Additionally, a short discussion of the data, computational, and research advancements that have led to the ability to build and successfully apply tag-integrated models is provided. Many people agree with Isaac Newton‟s quote that scientific progress is only possible by „standing on the shoulders of giants‟ (i.e., our scientific predecessors). While this claim is certainly true, the rate of progress and data assimilation in modern science appears to have developed a tendency to move forward without completely understanding where current research originated. This often results in a lack of understanding and an inability to properly analyze the intricacies of modern scientific models, while simultaneously devaluing the work of one‟s predecessors. In the case of fisheries science and especially stock assessment understanding the past necessitates going back to Beverton and Holt once again.

2. History

2.1 Early Migration Research

Although fishery scientists have understood that most marine fish species are capable of long range movements, the extent and details of many fish migrations remain unknown. The temporal changes in fish abundance due to migrations have intrigued scientists since the days when the field of ecology was first emerging. Johann Anderson (1746) presented the idea of panmictic stocks that underwent large-scale migrations. The main thesis was that herring would partake in migrations from their „home‟ under the polar ice cap in search of food when the population outgrew the available prey sources, and would arrive at the various worldwide fishing grounds at different periods during this migration (Wegner, 1996; Chambers and Trippel, 1997; Sinclair 2009). The „migration‟ theory remained a prominent viewpoint well into the early stages of the 20th century when work by Heincke (1898) with herring and by Hjort (1914) with cod

2 demonstrated that different spawning „races‟ or populations existed within a given species, which underwent much shorter spawning and feeding migrations than proposed by Anderson (1746). This research undermined and refuted the „migration‟ theory and ushered in the slow transition to „population thinking.‟ This theory claimed that fishery fluctuations were caused by year-class variability within geographically distinct populations and not long scale migrations of the entire species (Chambers and Trippel, 1997; Secor, 2002; Sinclair, 2009). The importance and baffling nature of fish movements was reflected during the inaugural meeting of the International Council for the Exploration of the Seas (ICES) in 1902. During this initial meeting one of first three committees established was to investigate the “Migrations of the Principal Food-fishes of the North Sea” (Anderson, 2002). Despite the committee focusing primarily on recruitment variability, the cause and effect of migrations remained an important research priority for many fisheries institutions worldwide.

2.2 The Initial Stages of Fisheries Modeling

In the 1920s ecology slowly began to turn towards mathematics in order develop modeling tools to predict and understand nature in much the same way as physics had done to understand the physical world. The transition was slow and many ecologists felt, as Russian scientist Nikolai Knipowitsch stated, that “it is completely unacceptable…as a biologist, to reach a conclusion on the basis of formulae” (Smith, 1994). However, the trend towards theoretical modeling in biology had begun and could not be reversed. In fisheries, the stage was set by F. I. Baranov (1918) with the publication of his catch equation, although it was not seen by the western world until decades later. Russell (1931) built on Baranov‟s theory resulting in his energy balance equation regarding fish population growth. Thompson and Bell (1934) further investigated the potential fishery yield from a given stock and calculated what the expected yield should be based on different combinations of natural and fishing mortality. This was followed by von Bertalanffy‟s (1938) growth equation, Michael Graham‟s (1939) introduction of maximum sustainable yield (MSY), and Ricker‟s (1944) work on instantaneous mortality rates. Finally, a short publication in Nature by Henry Hulme, Raymond Beverton and Sidney Holt (Hulme et al., 1947) synthesized much of the previous work into a single yield equation. At this point the stage was set for the biggest breakthrough, and what many consider the foundation of, theoretical fisheries science in the form of Beverton and Holt‟s (1957) On the Dynamics of Exploited Fish Populations (Hilborn, 1994; Anderson, 2002; Angelini and Moloney, 2007) [for a full review of the history of modeling in fisheries see: Smith, 1994 or Angelini and Moloney, 2007]

2.3 The Fisheries ‘Bible’

Mathematical modeling was not a well accepted approach by many in the contentious field of fisheries science. In fact, one of the pioneers of population dynamics modeling, F. I. Baranov, was attacked by the Russian fisheries community following the publication of his catch equation. This resulted in his being ostracized by the scientific community with some going as far as to call him a „saboteur‟ (Ben-Yami, 2010). As is the case with many famed scientists--such as Galileo--whose ideas predate the public and scientific community‟s ability to comprehend and accept them, history has exonerated and proven Baranov a visionary. Luckily, in 1957 when Beverton and Holt published their seminal work, the trend of incorporating mathematics into

3 biology had become a standard practice, albeit one that was accepted begrudgingly by many biologists. This was portrayed by Cren‟s (1959) critique: “Whether we like it or not, however, this book is another demonstration that it is becoming increasingly difficult to make fundamental contributions to ecology without invoking the aid of at least some mathematics.” Instead of facing alienation by the scientific community, Beverton and Holt, who open the book by stating “we make no apology for the fact that much of what is to follow is mathematical in nature,” have been exalted as pioneers and founders of quantitative fisheries science (Pitcher and Pauly, 1998). The original publication run took over three years to typeset and required the head of the Lowesoft Fisheries Laboratory, Michael Graham, threatening to retire in order to convince the United Kingdom Stationary Office to print it (Anderson, 2002; Holt, 2008). Initially, 1,500 copies were printed and sold for ₤6 6s (~US $17.64; Cren, 1959; Rounsefell, 1959; Anderson, 2002). Critiques at the time of first publication were generally positive, but weary of the “highly theoretical and purely mathematical approach” (Rounsefell, 1959). However, most reviewers agreed that the book was a “veritable goldmine of information for biometricians interested in population dynamics” (Rounsefell, 1959) and that “all fishery biologists should read and if possible own this book” (Cren, 1959). One even went on to describe it as “the bible of fishery biologists the world over” (Cren, 1959), a name that has stuck for over half a century. This classic text was reprinted in 1993 and sold for ₤50 (the 2004 reprint can now be purchased for $80), while garnering much of the same praise as the original release (Hilborn, 1994). Over the last 50 years Beverton and Holt‟s book has earned its title as the „fisheries bible‟ and many would agree with Hilborn‟s (1994) claim that it “probably constitutes the single most important contribution to fisheries science yet published.” The „fisheries bible‟ covers an amazing breadth of topics. Originally much of the focus was on the catch equation, yield-per-recruit analysis, stock-recruit functions, and other „fundamentals of the theory of fishing‟. However, they also carried out numerous in depth explorations into possible extensions of the simple methods. Many scientists in the field have found themselves thinking they were “breaking new ground” only to be “making minor additions to a well-ploughed field” tilled over decades ago by Beverton and Holt (Hilborn, 1994). Others like Steve Murawski had his “thesis published and a couple papers out, and then I started flipping through the last chapter of Ray‟s book (Beverton and Holt, 1957)…and all the good stuff that I thought I had invented [was already published] and [I] certainly had to take [it] back in subsequent papers” (Anderson, 2002). Such mistakes are common because few scientists can believe that using slide rules and rotary calculators (or most importantly their own brains and long-hand calculus), Beverton and Holt were able to set out the theoretical principles of complex bioeconomic and movement models (Anderson, 2002; Holt, 2008). In the current age of laptops, the thought that Beverton and Holt would sometimes spend „several months‟ calculating yield curves is indeed a „heroic effort‟ (Cushing and Edwards, 1996). However, because “whenever we undertake a yield-per-recruit or virtual population analysis we are building on the sure foundations of Beverton and Holt” (Ramster, 1996), it is essential that this original basis of current complex models is fully understood. By reviewing the foundations of these models, it allows a better understanding of how they work and the possible pitfalls that one‟s predecessors may have encountered. At the very least it pays tribute to those that did the „legwork,‟ which is so easy to take for granted in the modern computer age.

3. Movement Modeling in Population Dynamics

3.1 Application in Terrestrial Biology

Although Beverton and Holt (1957) were the first to provide a detailed account of incorporating movement into the population dynamics of fish species, Skellam (1951) was the first to apply the principles of random diffusion to the movement of animal populations. The main principal of his work was that the movement of animals was generally random and could be treated in the same way as the physical diffusion of gases. Skellam (1951) demonstrated that the spread of muskrat (Ondatra zibethicus) across mainland Europe from a single farm in Prague could be modeled using a 2-D random diffusion model. However, the properties of random diffusion did not necessarily hold for every movement, but instead depended on what temporal and spatial scales one looked at the movement of the population. When looking at the spread of the muskrat over short temporal and spatial scales the movement was not necessarily random, but when looking at the same data over a distance of hundreds of miles and multiple years the random diffusion analogy held up well. Skellam (1951) covered a multitude of dispersal topics showing how movement could be combined with a logistic population growth function. Additionally, the model was generalized into a discrete time model with application to the dispersal of seeds. Even though discrete time models are now the common temporal currency used for modern stock assessment models, this was not examined by Beverton and Holt (1957). The work of Skellam (1951) was unique not only because it developed a way to incorporate movement into population models, but because it linked the actions of individuals (movement) with the expansion (growth) of a population over both time and space (Toft and Mangel, 1991).

3.2 Beverton and Holt’s Movement Models

Building on the animal dispersion tenets developed by Skellam (1951), Beverton and Holt (1957) hypothesized that fish movements could be related to the random diffusion of gases in terms of their general foraging strategy: “…we can imagine a bottom-living fish such as a plaice moving in a certain direction until it encounters a patch of food organisms, and after spending some time more or less stationary while feeding on these continuing the search for food in a direction which is random with respect to that in which it first approached the patch of food organisms. In this case the „inter-patch‟ movement would be analogous to the „mean-free-path‟ in the kinetic theory of gases and would be the level at which the random direction component is introduced.” (Beverton and Holt, 1957; p. 137) The approach was applied to a demersal species, North Sea plaice (Pleuronectes platessa), for which movement of the population as a whole involves local and random dispersion of individual fish. Thus, the localized movement of individual fish is responsible for the continual mixing of adjacent populations. The general distribution of the population is assumed to be non- uniform, with individuals found throughout the population‟s range where habitat and environment are regarded as homogeneous. As described by Skellam (1951), individual movement is not assumed to be random, but the frequency of individual movements determines the spatial and temporal scale at which movement can be considered to be randomly diffusive.

For example, North Sea plaice were shown to disperse at random on a yearly timescale and a distance of about 80 miles.

3.2.1 The Dispersion Model

Beverton and Holt developed two different movement model formulations. The first followed Skellam‟s (1951) framework closely and was used for calculating the change in concentration from a given point (as is commonly used in computing the rate of dispersion of a tagging cohort released at a single location). This was termed the „dispersion‟ model. The change in concentration (C) per unit of time (t) of a group of fish within a small two-dimensional (x and y coordinates) area can then be described by: C D 2C 2C t 4 x 2 y 2

V 2 D Vd nd2 n Equation 1 in which D is the dispersion coefficient, V is the velocity of movement which takes into account the time interval between random movements, n is the number of random movements per unit time, and d is the length of the mean free path. In theory, the dispersion coefficient should be dependent on the concentration, since a decreased concentration will increase the mean free path, thereby increasing the dispersion factor. In reality, it is believed that the mean free path is actually affected by external factors such as predators or prey and mainly independent of population sizes except at extremely high concentrations. The dispersion model was successfully applied to indices of abundance for ages 1 to 5+ of North Sea Plaice taken from a line of stations stretching from inshore nursery grounds along the Dutch coast to the main fishing grounds from 1921 to 1928. Equation 1 was subsequently applied (assuming only unidirectional movement offshore) to the data in order to determine if the movement was indeed due to random diffusion and to see if the model could recreate the observed population distributions. The results showed that D was relatively constant for all ages and had a value of 552 miles2/year.

3.2.2 The Box-Transfer Model

Although the dispersion model was shown to be a reasonable approach to determine movement rates and the effects of dispersal on population densities, it requires many accurate observations of the position and time of fish captures. Data obtained from fisheries are often not reliable enough to fit such a model. Similarly, it is usually more important to focus on the general distribution of fish within the entire exploited region, than with the density at a given point. In order to do so, it is important to include varying fishing mortalities and movement rates at different points within the exploited population. Beverton and Holt (1957) used an approximation to describe the change in the number of fish due to movement across the boundary of a given region:

dN N . dt Equation 2 In this equation, the dispersion coefficient has been replaced by a transport coefficient (τ), which describes the instantaneous rate at which the abundance of fish in a region changes due to diffusion across the zone‟s boundary. The purpose of this simplification is that it retains many of the basic tenets of random diffusion, but is better suited for fish movement and the type of data that is usually available. It is important to note that the transport coefficient is the finite difference equivalent to the dispersion coefficient with respect to distance. Transport coefficients are often assumed to be the same within regions, but different between regions. All regions are usually assumed to be of equal size and shape, so that transport coefficients can be compared. For the case of a two region model, the change in abundance in each region due to transport between regions, fishing mortality and natural mortality (Beverton and Holt, 1957) is:

dN1 (F1 M1 1 )N1 2 N 2 dt dN 2 (F M )N N dt 2 2 2 2 1 1 Equation 3 where the subscript refers to regions 1 and 2, F is fishing mortality, M is natural mortality and N is population size. Although not explicitly developed by Beverton and Holt (1957), the finite difference form of the transport coefficient model with respect to time is often used in stock assessment models. This transfer coefficient or „box-transfer‟ model is given by:

(F1, y M1, y ) (F2, y M 2, y ) N1,y 1 T1 N1,y e T2 N 2,y e

(F2, y M 2, y ) (F1, y M1, y ) N 2,y 1 T2 N 2,y e T1 N1, y e

Equation 4 in which T1 is the transfer coefficient representing the proportion of fish that move from area 1 to the other area. The relation between the transport coefficient (τ) and the transfer coefficient (T) is: T e . Equation 5 When converted to a difference equation format (i.e., from continuous time to a discrete time format) the box-transfer model is essentially a discrete Markovian movement model. Because this type of diffusive movement is assumed to be a random process, it is a memoryless system [i.e., future states (regions) depend only upon current states and are completely independent of past states]. In addition, fish have a probability of transition or movement between regions. Thus, Beverton and Holt‟s (1957) box-transfer movement model is a simple Markov chain. This model provides a basic theoretical construct from which estimates of movement can be made using mark-recapture data and demonstrates how movement can be accounted for directly in a population dynamics or stock assessment framework. Although both aspects would

7 eventually receive attention, initially the aspect of Beverton and Holt‟s work that gained the most immediate recognition was how movement rates could be estimated from tagging studies using the same general equations listed above. Including movement in a stock assessment model requires movement estimates from associated tagging models. Therefore, tagging studies had to be refined so that more accurate estimates of movement could be made before movement could be included in stock assessments.

3.2.3 Assumptions

In order to fully understand how a model works and under what circumstances it can be effectively applied, it is necessary to be aware of any assumptions that are made during its development. This is an issue that is often overlooked in complex modern models, especially those that are developed over a long period of time by multiple users. It is difficult enough to keep track of all of the assumptions in a given tagging or stock assessment model where the general framework is usually developed by a predecessor long ago. It becomes even more difficult when additional complexity is incorporated, such as adding movement between geographic zones. For this reason it is a good idea to revisit the main assumptions that Beverton and Holt (1957) made in the development of both the dispersion and box-transfer model. The main assumption of both models traces back to the mathematical analogy used to define fish movement--that is, random diffusion. The main factor here is that the movement of fish must indeed be random. As mentioned, not every movement of every fish must be random, but on „average‟ over the given temporal and spatial scales of interest the movements must be random. This does not preclude fish that exhibit directed migrations, but in such instances it would be necessary to incorporate a directional (advection) component to explicitly account for this factor. When using the pure diffusion models it should thus be explored whether the species of interest is characteristic of plaice, in which: the general distribution is non-uniform, individuals are typically found throughout the range at any given moment in time, and fish typically do not move as a unit. This can be compared to a species such as herring where there is a strong tendency to aggregate and movement is often as a unit. In the latter case movement is not the result of local dispersion of individual fish, and thus cannot accurately be modeled with this type of movement model. Additionally, in order for the principles of diffusion to hold, the timescale of the random movements should be small in comparison to the temporal unit by which changes in the general distribution of the population are measured. For the dispersion model, the dispersion coefficient is assumed to be a function of velocity of the fish and the mean free path. One of the difficulties with the analogy used to apply the laws of diffusion is that in physics the mean free path is determined by the concentration of gas particles, whereas in the application to fisheries it is ill-defined. In the example given by Beverton and Holt (1957), the mean free path is a function of prey density, but it is suggested that it could be a function of many factors such as interaction with predators or not specifically related to any particular factor at all. It was thus suggested that the mean free path was independent of velocity, which implicitly assumes that the dispersion coefficient was solely dependent on the velocity of movement. These assumptions were the only ones made by Beverton and Holt (1957) for the continuous time dispersion model. However, for many Advection-Diffusion-Reaction (ADR) models that are based largely on the dispersion coefficient framework, a number of additional assumptions often exist. In theory, the ADR approach does not suffer from scale dependencies

8 that are inherent in the box-transfer model, but in reality similar assumptions are needed to make ADR models tractable. As in a box-transfer model, ADR models require a finite difference approximation that bins data by spatial area and time period. Oftentimes, in order to make the model run in a timely fashion on modern computers, the spatial grids become quite coarse. Similarly, the timestep usually follows suit so that enough observed tag recoveries are available per region to estimate parameters. Overall, many applied ADR models actually suffer from the same simplifying assumptions as box-transfer models (Goethel et al., 2011). Beverton and Holt (1957) admit that the application of the box-transfer model is a departure from the somewhat rigorous analogy of diffusion adhered to in the dispersion framework. However, the dispersion model does not fit for many fisheries applications where the goal is to assess the general distribution and movement of fish relative to effort. The main difference between the two frameworks, as mentioned previously, is that the transport coefficient is the finite difference equivalent to the dispersion coefficient with respect to distance. One important consequence of this is that τ is a function of movement, but also of the size and shape of the region that it corresponds to, while D is only a function of movement (velocity). Due to this factor Beverton and Holt (1957; p.141) suggested that the population be divided into regions that are “as small as is consistent with the accuracy of commercial statistics of catch and effort, since the smaller the size the more faithful will be the representation obtained.” Another important approximation is that the density of fish at the boundary must be in a nearly constant ratio to the average density throughout the region. The rate of movement across a boundary is assumed to be proportional to the average density within the region, implying immediate mixing once a fish crosses the boundary, which corresponds to infinite velocity. Actually, it is only the fish located near the boundary that will affect the rate of movement between regions. Although the box-transfer model may be a gross simplification of the dispersion model, the greater utility of the box-transfer concept can be appreciated in the context of the even simpler „unit stock‟ concept (Cadrin and Secor, 2009). Most conventional stock assessment models assume no movement across stock boundaries, so allowing for transfer across boundaries is a meaningful step towards realism and understanding of population dynamics processes that can be supported with existing information. Despite the assumptions inherent in the box-transfer framework, it actually can be considered analogous to how Newton‟s Law of Cooling is a simplification of true random diffusion (Beverton and Holt, 1957). Basically, Newton‟s Law of Cooling is to random diffusion as the box-transfer model is to the dispersion framework, mainly it is discretized in regards to distance (Newton‟s Law assumes no temperature gradient occurs within the radiating body and the box-transfer model assumes no gradient in fish density throughout a geographic zone). Although these assumptions are impossible to meet completely, especially that of instantaneous movement, the transport coefficient model provides a framework that can often usefully approximate the dynamics of a given set of regions (Hilborn, 1990; Porch, 1995).

3.2.4 Extensions to the ‘Simple’ Movement Models

The dispersion and box-transfer models provide the basic tenets for including movement into fisheries population dynamics models. These two model formulations have been the basis for much of the work that has been accomplished in terms of the development of multi-area or spatially explicit tagging and stock assessment models. However, it is not surprising, given how

9 thorough and visionary Beverton and Holt (1957) were, that numerous extensions and associated theoretical explorations were provided. The first extension involved how the dispersion rate could vary with food abundance. This extension was actually a more thorough investigation into the basic assumption that was used to apply random diffusion to fish—mainly that movement is based on the search for food, and thus the dispersion coefficient might be expected to change based on prey abundance. Most of the work in this section involved calculating the dispersion coefficient based on the size and distance between food patches. The main outcome of this exploration was that an increase in abundance of prey items would have a tendency to decrease the rate of dispersion by decreasing the mean free path. This line of reasoning has been examined to some extent in current ADR models by setting the advection [i.e., directional component; as compared to the simple diffusion (i.e., random) component] proportional to a habitat preference index—usually based on either abiotic (e.g., sea surface temperature) factors or prey density (Bertignac et al., 1998; Faugeras and Maury, 2005). However, Porch (2004) developed a similar model within a diffusion framework assuming that the transport coefficient between habitat patches was proportional to the distance between patches and the „intrinsic‟ attraction to the patch. The final addition that was explored was to look into the effect that directed migration played on dispersion rates. The principal idea was that in many instances dispersion was not purely random, but also contained a directed term. This factor has since come to be termed the advection component of movement. Beverton and Holt (1957) developed this extension through the assumption that movement contained a directed component based on dominant currents so that the transport coefficient could be broken down into upstream and downstream constituents. The main hypothesis maintained that most spawning migrations are counternatant, which is likely due to the mean free path being longer in the upstream direction. Using circular statistics they were able to estimate what the various rates of advection might be, but overall the ideas presented were mostly qualitative. This work was first extended by Jones (1959) in his analysis of tag returns for haddock (Melanogrammus aeglefinus) off the Scottish coast using the assumption that there was a “…superimposed directional component of the movement such that the centre of density of the whole fish group actually moves with some velocity…” It has since become the basis for ADR models, which assume that movement is based on both a passive diffusion and active advection (i.e., directed) component (e.g., Sibert and Fournier, 1994; Sibert et al., 1999).

3.3 Advancements Leading to Successful Application of Beverton and Holt’s Ideas

Beverton and Holt (1957) end the section on dispersal by saying that care should be taken when applying their movement models due to the approximate nature of the applications, while admitting that it is unlikely that these models would be used in the near future due to the problem “that neither knowledge of the mechanisms of dispersion nor accuracy of data and commercial statistics is sufficient to justify the labour involved in a rigourous treatment…” (p. 141). However, Jones (1959) was able to use the dispersion model to look at the movement of released haddock tags and estimate the rate of advection and diffusion of the cohort from the initial release point. The box-transfer framework was later applied to model observed and predicted tag releases and estimate the rate of movement between geographic zones of tagged fish (Ishii, 1979; Sibert, 1984; Hilborn, 1990). However, a number of barriers existed to applying these models in

10 a stock assessment framework, including lack of: reliable tagging and catch statistics, complex statistical models, and computing power. Most of these issues were interrelated in that a development within one category required a similar development in another (e.g., development of complex statistical models required increased computing capacity in order to run the now more complex assessment models, etc…). Beverton and Holt (1957) were correct with their initial statement, though. At the time of the development of these models, the temporal and spatial accuracy of tag recoveries from commercial catch statistics was not reliable enough to develop an accurate movement model. In many regions worldwide, it could also be argued that the scientific outreach with fishermen was lacking. Until recently, the distrust in many fisheries communities was so high that attempting such models would have been impossible. With recent technological developments [e.g., Global Positioning Systems (GPS)] and the general increase in cooperative research worldwide, these issues have generally become a thing of the past. In addition, the development of acoustic, satellite, and telemetry tags have allowed scientists to learn exact fish migration pathways. These advancements have thereby settled Beverton and Holt‟s other fear that little was known about „the mechanisms of dispersal.‟ The individual-based information on fish movement that is currently being collected from deployment of electronic tags is expanding our understanding of the motivations and mechanisms of fish dispersal (e.g., Nielsen and Arrizabalaga, 2009). In addition to the movement rates needed to estimate transfer coefficients, information on habitat use, movement behaviors and patterns will eventually help to modify simple movement models. For example, North Sea plaice, the example used by Beverton and Holt, have been observed to exhibit complex off-bottom movements and selective tidal drift (e.g., Hunter et al., 2004). As the field of stock assessment grew out of Beverton and Holt‟s (1957) treatise, so did the development (for fisheries scientists) of complex statistical models. The complimentary advances in computers and computing power led to stock assessment coming of age in the 1970s and 1980s. As all these factors came together, assessment models were built up step by step. For this reason, stock assessment slowly advanced from virtual population analysis, which recreated abundance from only observed catch-at-age data, to modern statistical catch-at-age (SCAA) models, that recreate abundance using multiple data sources (e.g., catch-at-age, survey abundance, stock-recruit data, etc…) and complex statistical methods. Although assessment scientists were aware that fish moved and that „closed‟ population assumptions were inherently violated by most species, the step by step nature of model development (and technological advancement) precluded a jump to complex models involving multiple areas and movement. By the late 1990s, all three of these factors finally came together to allow the development of the movement models first outlined by Beverton and Holt. Tagging datasets were available with reliable observations of position and time of tag release and recapture, complex statistical stock assessment models had been fully developed with the ability to incorporate numerous data sources (in the form of „integrated‟ SCAA models), and computer power had reached a stage where estimating hundreds of assessment parameters simultaneously was now a tractable problem.

3.4 Incorporating Movement in Stock Assessment

Much of the initial work with movement modeling occurred with tagging models, which estimated movement rates between geographic zones by matching observed and predicted tag

11 recaptures by region within the objective function. Ishii (1979) investigated the movement of tagged yellowfin tuna in coastal Mexico, and Sibert (1984) used a similar approach to model the transfer of tagged skipjack tuna in the South Pacific. Hilborn (1990) provided a generalized tagging model based on Beverton and Holt‟s (1957) original work that calculated predicted tag returns for multiple regions using a maximum likelihood formulation to estimate model parameters such as transfer and harvest rates. It was not until estimates of movement rates between stock areas were available from tagging studies that stock assessment models could begin incorporating movement (at least until computing power increased to the point that tagging and assessment models could be combined into a single framework). However, Aldenberg (1975) investigated the effect of movement on a hypothetical species containing two stocks within a VPA framework. The work was limited due to a lack of an analytical solution to the 2 stock VPA problem, but simulation experiments showed that bias in recruitment estimates occurred when movement was ignored (i.e., when the closed population assumption of a single stock VPA was used when movement was actually occurring between stocks). Migratory catch-age analysis was developed by Quinn et al. (1990) and allowed for movement of Pacific halibut between multiple stock areas. In this case, estimates of movement rates were input directly from a tagging model, and results indicated that statistical bias was smaller when movement was accounted for compared to when it was ignored. A number of studies used equation 4 to investigate the effect of movement of bluefin tuna between the eastern and western Atlantic stocks again using estimates of movement from tagging studies (Butterworth and Punt, 1994; National Research Council, 1994). The lack of ability to analyze tagging data directly by these early spatially explicit stock assessment models led to a number of problems, including: loss of information, inconsistencies between modeling assumptions, difficulty in determining error structure, difficulty in including uncertainty, and reduced diagnostic ability (Maunder, 2001).

3.4.1 Tag-Integrated Models

In the late 1990s, a new form of assessment model developed from integrated SCAA models, which were termed „tag-integrated‟ models due to their ability to analyze tagging data directly within the assessment (Goethel et al., 2011). The main feature of these models was that the tagged population and untagged population were modeled simultaneously within the assessment and the single combined likelihood function contained information from both components of the population. Mainly, tagging models [such as Hilborn‟s (1990) model] were run as a sub-model within the assessment where the objective function comparing observed and predicted recaptures was now a single component of a larger objective function that also contained components comparing observed and predicted catch and abundance indices from the main assessment model. The statistical model would then estimate the parameters of the assessment model and the tagging sub-model simultaneously, while sharing parameters such as movement rates and fishing mortality between the two populations (i.e., tagged and untagged). As mentioned, it was a multi-pronged breakthrough in both statistical models and increased computing power that allowed such models to be developed. Maunder (1998) created the first „tag-integrated‟ model with his integrated tagging and catch-at-age analysis (ITCAAN) model. This was applied to snapper (Pagrus auratus) in New Zealand with multiple stocks, but was extremely flexible and easily adapted to various species.

Work with bluefin tuna (Thunnus thynnus) led to the development of the VPA 2-box model by Porch et al. (2001) and later to the multi-stock age-structured tag-integrated stock assessment model (MAST; Taylor et al., 2009). The latter is currently one of the most advanced stock assessment models. It contains four geographic zones and incorporates multiple tagging datasets and tag types (i.e., conventional tags, archival tags, and satellite tags). Multifan-CL is a length- based tag-integrated model developed for western Pacific yellowfin tuna incorporating 7 spatial units (Hampton and Fournier, 2001). Goethel et al. (2011) provide a detailed description of the historical development of movement modeling in stock assessment beginning with its origins in Beverton and Holt‟s (1957) basic models up to modern tag-integrated models, and provide a generalized „tag-integrated‟ modeling framework that is adaptable to almost any data and biological situation for marine fish species.

4. Discussion

Deep inside the densely packed pages of Beverton and Holt‟s (1957) „fisheries bible‟ lies a rather short and often overlooked section on incorporating movement into population dynamics equations. Although often ignored, this work has become the basis for many of today‟s most advanced stock assessment models. It is understandable how, at the time of its publication, this work was generally glossed over and forgotten. Yet, due to the recent development of „tag- integrated‟ stock assessment models based on the tenets established by Beverton and Holt (1957), it is important to understand the assumptions behind the original movement models in order to learn exactly when and how they can be accurately applied. At the same time, it is important to understand the origins of these models and why they were developed. The main assumption of Beverton and Holt‟s (1957) original movement models was that fish could be viewed in much the same way as a gas particle. Hence, dispersion of fish was analogous to the random diffusion of gases and the same mathematical treatment used for both. For many species this analogy appears to hold quite well. However, it is important when applying a movement model based on this assumption, to make sure it is applicable for the species of interest. Beverton and Holt (1957) warned that many species do not fit this assumption due to behavioral tendencies such as schooling or long distance directed migrations to spawning areas. Many alternate assumptions regarding movement have been developed, including habitat attraction models (Porch, 2004) and natal homing models (Porch et al., 2001), in order to deal with behaviors where random diffusion does not fit the observed movement tendencies. When attempting to develop movement models, such as tag-integrated stock assessments, it is important to check the validity of all assumptions. It is easy to develop a model of this type using a movement coefficient based on random diffusion, since this is how other models have been developed, without checking the biological realism of this assumption. In reality, only a few demersal fish, such as some of the flounder species, likely adhere to this assumption. A prominent problem in modern stock assessment models is that of „historical inertia.‟ Mainly, that stock assessment scientists, often due to data limitations and time constraints, have a tendency to simply follow and use the assumptions set out by those working on similar models before them without considering the theoretical basis, assumptions or implications. A good example of this is the assumption that the instantaneous natural mortality rate (M) is equal to .2. This rate has been used for innumerable species for decades. Many scientists have become accustomed to using it, which has resulted in the „M=0.2‟ assumption gaining an inertia that has

13 become difficult to break. Even for species where it initially may have been accurate, it is doubtful that this is still the case since environmental changes have likely altered rates of natural mortality. It can be argued that assumptions of this type are sometimes necessary due to the inability to estimate all parameters and the lack of available data for many species. However, making such false assumptions causes a severe underestimation of uncertainty (Hilborn and Liermann, 1998) and goes against the attitude of the pioneers of fisheries science whom sought to challenge all theories that were treated as dogmatic (a good example of this are those in the late 1800s that fought against the theory that fisheries were inexhaustible). One of the main sources of „historical inertia‟ plaguing stock assessment today is the institutional reluctance to adopt new assessment techniques. The idea that assessment models which assume closed populations (e.g., single stock VPAs) can be accurately applied should be severely questioned. Although these models have a strongly entrenched history and sometimes provide adequate results for fisheries management, it has to be wondered if assuming that fish do not move between stocks (when studies in many fisheries demonstrate that they clearly do show extensive movement) inherently creates an underestimation of uncertainty similar to fixing M equal to 0.2. Both Beverton (Anderson, 2002) and Holt (2004) themselves question the reliability of VPAs, which makes one wonder why it is still the „default‟ stock assessment method in many fishery management systems today. The notion of attempting to break the „historical inertia‟ of modern stock assessment methods does not indict a given methodology such as VPA, but simply suggests that these frameworks should be reinvestigated periodically as new data and information is collected on a species by species basis. Just because a given model worked a decade ago does not mean it is still the best avenue for future assessments. All stock assessment assumptions should be tested every time a model is run in order to see if the assumptions are still being met and if the given framework does indeed provide the „best available science.‟ However, questioning whether a closed population assumption still holds (and testing whether a tag-integrated model would provide a better assessment than a closed- stock model), is only one of many assumptions that need to be investigated. There is no doubt that Beverton and Holt would agree that treating any given idea in fisheries science as dogma is dangerous and goes against the basic tenets of the scientific method. Even as we look back and admire the work in the „fisheries bible‟ we must remember that as our knowledge of fisheries improves we can still admire and appreciate the work that they did while we simultaneously question its validity. When new data becomes available through the use of cutting edge technology (e.g., Vessel Monitoring Systems with GPS that give spatially explicit information on catch locations, and satellite tags that can determine exact migration routes of tagged fish), it might turn out that building on the work of Beverton and Holt (1957) will mean proving that some of their original assumptions are violated for some applications. There is nothing wrong with this. The true insult would be to ignore altogether the stepping stones and solid foundation that they provided for us over 50 years ago. It would be a disgrace if future generations were to look back at the work of Beverton and Holt with what Ricker (2006) terms “the Doctor Watson effect” wherein “everything becomes simple and obvious once is has been discovered.” For the most part, though, it appears that, as Vaughan Anthony explains in his 1994 introduction to Ray Beverton‟s second lecture at Woods Hole, “when people talk up new ideas, they say, „Oh it‟s in the back of Beverton‟s [and Holt‟s; 1957] book, we just haven‟t gotten there yet‟” (Anderson, 2002). The important aspect as we move forward in fisheries science is to make sure that when we do get „there‟ we remember who got there first and what Beverton and Holt can still teach us.

5. Acknowledgements

This work was carried out with funding from a NOAA/Sea Grant fellowship, the Marine Fisheries Institute (MFI) of Massachusetts, and the University of Massachusetts-Dartmouth School for Marine Science and Technology (SMAST). This paper was stimulated by discussions with Terry Quinn II.

6. References

Aldenberg, T. Virtual population analysis and migration, a theoretical treatment. ICES C.M. 1975/F:32: 14 p. (mimeo) (1975).

Anderson, E. D. (editor). The Raymond J. H. Beverton lectures at Woods Hole, Massachusetts. Three Lectures on Fisheries Science Given May 2-3, 1994. U.S. Dep. Commer., NOAA Tech. Memo. NMFS-F/SPO-54: 161 p. (2002).

Anderson, J. Nachrichten von Island, Grönland und der Straße Davis. Hamburg: 328p (1746).

Angelini, R. and C. L. Moloney. Fisheries, ecology and modeling: an historical perspective. Pan-Amer. J. Aquat. Sci., 2(2): 75-85 (2007).

Baranov, F. I. On the question of the biological basis of fisheries. Nauchn. Issled. Ikhtiol. Inst. Izv., 1: 81-128 (1918).

Ben-Yami, M. From F.I. Baranov to fishing quotas: on the problems of the contemporary fisheries management. Plenary lecture delivered on May 20, 2010 to the Scientific Conference at Kaliningrad State Technical University. (2010).

Bertalanffy, L.von. A quantitative theory of organic growth (inquiries on growth laws II). Human Biology, 10(2): 181-213 (1938).

Bertignac, M., P. Lehodey and J. Hampton. A spatial population dynamics simulation model of tropical tunas using a habitat index based on environmental parameters. Fish. Oceanogr., 7: 326-334 (1998).

Beverton, R. J. H. and S. J. Holt. On the Dynamics of Exploited Fish Populations. Fisheries Investment Series 2. 19. U. K. Min. of Agr. and Fish. Chapman and Hall: London (1957).

Butterworth, D. S. and A. E. Punt. The robustness of estimates of stock status for the western North Atlantic bluefin tuna population to violations of the assumptions underlying the associated assessment models. Col. Vol. Sci. Pap. ICCAT., 42(1): 192-210 (1994).

Cadrin S. X. and D. H. Secor. Accounting for spatial population structure in stock assessment: past, present and future. In: The Future of Fishery Science in North America (B. J. Rothschild and R. Beamish, Eds.). Springer Verlag: p. 405-426 (2009).

Chambers, C. and E. A. Trippel. Early Life History and Recruitment in Fish Populations. Chapman and Hall: London (1997).

Cren, E. D. Le. Rev. of On the Dynamics of Exploited Fish Populations, auth. R. J. H. Beverton and S. J. Holt. J. Anim. Biol., 28(1): 157-159 (1959).

Cushing, D. H. and R. W. Edwards. Raymond John Heaphy Beverton, C.B.E. 29 August 1922-23 July 1995. Biogr. Mems. Fell. R. Soc. 42: 25-38 (1996).

Faugeras, B., and O. Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: application to the Indian Ocean skipjack tuna fishery. Math. Biosci. Eng., 2(4): 719- 741 (2005).

Graham, M. The sigmoid curve and the overfishing problem. Cons. Int. Explor. Mer, Rapp. Et Proc.-Verb., 110: 15-20 (1939).

Goethel, D. R., T. J. Quinn II, and S. X. Cadrin. Incorporating spatial structure in stock assessment: movement modeling in marine fish population dynamics. Rev. Fisher. Sci., 19(2): 119-136 (2011).

Hampton, J. and D. A. Fournier. A spatially disaggregated, length-based, age-structured population model of yellowfin tuna (Thunnus albacares) in the western and central Pacific Ocean. Mar. Fresh. Res., 52: 937-963 (2001).

Heincke, F. Naturgeschichte des Herings I. Die Lokalformen und die Wanderungen des Herings in den europaischen Meeren. Abhandlungen des Deutschen Seefischerei- Vereins, 2(1). 136 p (1898).

Hilborn, R. Determination of fish movement patterns from tag recoveries using maximum likelihood estimators. Can. J. Fish. Aquat. Sci., 47: 635-643 (1990).

Hilborn, R. Rev. of On the Dynamics of Exploited Fish Populations, auth. R. J. H. Beverton and S. J. Holt. Rev. Fish Biol. Fisher., 4(2): 259-260 (1994).

Hilborn, R. and M. Liermann. Standing on the Shoulders of giants: learning from experience in fisheries. Fish Biol. Fisher., 8(3): 273-283 (1998).

Hjort, J. Fluctuations in the great fisheries of northern europe. Cons. Int. Explor. Mer, Rapp. Et Proc.-Verb., 20: 1-228 (1914).

Holt, S. J. Foreword. On the Dynamics of Exploited Fish Populations. By R. J. H. Beverton and S. J. Holt. Blackburn Press: Caldwell, NJ (2004).

Holt, S. J. Three lumps of coal: doing fisheries research in Lowestoft in the 1940s. Lecture delivered on April 25, 2008 to a CEFAS seminar in Lowestoft, Suffolk, U.K. (2008).

Hulme, H. R., R. J. H. Beverton and S. J. Holt. Population studies in fisheries biology. Nature, 159: 714-715 (1947).

Hunter, E., J. D. Metcalfe, G. P. Arnold, and J. D. Reynolds. Impacts of migratory behavior on population structure in North Sea plaice. J. Animal Ecol. 73: 377-385 (2004).

Ishii, T. Attempt to estimate migration of fish population with survival parameters from tagging experiment data by the simulation method. Inv. Pesq., 43: 301-317 (1979).

Jones, R. A method of analysis of some tagged haddock returns. ICES J. Mar. Sci., 25: 58-72 (1959).

Maunder, M. Integration of tagging and population dynamics models in fisheries stock assessment, University of Washington. PhD: 306p (1998).

Maunder, M. Integrated tagging and catch-at-age analysis (ITCAAN): model development and simulation testing. In: Spatial Processes and Management of Marine Populations. (G. H. Kruse, N. Bez, A. Booth, M. W. Dorn, S. Hills, R. N. Lipcius, D. Pelletier, C. Roy, S. J. Smith, and D. Witherell, Eds.). University of Alaska Sea Grant: Fairbanks, AL. AK-SG-01-02 (2001).

Nielsen, J.L. and H. Arrizabalaga (eds.). Tagging and Tracking of Marine Animals with Electronic Devices. Springer (2009).

NRC (National Research Council). An Assessment of Atlantic Bluefin Tuna. National Academy Press: Washington, D.C. 148p (1994).

Pauly, D. Foreword. On the Dynamics of Exploited Fish Populations. By R.J. H. Beverton and S. J. Holt. Blackburn Press: Caldwell, NJ (1993).

Pitcher, T. J. and D. Pauly. Editorial: the Beverton and Holt jubilee issues, 1947-1997. Rev. Fish Biol. Fisher., 8(3): 225-227 (1998).

Porch, C. E. Trajectory-based approaches to estimating velocity and diffusion from tagging data. Fish. Bull., 93: 694-709 (1995).

Porch, C. E. An age-structured assessment model for red snapper that allows for multiple stocks, fleets and habitats. S.E.F.S.C., Sustainable Fisheries Division., SEDAR7-DW- 50 (2004).

Porch, C., S. C. Turner, and J.E. Powers. Virtual population analyses of Atlantic bluefin tuna with alternative models of transatlantic migration: 1970-1997. Col. Vol. Sci. Pap. ICCAT., 52(3): 1022-1045 (2001).

Quinn, T. J., II, R. B. Deriso, and P. R. Neal. Migratory catch-age analysis. Can. J. Fish. Aquat. Sci., 47: 2315-2327 (1990).

Ramster, J. Professor Ray Beverton 29 August 1922-23 July 1995. ICES J. Mar. Sci., 53: 1-9 (1996).

Ricker, W. E. Further notes on fishing mortality and effort. Copeia, 1944(1): 23-44 (1944).

Ricker, W. E. One man‟s journey through the years when ecology came of age. Environ. Biol. Fishes, 75: 7-37 (2006).

Rounsefell, G. A. Rev. of On the Dynamics of Exploited Fish Populations, auth. R. J. H. Beverton and S. J. Holt. Limnol. Oceanogr., 4(2): 230-231 (1959).

Russell, E. S. Some theoretical considerations on the “overfishing” problem. J. Cons. Int. Explor. Mer, 6(1): 3-27 (1931).

Schwarz, C. J. Estimation of movement from tagging data. In: Stock Identification Methods (S. X. Cadrin, K. D. Friedland, and J. R. Waldman, Eds). Elsevier Academic Press, San Diego (2005).

Secor, D. H. Historical roots of the migration triangle. ICES Mar. Sci. Sym., 215: 329-335 (2002).

Sibert, J. R. A two-fishery tag attrition model for the analysis of mortality, recruitment and fishery interaction. Tuna and Billfish Ass. Prog., Nuomea, New Caledonia, South Pacific Commission: 27p (1984).

Sibert, J. R. and D. A. Fournier. Evaluation of advection-diffusion equations for estimation of movement patterns from tag recovery data. In: Proceedings of the First FAO Expert Consultation on Interactions of Pacific Ocean Tuna Fisheries. (R. Shomura, J. Majkowski, and S. Langi, Eds.). Noumea, New Caledonia. FAO Fish. Tech. Paper. 336: 108-121 (1994).

Sibert, J. R., J. Hampton, D. A. Fournier, and P. J. Bills. An advection-diffusion-reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (Katsuwonus pelamis). Can. J. Fish. Aquat. Sci., 56: 925- 938 (1999).

Sinclair, M. Herring and ICES: a historical sketch of a few ideas and their linkages. ICES J. Mar. Sci. 66: 1652-1661 (2009).

Skellam, J. G. Random dispersal in theoretical populations. Biometrika., 38: 196-218 (1951).

Smith, T. D. Scaling fisheries: the science of measuring the effects of fishing, 1855-1955. Cambridge studies in applied ecology and resource management. Cambridge University Press: Cambridge, 395p. (1994).

Taylor, N., M. K. McAllister, B. Block, and G. Lawson. A multi stock age structured tag integrated model (MAST) for the assessment of Atlantic bluefin tuna. Col. Vol. Sci. Pap. ICCAT., 64(2): 513-531 (SCRS/2008/097) (2009).

Thompson, W. F. and F. H. Bell. Biological statistics of the Pacific halibut fishery. Rep. Internat. Fish. Comm., 8 (1934).

Toft, C. A. and M. Mangel. Discussion: from individuals to ecosystems; the papers of Skellam, Lindeman and Hutchinson. Bull. Math. Biol., 53:121-134 (1991).

Wegner, G. 250 years polar migration theory of herring and actual aspects. Inf. Fischwirtseh., 43(3): 120-126 (1996).