THE USE OF MATHEMATICAL MODELS IN FISHERIES

13.6 THE USE OF MATHEMATICAL MODELS IN FISHERIES

Models of exploited fish populations have devel- oped along several different paths determined by the amount of detail included. Early attempts were constrained by the limited computing power that was available just after the Second World War. There was heavy dependence on integral calculus with equations that could be solved and there was

a drive to find simple functions for growth and mortality. These early attempts have left a legacy that has been used to develop some sophisticated models, which use methods that were not avail- able to the pioneers. In this handbook we have acknowledged the legacy by devoting chapters to surplus yield models (Chapter 6, this volume) and dynamic pool models (Chapters 7 and 8, this vol- ume). In addition other chapters deal with tradi- tional models of growth (Chapter 5, Volume 1) and recruitment (Chapter 6, Volume 1). The legacy pervades the practice of fisheries management but fishery scientists have gone beyond the early monolithic models, which tended to ignore differ- ences between fisheries and fish stocks. Most as- sumed that the Beverton and Holt dynamic pool model could be applied to any species, so long as one could estimate the required parameters. In the remainder of this section we attempt to show how fishery scientists have developed detailed models to cope with the reality not always covered by the original formulations. In many ways, the submod- els discussed are really only filling in the cracks that have appeared in the original edifice. There is definitely room for fresh thinking as described in the various chapters which deal with and develop the legacy.

In this section it is helpful to make a distinction between ‘full’ population models that allow fore- casting of yields and stock dynamics under various fishing scenarios, and ‘submodels’ or ‘ingredients’ that go into these. For example, many models in- corporate body growth rates, and this information is either input directly from data, or represented by an equation, which represents the data. This equa-

277 tion enters as a submodel. In many cases these sub- where p is an index of ‘predator type’ as is done

Choosing the Best Model for Assessment

models have been introduced to deal with detail in multispecies VPA (Andersen and Ursin 1977; that the earlier models did not account for. There Shepherd and Pope Chapter 7, this volume). M 0 are no hard-and-fast rules here, because, for exam- is the natural mortality not caused by predation, ple, an equation that describes relationships be- such as disease, spawning stress and old age and tween stock size and recruitment might serve as NP is the number of predators. M p may be mod- an end in itself when one wants to predict optimal elled as suggested by Andersen and Ursin (1977): exploitation rates of semelparous species. Alterna- tively it might be a component of a full statistical

NRS

catch-at-age model. But for presentation purposes, M p

(13.13) we adopt this distinction.

p pr p = ,

 NwS j jjp , j = 1

jP =

13.6.1 Simple models of fish

where S j,p is a measure of the suitability of prey j as

stock dynamics

food for predator p and R p is the food ration for predator p. The multispecies VPA was an exten-

Most fisheries models are based on differential sion of the single-species VPA, which aimed at the or difference equations, here exemplified by the estimation of M p from the investigation of stom- model of the death process which is central to age- ach contents of predators (Sparre 1991; Shepherd structured models of exploited populations:

and Pope, Chapter 8, Volume 2). dN t =-

ZN t t or when integrated dt

13.6.2 Models of fishing

N t = N 0 exp ( - Zt ) , (13.10) Like natural mortality, fishing mortality may be divided into fleet components:

where N(t) is the number of survivors from a cohort at time t and Z is the coefficient of total mortal-

ity. In fisheries, Z is usually assumed to remain NFs

(13.14) constant during a standard time interval such as

F = Â F Fl ,

Fl = 1

a month, a quarter or a year. Using a time step of length Dt, we get a difference equation, where Z is where Fl is index of fleet and NF is the number assumed to remain constant from time t to time of fleets. A fleet is a group of fairly homogenous t + Dt:

vessels with respect to size, fishing operations and fishing grounds.

D N t = N t - N t + D t = N t [ - 1 exp ( - Zt t D ) ] .

F Fl may be further modelled, for example by: It is important to notice that Z refers to the en- F Fl,St,a =E Fl Q St,Fl S Fl,St,a ,

(13.15) tire stock. If the stock is divided into fractions ac- cording to geographical areas, there is need for where St is an index of a stock and a is an index of

redefining Z when it becomes area-specific. This age groups. Q is the catchability of fleet Fl catching will be discussed below in connection with ‘areas stock St. Here the F is given the index St to empha- and migration’.

size that almost all fisheries are mixed fisheries The total mortality, Z, is composed of fishing catching more than one species, a fact that is often

mortality F, and natural mortality M, which again not accounted for in routine stock assessments can be further divided, for example:

and certainly not in fisheries where the manage-

ment system is still relatively primitive. M = M 0 + M p

NP

 The gear selection (S) by trawls can be modelled , (13.12)

p = 1 by the logistic curve:

1 + exp ( s 1 + s 2 L St a , )

L 50 % where N is the number of age groups and w a is the

mean body weight at age a. This is the basic form of where L

the dynamic pool model, derived originally in con- St,a is the mean body length of age group

a (see MacLennan 1992 for a review of gear selec- tinuous form as an integral equation, by Beverton tivity). L

and Holt (1957) (see Shepherd and Pope, Chapter 8, 50% and L 75% are the lengths at which 50% and 75% of the fish entering the gear is retained, this volume for further details). respectively. The logistic curve takes values

The value of the catch can be estimated by ex- between zero and one, starting with zero and tending the model: approaching one as L increases (e.g. Hoydal et al.

1982). For some other gears like a gill-net a dome- N

(13.21) shaped curve may better reflect the selectivity of

V Fl = Â C Fl a , w a Pr Fl a , ,

the gear (Hovgård and Lassen 2000).

A simple relationship between mesh size (m) in, where Pr is the price per unit weight. Bioeconomic for example, the cod-end of a trawl and the reten- modelling is discussed in detail by Hannesson tion of fish of a length L is

(Chapter 12, this volume). The input vector, Z t , of the general model, Y t = G(X ,Z

t , P) could include the fishing effort, the

L =S m .

50% Fl,St Fl

mesh size and the price, all of which are human in- puts to the fisheries system. The yield and the

The catchability may be modelled by a fleet and value of the yield may be the output, and the para- stock-specific constant, but it may also be assumed meters are those listed above which are not vari- to be a function of stock biomass and the year, ables: G [X

t , (Effort, Mesh size, Price), P] = (Yield, accounting for the technical development in fish- Value). Fishing mortality may also be considered

ing techniques: output, as it is used as a measure for the perfor- Q ,

mance of the fisheries system.

= Q B a by 0 b

Fl year

The development of fishery models has made extensive use of the catch equation (13.19). It lies

where Q 0 , b, a and b are constants and y is at the heart of Beverton and Holt’s (1957) yield- year. This is just one among many other math- per-recruit model, but since then it has been used ematical models one can imagine and it assumes extensively in the development of methods to that catchability increases with time in a linear estimate population size and fishing mortality. fashion.

Equation (13.19) is the basis of VPA, which is de- The catch can be estimated as follows. Assume scribed in detail in Chapter 7 (this volume) by

that the output from the model might be the catch Shepherd and Pope. The equation also lies at the of age group a fish caught by fleet Fl. If we assume

root of statistical catch-at-age methods (Hilborn

F and Z to remain constant in time period Dt and Walters 1992). These derive an equation, (Baranov 1918), and we skip the stock index, which, in its simplest form, allows the estimation then the ‘catch equation’ gives the catch as:

of initial cohort size, fishing mortality and the catchability coefficient from a multiple linear re-

(13.19) gression of catch on effort. Doubleday (1976) pro- Z a duced a non-linear equation relating catch to effort

but a linear, and therefore easier to fit, version was The yield in weight of the catch accumulated over derived by Paloheimo (1980). His equation for a age groups may also be estimated as

single cohort is

Choosing the Best Model for Assessment

) - q ËÁ E y ¯˜

( ˆ ya

ËÁ kya  E

13.6.3 Body growth

1 When estimating yield it is often useful to have a - Ma Ê Ë - ˆ

(13.22) way of estimating weight at age. This has been re- viewed by Jobling (Chapter 5, Volume 1), who pre-

sents the von Bertalanffy equation (von Bertalanffy where C a,y is the catch of age a fish in the yth year,

y is the fishing effort in the yth year, R y -a is the body weight (or length) and age. Recall that length, number of fish recruited at age 0 in year y - a, q is L catchability and M is natural mortality. The equa-

E 1934), which expresses the relationship between

t , is given as follows:

tion is in the form

(13.25) Y =b 0 +b 1 X 1 +b 2 X 2 ,

L t = L 1 - e Ktt ( - 0 • ) ( )

(13.23) and the length–weight relationship is which is the equation for a multiple regression. In W = aL b

this case b 0 = log(R q

y -a ), b 1 = q, and b 2 = M. The

equation can be fitted using a standard multiple where L t and W t are length and weight at time t, L regression package which will give estimates of

is the asymptotic length at t = •, K is the rate at Rq , q and M, which is unlike VPA and cohort which L t approaches L • and t 0 is the age at which analysis. There is no need to guess F for any ages the fish would be zero length. These equations as-

and normal confidence estimates on the parame- sume a one-to-one relationship between age and ters are possible. In this sense these statistical length, which may not be the case. When this is catch-at-age methods are superior to VPA. Even so, true, age–length keys (ALK) can be used (Pitcher, the validity of the method is entirely dependent on Chapter 9, this volume). For example the assumption that equation (13.10) correctly de-

scribes the decay of survivors and that the stock ALK ia , = Pr ( L i £ length < L i + 1 age = a ) . (13.27) numbers are proportional to the catch per unit

of effort (CPUE). Actually, equation (13.22) is This gives the probability that an age group of fish equivalent to

has a length belonging to length group i. For the weight at age we get

E Î kya =- Â Z ˚

AWK ia , = Pr ( W i £ weight < W i + 1 age = a ) . (13.28) so that the equation essentially says that the CPUE

Keys like the ALK could be applied almost is proportional to stock numbers. The most prob- everywhere in the models, and they would always lematic element in this appealingly simple model represent a more realistic description of the real is the definition and measurement of effort. Total world, but the cost would be a very large number of effort may be composed of effort from many differ- parameters and cumbersome models. A spread- ent vessel types and sizes, all of which can be mea- sheet implementation of a fishery model, on Excel sured in many alternative ways such as sea-days, for example, can use ALKs easily. fishing days, trawling hours and number of hooks.

Usually, one needs to keep track of the body Furthermore, due to technical development, one weightofallthecohorts,soanextra index is needed, unit of effort in year y may be different from a unit L a,t and w a,t , the length and weight at time t of age of effort in, say, year y -5. For this reason equation group a. The body sizes at age may be assumed to re- (13.18) is problematic to apply in practice. For main constant from year to year, but they may also further details of this approach see Fournier and vary. In that case one more index is needed, w y,a,t : Archibald (1982).

the weight of age group a at time t in year y.

Chapter 13

In ICES, size-based models are assumed to be

the same as age-structured models. The entire the- Mat a =

1 + exp S 1 M + SL 2 ( M ◊ a )

ory of size-structured models may be reformulated

ML 50 %

with body length as the basic observation (Chapter

1 =() ln 3 ; ML 75 % - ML 50 %

9, this volume, and Sparre and Venema 1992). This approach is particularly useful in the tropics where M

S =- 2 .

(13.31) addition, in all areas, shellfish do not have

it is more difficult to read age from otoliths. In

ML 50 %

otoliths. For these and other species groups, ML 50% and ML 75% is the length at which 50% and length-based models are more useful (Pitcher, 75% of the fish are mature, respectively. Chapter 9, this volume).

SSB yt , = Â N yat ,, W Mat at , a .

13.6.4 Biomass of stock, catch, discards and landings and

Where there are large differences between the

value of landings

sexes, we may need to have two models, one for females and one for males. If the maturity ogive is

The von Bertalanffy equation has only three differentiated between sexes, all other equations parameters, whereas each age and size class in an (for example, stock numbers) must also be separate age–length key is a parameter. For this reason, for females and males. careful thought needs to be given before using an

A more realistic approach would be to use a age–length key.

length–maturity ogive to estimate the proportion The stock biomass at time t in year y may be of fish aged a that were mature. This empirically

derived from the one-to-one equation: based estimate could then be used to calculate

spawning stock biomass. This method should only

B yt , = Â N yat ,, w at , ,

aa = max

be used if the extra complication yields results

a = 0 which are a significant improvement on the sim- pler solution shown in equation (13.32).

or it may be derived from the age–weight key, The catch may be divided into landings and which illustrates the extra computations needed discards, which implies a model for discards. The when using such a key:

logistic curve can also be used here to model the fraction of fish caught, and which are discarded

 + N ,, +

N NW

W i W i 1 (D a ), as a function of body length:

B yt ,

 yat AWK ia ,

D a =- 1 ; where N is the number of age groups and NW is

1 + exp ( S 1 D + SL 2 D a )

the number of weight groups. To compute the S 1 D DL 50 =() % ln 3 ; spawning stock biomass (SSB), a relationship is

DL 75 % - DL 50 %

needed between either age or length or weight,

(13.33) the easy solution and use a one-to-one relation-

S 2 D =-

and maturity. For that purpose we might go for

DL 50 %

ship between length (or age) and the fraction of DL 50% and DL 75% is the length at which 50% mature fish. We may use the logistic curve to esti- and 75% of the fish caught are discarded, res-

mate the proportion of mature fish, which was also pectively. Note that (1 - D a ) is the fraction landed. used for trawl gear selection earlier (equation The total quantity discarded by a fleet D Fl is 13.16).

given by

Choosing the Best Model for Assessment

D Fl = Â C Fl a , wD a a .

when the relevant data are available, for example,

a = 0 through logbook schemes. Some of the problems faced in stock assessment might be reduced by the

To model the effect of a minimum landing size on introduction of models accounting for geographi- discarding we could also use the logistic curve, so cal and seasonal aspects. illustrating that there are many different applica-

Geographical aspects may be dealt with in a tions of this curve in fisheries.

simple manner by introducing a finite number of

The SSB used as a measure of the performance areas, and seasonal aspects by introducing a finite of the system and the level of discards can be added number of seasons. These could be defined in to the list of outputs. Discarding is a political issue weeks, months or quarters of the year. As an ap- as the rejected fish are considered a waste of poten- proximation we then assume that within a time tial catches (Kaiser and Jennings, Chapter 16, this period, all parameters remain constant, fish are as- volume). Therefore discards can be considered as a sumed to be homogeneously distributed within an part of output. This illustrates that there are no area, and nothing moves between the areas during uniqueness rules for nominating certain variables

a time period. All movements are assumed to as output, and others as variables.

occur at the end of the time period, and take no

Fisheries landing statistics are not given by age time to happen. Although an approximation to the group but by commercial size category. The com- real world, any precision can be modelled by se- mercial size categories may be based on body lecting areas and time periods at a scale that can weight, body length or some other criterion, such

be supported by the data. For a detailed discussion as the number of tails per pound for shrimps. These of the migration of commercial fish species, see groupings can be transformed into age groups or Metcalfe et al., Chapter 8, Volume 1. size groups by keys similar to the ALK or they may

The Migration Coefficient, MC, from area A to

be modelled under the assumption of a one-to-one area B is defined as the fraction of the animals in relationship using a growth equation and the area A which moves to area B. In this definition, length–weight relationship.

the ‘movements’ include the ‘move’ from area A to

The real choice to make is not so much the area A, which covers the case where the animals do mathematical model, for example the logistic not move. The migration coefficient depends on model for gear selection, but rather whether or not the starting area (F Ar ) and the destination area (T Ar ). to include a submodel for gear selection. The gear Note that the sum of migration coefficients over selection must be included if the objective of the destination areas is always equal to 1.0, as the model is to predict, for example, the effect that starting area is also considered a destination area: mesh-size regulations might have. In other cases

one may have a choice of whether to include the N Â

13.6.5 Models for spatial and

where a = age group, q = quarter or any time period. For a theoretical discussion of migration in con-

seasonal aspects of fisheries

nection with age-based fish stock assessment Stock abundance and consequently catch rates the reader is referred to Quinn et al. (1990). These may vary from area to area. For this reason fishing authors also discuss the estimation of migration effort will also vary between areas, and distribu- parameters. There are also more sophisticated tion of resources and fishing areas will vary be- models, based on differential equations (Olivier tween seasons. Most management agencies do not and Gascuel, 1999). account for seasonal and geographical variations

The concept of mortality, as it is traditionally but we will introduce here simple models dealing applied, refers to the entire stock, not to a fraction with areas and seasons. These can be implemented of a stock. Consequently, the number of deaths in a

Area one (Ar = 1)

Area one (Ar = 1)

N( y, Ar = 1,q) N1( y, Ar = 1,q) N( y, Ar = 1, q+1) N1( y, Ar = 1, q+1)

a=0

death process

a=0

a=0

death process

a=0

a =1

death process

a =1

a =1

death process

a =1

a =2

death process

a =2

a =2

death process

a =2

Fig. 13.3 The flow of stock Area two (Ar = 2)

Migration (takes zero time)

Area two (Ar = 2)

numbers (N(year, area, quarter) = N(y,Ar,q) depicted as small

N( y, Ar = 2,q) N1( y, Ar = 2,q) N( y, Ar = 2, q+1) N1( y, Ar = 2, q+1) boxes), between model-

components from quarter q to a=0

death process

a=0

a=0

death process

a=0

quarter q + 1 (during quarters 1, 2 and 3) and between two areas

(depicted as large grey boxes)

a =1

death process

a =1

a =1

death process

a =1

in a hypothetical example with one stock and three age groups

a =2

death process

a =2

a =2

death process

a =2

(a = 0, 1 and 2). Migrations and recruitments are depicted

as thin arrows and the death (except for transition between quarter 4 and quarter 1)

Transition between quarters

process by fat arrows. For further explanation see text.

sub-area of the total area occupied by a stock, To explain the multi-area version of the expo- should not be associated with a mortality, but nential decay model (see equation 13.10), we use with some other concept. Here we use the term the quarter as the time period. The stock at the end area-specific mortality, as the concept required of quarter q of year y in area Ar (before migration) is to describe the death process within a sub-area. modelled by: Area-specific mortality is closely related to mor- tality as normally understood. Let Z stock,y,a,q indi-

cate the traditional total mortality of the stock. (13.37) The relationship between Z stock,y,a,q and the area-

N 1 y a Ar q ,, , = N y a Ar q ,, , exp ( - Z y a Ar q ,, , D t ) .

specific total mortalities, Z y ,a,Ar,q , is Note that the indices of N and N1 remain un-

changed when considering the death process Ï

during a quarter of the year. In the model the Z stock, , , yaq

NA

exp ( - Z y a Ar q ,, , D tN ) y a Ar q ,, , ¸

transition between quarters occurs between just =- Ì

ln ÔÔ Ar

before migration and just after migration. The use Ô

NA

N y a Ar q

Ar  = 1 ˛ Ô

of indices in relation to the transition between (13.36) components is illustrated in Fig. 13.3 for quarters

1, 2 and 3.

where N y ,a,Ar,q = stock number at the beginning of The number at the beginning of quarter q in quarter q of year y in area Ar and NA is the number year y in area Ar (just after migration, see Fig. 13.3), of areas.

if q = 1,2,3 is given by

Choosing the Best Model for Assessment

NA

N y a TAr q ,, , + 1 = Â MC , a FAr TAr q , , N 1 St y a Ar q ,,, , (13.38)

fish (Myers, Chapter 6, Volume 1). This is a

FAr = 1 central problem of fish population dynamics, since it represents nature’s regulation of population

if q = 4 (and a < a max ) then size, whether or not the populations are being exploited. Obviously, there can be no recruits if no

adult fish are left to mature, spawn and produce N y + 1 , a + 1 , TAr , 1 = Â MC , a FAr TAr , , 4 N 1 y a Ar ,, , 4 (13.39) eggs, which hatch and grow to become recruits.

NA

FAr = 1 Unfortunately, this is about the only thing we know for sure about the stock–recruitment rela-

where: a max is the oldest age group. tionship (for a discussion see, for example, Gilbert MC a ,Ar1,Ar2,q = the migration coefficient for 1997, Hilborn 1997, Myers 1997 and Myers, Chap- stock St moving from area Ar1 to area Ar2 in quar- ter 6, Volume 1). ter q of age group a and N1 St ,y,a,Ar,q = the number of

The problem of recruitment is so important fish in stock St, at the end of quarter q of year y in that it has a chapter to itself (Myers, Chapter 6, area Ar (before migration).

Volume 1). It is difficult to recommend any of the Stock number at the end of quarter q of year y in stock–recruitment models, as very few of them area Ar (before migration) in the oldest age group have been shown to provide a convincing fit to

a max , which is here modelled as a plus group, that observations. Many fisheries scientists use the is, it contains all the age groups, (a max ,a max + 1, a max Beverton and Holt model, when a stock and re- + 2, . . . a max + •). If q = 1,2,3 and a = a max then the cruitment model has to be used, but there is no jus- equation is the same as for the younger tification for this use other than familiarity. age groups. Only if q = 4 and a = a max is there a dif- Perhaps the best approach is to use ‘no model’, ference compared to the younger age groups:

meaning that the model is replaced by a stochastic factor multiplied by a constant. Such an approach

NA

does not pretend to describe a biological mecha-

N y + 1 , a max , TAr , 1 = Â MC a max , FAr TAr , , 4 N 1 ya , max , Ar , 4

nism, which we do not yet understand. Examples

= 1 NA

FAr

of stock–recruitment curves and a discussion of

+ Â MC a max - 1 , FAr TAr , , 4 N 1 ya , max - 1 , FAr , 4 .

(13.40) mechanisms of density-dependent mortality are FAr = 1 given in detail by Myers (Chapter 6, Volume 1).

Spatial features in general involve mapping and other means of presentation, as well as spatial

13.7 A KEY TO MODELS

analysis, which all falls under the category of GIS (Geographical Information Systems). However, In Table 13.1 we present a key to some of the GIS is too big a field to be covered here and the in- main models used in fisheries. This key is far from terested reader is referred to the literature on appli- exhaustive, and we wish to emphasize that while it cation of GIS in fisheries (Castillo et al. 1996; Fox lists major techniques, no model can be applied and Starr 1996; Meaden and Do Chi 1996; Meaden without thinking very carefully about the model’s and Kemp 1997; Starr and Fox 1997; Denis and assumptions and applicability to each particular Robin 1998; Foucher et al. 1998).

situation. We refer readers to the relevant chapters of this book as well as Quinn and Deriso (1999) for

13.6.6 Some stock–recruitment

detailed explanations of these assumptions as

models well as full mathematical derivations. A recent

textbook by Jennings et al. (2001) also provides an The stock and recruitment problem is the search overview of methods. As noted earlier, some of the for a relationship between parental stock size and models presented here may meet the objectives of the subsequent recruitment in numbers of young giving advice for fishery management by them-

Table 13.1 Key to major groups of fisheries models, including primary chapters of this volume where described (John D. Reynolds assisted with the development of this key).

Goals and data available

Primary chapter Aim to model individuals?

Model

Aim to model interactions between fishers?

Huse et al. Aim to model interactions between fishes?

Game theory, individual-based models

Huse et al. Aim to model a single species?

Game theory, individual-based models

Aim to reconstruct past events? Historical data on catches and age composition?

Shepherd and Pope, 7 Historical data on catches, age composition + auxilliary (e.g. stock and

VPA, cohort analysis

this chapter recruitment)?

Statistical catch-at-age models

Aim to forecast? Chapter 13 Data on catches and effort only? Stock in equilibrium with fishery? (Be careful!) CPUE and effort linked by straight-line regression?

Schnute and Richards CPUE and effort linked by negative exponential curve?

Schaefer production model

Schnute and Richards CPUE and effort linked by non-linear curve?

Fox production model

Schnute and Richards Stock not assumed to be in equilibrium with fishery? All error in model and not in data?

Pella and Tomlinson production model

Schnute and Richards All error in data and not in model?

Process-error production model

Schnute and Richards Data on catches, effort, age and size composition, stock and recruitment?

Observation-error production model

Shepherd and Pope 8, this chapter Aim to model multispecies?

Yield-per-recruit, dynamic pool, stat. catch-age models

Shepherd and Pope 7 Aim to forecast? Data on catches and effort only? Data from one location over previous years?

Aim to reconstruct past events?

MSVPA

Jennings et al. (2001) Data from many locations?

Temporal multispecies production model

Spatial multispecies production model

Data on catches, effort, age and size composition, stock and recruitment? Aim to examine multiple species caught in same gears?

Shepherd and Pope 8, this chapter Aim to examine predator–prey interactions among species?

Multi-species yield-per-recruit, stat. catch-age models

Trophic models

Pauly and Christensen Pauly and Christensen

The first triplet presents a choice between models of individuals, single species and several species. Most models in fisheries science have dealt with either population dynamic models of species or, more recently, the population dynamics of many species. Population dynamics treats indi- vidual animals as replicates of each other and models accounting for changes in numbers with time only recognize reproduction and death as individual properties. Even then, each individual has the same chance of giving birth and dying. The development of behavioural ecology over the past

25 years has shown the importance of recognizing that two individuals in a population are not the same and within a population there can be more than one strategy for reproduction, feeding and use of space (Krebs and Davies 1997). This conceptual advance, coupled to the power of modern com- puters, makes it possible to develop models that are based on the actions of individuals. Three powerful methods have emerged that simulate the way in which organisms have adapted themselves to their environment or interact with each other. Genetic algorithms mirror many of the processes that occur during natural selection and neural net- works mimic the way in which nervous systems learn about the world. Game theory handles the way in which the reward gained by an individual when interacting with others depends on the strategy used by it and its opponent. The first entry in Table 13.1 recognizes that these new methods are now being applied to problems of interest to fisheries biologists but at present the approach is not used at the stock assessment level. Fisheries scientists who still have to confront the reality of delivering a stock assessment to management authorities may still be reluctant to use the models outlined by Huse et al. (Chapter 11, this volume) but they should certainly look to the area as one from which a totally new set of methods may emerge over the next 20 years.

Multispecies models may account for ‘techni- cal interactions’, due to more than one species being caught by the same gear (e.g. Misund et al., Chapter 2, this volume), or ‘biological interac-

tions’, principally through predator–prey relation- ships. For basic biology of foraging and predation, see the trio of chapters in Volume 1 by Mittelbach (Chapter 11), Juanes et al. (Chapter 12) and Krause et al. (Chapter 13). For ecosystem interactions and impacts see Polunin and Pinnegar (Chapter 14, Volume 1), Persson (Chapter 15, Volume 1), Jones et al. (Chapter 16, Volume 1) and Kaiser and Jennings (Chapter 16, this volume), and for incor- poration into fisheries models, see Shepherd and Pope (Chapters 7 and 8, this volume) and Pauly and Christensen (Chapter 10, this volume).

Many models aim to reconstruct past stock sizes and fishing mortality, in order to relate these to fisheries effort and hence predict the future. The simplest method is virtual population analysis, which is often implemented using a computa- tional simplification called cohort analysis (Shep- herd and Pope, Chapter 7, this volume). Cohort analysis may be based either on age classes or on length classes if age is not known. VPA/cohort analysis requires initial guesses of fishing mortali- ty and natural mortality, and while these esti- matesbecomelessimportant to the reconstruction of stock sizes and fishing mortality in early years, there is considerable uncertainty in estimates for the most recent years (Megrey 1989). Statistical catch-at-age models, discussed in this chapter, are more sophisticated but also more difficult to im- plement. Single-species VPA has a multispecies analogue (MSVPA) (Shepherd and Pope, Chapter 7, this volume), which has a voracious appetite for data concerning predatory habits of fishes (Sparre 1991; Magnússon 1995; Christensen 1996).

The key in Table 13.1 offers two main routes to forecasting stocks and yields in single-species assessments. One can collapse the effects of age structure and body growth rates, although they are still implicitly incorporated by summing over all age groups, and use production models to assess the behaviour of the entire stock to previous fishing effort (Schnute and Richards, Chapter 6, this volume). Then one can predict how both yields and stock sizes will be affected by fisheries. In the past, researchers were willing to make the assump- tion that such data were derived from stocks that were in equilibrium with fisheries. However, this

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Chapter 13

is a very dangerous assumption, as fishing effort usually increases steadily during the early years of a fishery, without giving stocks a chance to catch up through density-dependent responses (Hilborn and Walters 1992). More modern methods, re- viewed by Schnute and Richards (Chapter 6, this volume) overcome this assumption. Regression- based approaches which assume that the error is in the underlying model rather than in the data (process-error methods) tend to be less precise than methods in which the error is assumed to be in the data (observation-error; Polacheck et al. 1993). For certain purposes, the surplus production models may perform as well as the sophisticated age-based models discussed below. It is not generally the case that complicated models with many parameters have higher predictive power than do simple mod- els. This may not seem fair to the hard-working de- signers of age-based or size-based models, but if the purpose of the model is to predict CPUE, the sur- plus production model would work probably as well as the Thompson and Bell (1934) model dis- cussed earlier. But there is an increasing stream of requests from management bodies for an analysis of details. An example would be to assess the effect of technical management measures on catches (see Section 13.6.4). These questions cannot be ad- dressed with a surplus production model. If the key issue is the discarding of young fish, the model must contain some splitting of the stock biomass by size and some description of discard practice. We recommend that it is best to test both the simple surplus production models and the more compli- cated size-based models and compare and test the performance of each. Whenever the simple model performs as well as the complicated model, it should be used.

Simple forecasting methods that break stocks down into age classes and account for individual body growth are reviewed by Shepherd and Pope (Chapter 8, this volume). Yield-per-recruit and dynamic pool models can be easily calculated and readily understood, and are in widespread use. Statistical catch-at-age models (outlined in this chapter) are more flexible, but they tend to be more difficult for people other than the modeller to evaluate.

There are multispecies counterparts to the single-species forecasting methods (Daan and Sissenwine 1991). Thus, multispecies production models can be used when species are aggregated in catch statistics. Comparisons are then made between an index of fishing effort and catches in either multiple regions (spatial multispecies production models) or over multiple years from

a single site (temporal multispecies production models). If numbers and sizes at age are known, then multispecies yield-per-recruit and statistical catch-at-age models can be used. While multi- species models represent a conceptual advance over single-species models, the difficulty of gathering data to parameterize them has prevented their widespread adoption (Evans and Grainger, Chapter 5, this volume; FAO 1999).

A further conceptual advance is provided by trophic models, which attempt to include environ- mental data and aim to model the whole ecosys- tem, in principle starting with the sun’s radiation, photosynthesis and primary production. One can then model the entire transport of biomass be- tween components of the ecosystem and keep track of the food web, as explained by Pauly and Christensen (Chapter 10, this volume). Software such as Ecopath (Christensen and Pauly 1992; Pauly et al. 2000) makes the bookkeeping fairly straightforward. This approach builds on a ground- breaking model by Andersen and Ursin (1977), which contained all the major components of the ecosystem. The multispecies VPA was developed as a reduced version of this ecosystem. Thus, at some future stage, all the models may be merged into a single mega model.