A general theory of fish stock assessment models

Ecological Modelling 128 (2000) 165 – 180

A general theory of fish stock assessment models Yongshun Xiao * CSIRO Di6ision of Fisheries, GPO Box 1538, Hobart, Tas., 7001, Australia Accepted 29 October 1999

Abstract Fish stock assessment models are of four main types: simple production models, models of Deriso type, depletion models, and age-dependent models. They were derived independently, except for models of Deriso type, which were derived from an age-dependent model. For some fish stock assessments, practitioners have to make tremendous efforts to identify the ‘right’ model. In this paper, I develop a general age-dependent model, a general age-dependent production model and a general age-dependent depletion model, demonstrate that simple production models, models of Deriso type and depletion models are, under three or four assumptions, all deducible from an age-dependent model, and hence provide a general theory of fish stock assessment models and a basis for identifying models in studies of fish population dynamics. Also, I derive seven new production models, each in a differential equation, and where possible, give their analytical solutions; derive a new production model in a system of two differential equations (which, if conditioning on fishing effort and discretized, becomes Deriso’s system of difference equations) and give its analytical solution; and finally re-derive, in a more general context, Deriso’s system of difference equations and his delay difference equation and give its analytical solution. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Fish; General theory; Stock assessment; Models

1. Introduction Fish (a term used here for crustaceans, molluscs and whales as well as fishes) stocks are often assessed by simple production models (Schaefer, 1954; Pella and Tomlinson, 1969; Walter, 1973; Fox, 1975; Jensen, 1984; Prager, 1994), models of Deriso type (Deriso, 1980; Schnute, 1985; Horbowy, 1992), age-dependent production models (Megrey, 1989; Hilborn, 1990; Walker, 1992), depletion models (Leslie and Davis, 1939; De Lury, 1947), and various age-dependent estimation procedures such as Virtual Population Analysis (De Lury, 1947; Gulland, 1965; Murphy, 1965; Zhang and Sullivan, 1988) and Catch at Age Analysis (Doubleday, 1976; Paloheimo, 1980; Fournier and Archibald, 1982; Deriso et al., 1985;

* Present address: South Australian Aquatic Sciences Centre, SARDI, 2 Hamra Avenue, West Beach, SA 5024, Australia. Tel.: +61-8-82002434; fax: +61-8-82002481. E-mail address: [email protected] (Y. Xiao) 0304-3800/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 0 ) 0 0 1 9 9 - X

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Y. Xiao / Ecological Modelling 128 (2000) 165–180

Paloheimo and Chen, 1993). These five types of models or estimation procedures for fish stock assessment all originated independently, except for models of Deriso type, which were derived from an age-dependent model. Specifically, simple production models and depletion models are derived from direct assumptions about the total number or total biomass dynamics of a fish population and relate its present total number or total biomass directly to its previous total numbers or total biomasses. They are simple conceptually (although not necessarily simple, either biologically or mathematically) and generally cannot capture certain age-dependent characteristics of a fish population. By contrast, age-dependent production models relate the present total number or total biomass of a fish population to its previous total numbers or total biomasses through its age structure (i.e. fish numbers and biomasses at age and time). They seem more general representations of a population’s dynamics, but usually have many parameters to be estimated or specified. For useful perspectives on number-based age-dependent models in non-fisheries contexts, readers may wish to consult Lotka (1922, 1924), McKendrick (1926), Kermack and McKendrick (1927, 1932), Gurtin and MacCamy (1974), Hoppensteadt (1975), Sa´nchez (1978), Okubo (1980), Freeman and Strobeck (1982), Nisbet and Gurney (1982), Levin and Hallam (1984), and Webb (1985). All commonly used age-dependent estimation procedures are based on Beverton and Holt’s (1957) exponential population dynamics model, which relates the present number and biomass of a fish population to numbers and biomasses at previous ages and times (Megrey, 1989). Because these stock assessment models all deal, either explicitly or implicitly, with numbers, biomasses and their derivative quantities of a fish population, they should be deducible from an age-dependent model under certain assumptions. Indeed, Sinko and Streifer (1967) and Hoppensteadt (1975) reduced a number-based age-dependent model to what might be called an exponential production model and a logistic production model, respectively; Deriso (1980) reduced an age-dependent model to a difference equation in his work on the ‘harvesting strategies and parameter estimation for an age-structured model’; and Schnute (1985) reduced an age-dependent model to what might be called mixed simple production models and models of Deriso type (by combining population dynamics models with their observational models) in his work on the ‘general theory for analysis of catch and effort data’. Such a reduction, which makes explicit certain assumptions about the stock assessment models, is fundamental for selecting a model to study fish population dynamics. Box and Jenkins (1970) recommended choosing a model by four steps: (1) postulate a useful and appropriate class of models of a system from theory and practice; (2) use data and knowledge of the system to identify a particular tentative model from the class by restricting model parameters meaningfully; (3) estimate parameters in the resulting tentative model; and (4) perform diagnostics: if the tentative model is adequate, then use it for prediction and control; otherwise, return to step 2 and revise accordingly. This approach was advocated by Schnute (1985) in the fisheries context. Current age-dependent models are number-based. Biomass-based age-dependent models should also be developed. This is because biomass is often a more useful quantity: (1) fisheries managers set their objectives by weight in most marine fisheries; (2) fishers are, for economic reasons, usually interested in the total weight of their catch, and often record, report and process catch in units of weight; and (3) fisheries biologists often deal more with fish biomass than with number to avoid biasing a model by ignoring fish growth in weight. However, no general biomass-based age-dependent models have been developed, at least in the fisheries literature, except for Zhang and Sullivan’s (1988) biomass-based estimation procedure (i.e. VPA). In this paper, I develop a general number- and biomass-based age-dependent model, a general numberand biomass-based age-dependent production model and a general number- and biomass-based age-dependent depletion model; demonstrate that simple production models, models of Deriso type and depletion models are, under three or four assumptions, all deducible from an age-dependent model; and hence provide a general theory of fish stock assessment models and a basis for choosing a model for studies of fish population dynamics. Also, I derive seven new production models, each in a differential equation, and where possible, give their analytical solutions; derive a new production model in a system

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of two differential equations (which, if conditioning on fishing effort and discretized, becomes Deriso’s system of difference equations) and give its analytical solution; and finally re-derive, in a more general context, Deriso’s system of difference equations and his delay difference equation and give its analytical solution. All these models are process-based population dynamics models, while Schnute’s (1985) mixed simple production models and models of Deriso type were derived by combining population dynamics models with their observational models.

2. A general number- and biomass-based age-dependent model Let Ng (a, t)]0, Bg (a, t) ]0 and wg (a, t) ]0, − Ba0 5 aB , − Bt0 5 tB , denote, respectively, the number, the average biomass and the average body biomass, of fish of age a and sex g (for females, g= f; for males, g = m) at time t in a fish population, with the average age at birth a0 and reference time t0. Since the age of an animal is finite, Ng ( , t)= 0 and Bg ( , t)= 0. Because Bg (a, t) is related to Ng (a, t) by Bg (a, t) =wg (a, t)Ng (a, t), I will now deal only with biomass-based age-dependent models, which become number-based age-dependent models if wg (a, t)= 1. It should be noted that wg (a, t) can, of course, be seen as a general weight function, in which case Bg (a, t) is the effective (weighted) size of fish of age a and sex g at time t in a fish population. The total population size in biomass of fish of sex g at time t is defined as Bg (t) =

&



Bg (a, t) da,

t ]t0,

(1)

g =f, m.

a0

Three assumptions are then made about how individuals are removed from a fish population, about how they are introduced into it, and about the initial age distribution of the population. Assumption 1 The change in number of fish of age a and sex g at time t in a time interval of length Dt is lim

Dt “ 0

!

Ng (a+ Da, t+ Dt) − Ng (a, t) cg (a, t) = −mg (a, t)Ng (a, t)− Dt dg (a, t)Ng (a, t)

if (catch) , if (effort)

g=f, m

where mg (a, t)]0, dg (a, t) ]0 and cg (a, t) ]0 are, respectively, the instantaneous rates of natural mortality, fishing mortality, and catch of fish of age a and sex g at time t; ‘if (catch)’ denotes conditioning on catch and ‘if (effort)’ denotes conditioning on fishing effort. Expansion of Ng (a+Da, t+ Dt) in the neighbourhood of (a, t), passing to the limit Dt “ 0, and assuming that (da)/(dt) = 1 give lim

Dt “ 0

Ng (a+ Da, t+ Dt) − Ng (a, t) (Ng (a, t) (Ng (a, t) = + Dt (a (t

which leads to

!

cg (a, t) (Ng (a, t) (Ng (a, t) + = −mg (a, t)Ng (a, t) − (a (t dg (a, t)Ng (a, t)

if (catch) , if (effort)

g= f, m.

Similarly, the change in biomass of fish of age a and sex g at time t in a time interval of length Dt is defined as

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Bg (a+Da, t+Dt) −Bg (a, t) w (a + Da, t+ Dt)Ng (a+Da, t+Dt)−wg (a, t)Ng (a, t) = lim g . Dt Dt Dt “ 0 Taylor series expansion of Bg (a + Da, t + Dt), Ng (a+ Da, t+ Dt) and wg (a+Da, t+Dt) in the neighbourhood of (a, t), passing to the limit Dt “0, and assuming that (da)/(dt)= 1 yield (Bg (a, t) (Bg (a, t) (Ng (a, t) (Ng (a, t) (wg (a, t) (wg (a, t) + = wg (a, t) + + Ng (a, t) + . (a (t (a (t (a (t This gives lim

Dt “ 0



n

!



(Bg (a, t) (Bg (a, t) cg (a, t)wg (a, t) + = − [mg (a, t) −rg (a, t)]Bg (a, t)− (a (t dg (a, t)Bg (a, t)



n

if (catch) , if (effort)

g=f, m

(2)

with the average relative growth rate of fish of age a and sex g at time t (wg (a, t) (wg (a, t) 1 rg (a, t)= + . (a (t wg (a, t) Notice how the growth of a fish individual enters the population dynamics model! Thus, density-dependent growth of animals can be readily studied by hypothesizing about rg (a, t) as a function of Ng (a, t) and/or Bg (a, t). Eq. (2) is a system of two nonhomogeneous first order partial differential equations if conditioning on catch, and a system of two homogeneous first order partial differential equations if conditioning on effort. Its left-hand side states that the total change in biomass of fish of age a and sex g at time t is the sum of change due to a change in time and that due to a change in age. Gains and losses of individual fish through processes such as diffusion and advection can also be readily incorporated on the right-hand side of Eq. (2) but, for analytical simplicity, this will not be done in the present paper.

n

Assumption 2

&

The birth rate in number of fish of sex g at time t from all females Ng (a0, t) satisfies, say, Ng (a0, t)=



Sg (a, t)b(a, t)Nf (a, t) da,

t ]t0, g= f, m

(3)

a0

where Sg (a, t)]0 is the average sex ratio of the progeny of all female fish of age a at time t with Sf (a, t)+ Sm (a, t)= 1; b(a, t) ] 0 is the average fecundity of all female fish of age a at time t; and both Sg (a, t) and b(a, t) capture the effects of fish age, time and density dependence. Different birth schedules and births from specific age-groups of female fish can be readily implemented or examined by specifying appropriate functional forms of Sg (a, t) and b(a, t). A special case of Eq. (3) is used widely in human demography, genetics, epidemiology and, recently, in fisheries science. Two formulations of more common use in fisheries science are those of Ricker (1954), and Beverton and Holt (1957). They are all special cases of a more general formulation Bg (a0, t) =G(a, t, Bg (a, t)). In the following, no assumption is made about the functional form of Ng (a0, t) or Bg (a0, t), however. Assumption 3 The initial age distribution of the population is assumed to be known: Bg (a, t0)=wg (a, t0)Ng (a, t0),

a ]a0, g = f, m.

(4)

In the following, I will not particularize the functional form of Ng (a, t0) or Bg (a, t0). Eqs. (1)–(4) complete the definition of the problem. Similar assumptions have been made and studied extensively in mathematical biology and human demography, and their solutions are fairly well known, although they have not been used in the fisheries literature.

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Because of the structural similarity between the model conditioning on catch and that conditioning on effort in Eq. (2), it suffices to derive models conditional on catch only, which give models conditional on effort if cg (a, t) is replaced by 0 and mg (a, t) by mg (a, t)+ dg (a, t). Now, suppose that the solution of Bg (a, t) of Eqs. (1)– (4) for conditioning on catch is known. Let a= t+ c, or c= a− t, then wc (t) = Bg (t+ c, t)

t ]tc =max(t0, a0 −c), g = f, m

for a fixed value of c R. Since Bg (a, t) satisfies Eq. (2), dwc (t) = − [mg (t+c, t) −rg (t + c, t)]wc (t) − cg (t + c, t)wg (t+c, t), dt which, upon integration, becomes

 &

t

wcˆ (t) = wc (tc ) exp − −

&

[mg (s + c, s) −rg (s + c, s)] ds

 &

tc t

t

cg (s +c, s)wg (s + c, s) exp −

tc

t] tc,

g= f, m





[mg (j+ c, j)− rg (j+ c, j)] dj ds,

t]tc, g=f, m.

(5)

sˆ

If a B t, then cB 0, − c\ 0, then tc =a0 −c =t − a+ a0; if a] t, then c ]0, − c5 0, then tc = t0. In other words

 &



t Á (a , t− a +a ) exp − [mg (s+ a− t, s)− rg (s+a− t, s)] ds B g 0 0 à t−a+a 0 à t t à à − t − a + a cg (s + a −t, s)wg (s + a −t, s) exp − s [mg (j+ a−t, j) 0 à à Bg (a,t)= Í − rg (j +a − t, j)] dj ds a B tBg (a −t+ t0, t0) à à t t à exp − [m (s + a −t, s) −r (s + a −t, s)] ds − c (s+a− t, s)w (s+a− t, s) g g g g à t0 t0 à t à exp − [m (j +a − t, j) − r (j +a −t, j)] dj ds a ]t g g Ä s

&

 &



 &  &

 & 

(6)

Eq. (6) gives the biomass of fish of age a and sex g at time t and tracks down the fate of fish born before (a] t) and after (aB t) exploitation. From Eq. (6), we have

 &



t + Dt

Á B (a , t − a + a ) exp − [mg (s+a− t, s)− rg (s+a− t, s)] ds 0 à g 0 t−a+a 0 à t + Dt t + Dt Ã− cg (s + a −t, s)wg (s+a− t, s) exp − [mg (j+ a−t, j) à t−a+a 0 s à à − r (j +a − t, j)] dj ds a BtB (a−t+ t , t ) g g 0 0 à Bg (a +Dt, t+Dt)= Í t + Dt à exp − [mg (s + a −t, s)− rg (s+a− t, s)] ds à t0 à t + Dt Ã− cg (s + a −t, s)wg (s+ a− t, s) à t0 à t + Dt à exp − [mg (j +a − t, j)− rg (j+ a−t, j)] dj ds a ] t, g= g, m. Ä s

&

 & &  &



 &





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170

This gives the biomass of fish of age a + Dt and sex g at time t+ Dt. Comparison of this equation with Eq. (6) yields

 &

t + Dt

Bg (a +Dt, t+Dt)=Bg (a,t) exp − −

&

[mg (s + a− t, s)− rg (s+a− t, s)] ds



st. t + Dt

cg (s + a −t, s)wg (s + a− t, s)

 &

t

t + Dt

exp −



[mg (j +a − t, j)− rg (j+ a−t, j)] dj ds

(7)

g =f, m

s

which gives the biomass of fish of age a +Dt and sex g at time t+ Dt as a function of the biomass of fish of previous age a and sex g at time t. If conditioning on catch, the catch in biomass of fish of age a and sex g over the time interval [t, t+ Dt] is given by Cg (a, t; a +Dt, t+Dt) =

&

t + Dt

cg (s + a −t, s)wg (s +a− t, s) ds,

(8)

g= f, m.

t

It is interesting to note that cg (a, t)wg (a, t) enters Eq. (7) as its negatively exponentially weighted sum not as the total catch tt + Dt cg (s + a −t, s)wg (s + a −t, s) ds. Since 05mg (a, t) and 05 cg (a, t)wg (a, t), t5 s5 j5 t + Dt, then

&

t + Dt

t

]

&

cg (s+a −t, s)wg (s + a −t, s) ds

 &

t + Dt

cg (s +a− t, s)wg (s + a −t, s) exp −

t

t + Dt



[mg (j+ a−t, j)− rg (j+ a−t, j)] dj ds.

sˆ

Then, if the total observed catch is used in Eq. (7), a model bias results, the degree of which depends on the functional forms and values of cg (a, t), wg (a, t), mg (a, t), as well as on values of a, t and Dt. For mg (a, t)Dt B0.1, the bias can, however, be below 5%. Eqs. (6)–(8) make up the general biomass-based age-dependent model conditional on catch. If conditioning on effort, i.e. if cg (a, t) is replaced by 0 and mg (a, t) by mg (a, t)+ dg (a, t), then Eqs. (6) – (8) become the general biomass-based age-dependent model conditional on effort, with

 &  & t

[mg (s+ a− t, s)+ dg (s+a− t, s)− rg (s+a− t, s)] ds Á Bg (a0, t− a +a0) exp − t−a+a 0 à t à Bg (a,t)= Í a BtBg (a − t +t0, t0) exp − [mg (s + a− t, s)+ dg (s+a− t, s)− rg (s+a− t, s)] ds t0 à à a]t, g = f,m Ä

 &

Bg (a +Dt, t+Dt)=Bg (a, t) exp −

t + Dt







(9)

[mg (s + a− t, s)+ dg (s+a− t, s)− rg (s+a− t, s)] ds , g

t

= f, m, and

(10)

Y. Xiao / Ecological Modelling 128 (2000) 165–180

&

171

 &

Cg (a, t; a +Dt, t+Dt) s Á t + Dt à dg (s+a −t, s)Bg (a0, t −a +a0) exp − [mg (j+ a−t, j)+ dg (j+ a−t, j) t−a+a 0 à t t + Dt à = Í − rg (j+a −t, j)] dj ds a B t dg (s + a− t, s)Bg (a−t+ t0, t0) t à s à à exp − [mg (j+ a −t, j) + dg (j +a − t, j)− rg (j+ a−t, j)] dj ds a ] t, g= f, m. t0 Ä



 &

&



(11)

Note that various age-dependent estimation procedures such as Virtual Population Analysis (De Lury, 1947; Gulland, 1965; Murphy, 1965; Zhang and Sullivan, 1988) and Catch at Age Analysis (Doubleday, 1976; Paloheimo, 1980; Fournier and Archibald, 1982; Deriso et al., 1985; Paloheimo and Chen, 1993) result immediately from manipulating Eqs. (9) – (11).

3. Production models in differential/difference equations In this section, I deal mainly with production models in differential equations, from which those in difference equations result trivially. Eqs. (1)–(4) can be cast as an age-dependent production model in a system of integral equations (Xiao, 1997). By integrating both sides of Eq. (2) with respect to a from a0 to and by applying Eq. (1) and Bg ( , t)=0, I now cast them as an age-dependent production model in a differential equation

& &

&

Á Ã B: g (t) =Bg (a0, t)− [mg (a, t) −rg (a, t)]Bg (a, t) da− Í a0 Ã Ä



a0

cg (a, t)wg (a, t) da

if (catch)

dg (a, t)Bg (a, t) da

, g= f,m, if (effort)

a0

(12) solution of which as an initial value problem with Bg (t) t = t 0 = Bg (t0) yields an age-dependent depletion model Bg (t) =Bg (t0)+

&& &&

Á Ã −Í Ã Ä

&

t0

t



t0 t

a0

t0

t

&& t

Bg (a0, s)ds −

t0



[mg (a, s) − rg (a, s)]Bg (a, s) dads

a0

cg (a, s)wg (a, s) dads

if (catch)

dg (a, s)Bg (a, s) dads

if (effort)

,

g= f,m.

(13)

a0

Notice that, for data analysis, Eq. (12) or (13) must be coupled with Eq. (6) or (7), which in turn necessitates a proper specification of (1) the initial age distribution of a fish population Bg (a, t0) (i.e. numbers or biomasses of fish of age 0, 1, 2,…, before the start of the fishery); (2) the instantaneous rate of fishing mortality dg (a, t) and fish catch cg (a, t); and (3) the birth rate at time t Bg (a0, t). Readers intending to use an age-dependent production model both in a system of integral equations and in

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differential equations, and an age-dependent depletion model in their work may wish to consult Xiao (1997) for many ‘subtleties in, and practical problems with, the use of production models in fish stock assessment’, and for their solutions. In that work, he gives, for example, a maximum entropy specification of the initial age distribution of a fish population and formulae for calculating the instantaneous rate of fishing mortality dg (a, t) and fish catch cg (a, t). Under appropriate assumptions, Eq. (12) can be reduced to commonly used and numerous potentially useful production models, including simple production models and models of Deriso type, which need neither Eq. (6) nor (7) for data analysis. It must be noted that there might be an infinite number of ways for such a reduction; only one way will be given here for each model.



Bg (a0, t)= ag (t)Bg (t) 1 −

If

ag (t) − mg (t) + rg (t) Bg (t) ag (t)Kg (t)

n

where ag (t) and Kg (t) are recruitment parameters of fish of sex g at time t, mg (a, t)= mg (t) and rg (a, t)=rg (t), then Eq. (12) conditional on catch becomes the generalized Schaefer (1954) production model conditional on catch



B: g (t) =[ag (t)−mg (t) +rg (t)]Bg (t) 1 −



If Bg (a0, t)= ag (t)Bg (t) 1 −

n &

Bg (t) − Kg (t)



cg (a, t)wg (a, t) da,

a0

g= f,m.

n

ag (t) − mg (t) − dg (t) +rg (t) Bg (t) ag (t)Kg (t)

where ag (t) and Kg (t) are recruitment parameters of fish of sex g at time t, mg (a, t)= mg (t), rg (a, t)=rg (t) and dg (a, t)= dg (t), then Eq. (12) conditional on effort becomes the generalized Schaefer (1954) production model conditional on effort



B: g (t) =[ag (t)−mg (t) −dg (t) + rg (t)]Bg (t) 1 −

n

Bg (t) , Kg (t)

g= f,m.

If Bg (a0, t)= {ag (t) log(Kg (t)) −[ag (t) − mg (t) + rg (t)] log(Bg (t))}Bg (t) where ag (t) and Kg (t) are recruitment parameters of fish of sex g at time t, mg (a, t)=mg (t) log(Kg (t)) and rg (a, t)=rg (t) log(Kg (t)), then Eq. (12) conditional on catch becomes the generalized Fox (1975) production model conditional on catch B: g (t) =[ag (t)−mg (t) +rg (t)]Bg (t) log

  & Kg (t) − Bg (t)



cg (a, t)wg (a, t) da,

g=f,m.

a0

If Bg (a0, t)={ag (t) log(Kg (t)) −[ag (t) − mg (t) −dg (t)+ rg (t)] log(Bg (t))}Bg (t), where ag (t) and Kg (t) are recruitment parameters of fish of sex g at time t, mg (a, t)= mg (t) log(Kg (t)), rg (a, t)=rg (t) log(Kg (t)) and dg (a, t)= dg (t) log(Kg (t)), then Eq. (12) conditional on effort becomes the generalized Fox (1975) production model conditional on effort B: g (t) =[ag (t)−mg (t) −dg (t) + rg (t)]Bg (t) log If

!

 

Bg (a0, t)= Bg (t) ag (t) −[ag (t) − mg (t) + rg (t)]

Kg (t) , Bg (t)

g= f,m.

 " Bg (t) Kg (t)

p

where ag (t), Kg (t) and p are recruitment parameters of fish of sex g at time t, mg (a, t)=mg (t) and rg (a, t)=rg (t), then Eq. (12) conditional on catch becomes the generalized Pella and Tomlinson (1969) production model conditional on catch

Y. Xiao / Ecological Modelling 128 (2000) 165–180

  n &

B: g (t) =[ag (t)−mg (t) +rg (t)]Bg (t) 1 −

!

Bg (t) Kg (t)

p

173



cg (a, t)wg (a, t) da,

−

a0

If Bg (a0,t)= Bg (t) ag (t) − [ag (t) − mg (t) − dg (t) + rg (t)]

g= f,m.

 " Bg (t) Kg (t)

p

where ag (t), Kg (t) and p are recruitment parameters of fish of sex g at time t, mg (a, t)=mg (t), rg (a, t)=rg (t) and dg (a, t) =dg (t), then Eq. (12) conditional on catch becomes the generalized Pella and Tomlinson (1969) production model conditional on effort

  n

B: g (t) =[ag (t)−mg (t) −dg (t) + rg (t)]Bg (t) 1 −

Bg (t) Kg (t)

p

,

g= f,m.

I now derive four new production models from Eq. (12); model 1 includes models 2–4 as special cases. If mg (a, t)=mg (t), rg (a, t) =rg (t) and dg (a, t) =dg (t), then Eq. (12) becomes

&

Á cg (a, t)wg (a, t) da à B: g (t) =Bg (a0, t)− [mg (t) − rg (t)]Bg (t) − Í a 0 Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g= f, m,

solution of which as an initial value problem with Bg (t) t = t 0 = Bg (t 0) yields

 &   &

Bg (t)

t

 &

&

t

n  &



t

Á Bg (t0) exp − [mg (s) − rg (s)] ds + Bg (a0, s)− cg (a, s)wg (a, s) da exp − [mg (j) à t0 t0 a0 s à à − r (j)] dj ds if (catch) g =Í Ã t t t à B (t ) exp − [m (s) + d (s) − r (s)] ds + B (a , s) exp − [m (j)+ d (j)− r (j)] dj ds g 0 g g g g 0 g g g à t0 t0 s Ä if (effort)

 &

 &



If Bg (a0, t)= ag (t)Bg (t)/(bg (t) + Bg (t)) (Beverton and Holt, 1957), where ag (t) and bg (t) are recruitment parameters of fish of sex g at time t, mg (a, t) =mg (t), rg (a, t)= rg (t) and dg (a, t)= dg (t), then Eq. (12) becomes



n &

Á cg (a, t)wg (a, t) da à ag (t) B: g (t) =Bg (t) − mg (t) + rg (t) − Í a 0 bg (t)+ Bg (t) Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g=f,m

If Bg (a0,t)= Bg (t)e ag (t)[1-Bg (t)/bg (t)] (Ricker, 1954), where ag (t) and bg (t) are recruitment parameters of fish of sex g at time t, mg (a, t) =mg (t), rg (a, t) =rg (t) and dg (a, t)=dg (t), then Eq. (12) becomes

&

Á cg (a, t)wg (a, t) da à B: g (t) =Bg (t){e ag (t)[1 − Bg (t)/bg (t)] −mg (t) + rg (t)} − Í a 0 Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g= f,m

If Bg (a0, t)= ag (t)Bg (t)[1 −bg (t)gg (t)Bg (t)] − gg (t) (Deriso, 1980), where ag (t), bg (t) and gg (t) are recruitment parameters of fish of sex g at time t, mg (a, t)= mg (t), rg (a, t)=rg (t) and dg (a, t)= dg (t), then Eq. (12) becomes

Y. Xiao / Ecological Modelling 128 (2000) 165–180

174

&

Á cg (a, t)wg (a, t) da à B: g (t) =Bg (t){ag (t)[1−bg (t)gg (t)bg (t)] − gg (t) −mg (t)+rg (t)}− Í a 0 Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g

= f,m. I now derive production models in differential equations with a time delay, which may occur between fish spawning and recruitment. If Bg (a0, t)= [ag (t)− mg (t) +rg (t)] Bg (t)



n

ag (t) B (t) B (t−v) − g + g , ag (t) − mg (t)+ rg (t) Kg (t) bg (t−v)

where ag (t), Kg (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t) and rg (a, t) =rg (t), and dg (a, t) = dg (t)then Eq. (12) conditional on catch becomes the generalized Schaefer (1954) – Walter’s (1973) production model conditional on catch



B: g (t) =[ag (t)−mg (t) +rg (t)]Bg (t) 1 −

n &

Bg (t) Bg (t−v) + − Kg (t) bg (t−v)

If Bg (a0, t)=[ag (t) −mg (t) − dg (t) + rg (t)] Bg (t)





cg (a, t)wg (a, t) da,

g= f,m.

n

a0

ag (t) B (t) B (t−v) − g + g , ag (t)− mg (t)− dg (t)+ rg (t) Kg (t) bg (t−v)

where ag (t), Kg (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t),=mg (t), rg (a, t) =rg (t), and dg(a,t) =dg(t), then Eq. (12) conditional on effort becomes the generalized Schaefer (1954) – Walter’s (1973) production model conditional on effort



B: g (t) =[ag (t)−mg (t) −dg (t) + rg (t)]Bg (t) 1 − If Bg (a0,t)= [ag (t)− mg (t) + rg (t)]

!

n

Bg (t) Bg (t−v) + , Kg (t) bg (t−v)

g= f,m.

"

ag (t) log(Kg (t)) − log (Bg (t))+ bg (t−v)Bg (t−v) Bg (t), ag (t) − mg (t) +rg (t)

where ag (t), Kg (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t) log(Kg (t)) and rg (a, t) =rg (t) log(Kg (t)), then Eq. (12) conditional on catch becomes the generalized Fox (1975)-Walter’s (1973) production model conditional on catch B: g (t) =[ag (t)−mg (t) +rg (t)]Bg (t) log



 &

Kg (t)B bg g (t − v)(t−v) − Bg (t)

If Bg (a0,t)=[ag (t)−mg (t) − dg (t) + rg (t)]

!



cg (a, t)wg (a, t) da,

a0

g=f,m.

"

ag (t) log(Kg (t)) − log(Bg (t))+ bg (t−v)Bg (t−v) ag (t) −mg (t)− dg (t)+ rg (t)

Bg (t) where ag (t), Kg (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t) log(Kg (t)), rg (a, t) =rg (t) log(Kg (t)) and dg (a, t)=dg (t) log(Kg (t)), then Eq. (12) conditional on effort becomes the generalized Fox (1975)-Walter’s (1973) production model conditional on effort B: g (t) =[ag (t)−mg (t) −dg (t) + rg (t)]Bg (t) log





Kg (t)B bg g (t − v)(t−v) , Bg (t)

g= f,m.

Numerous other production models with one or more time delays can be derived trivially from Eq. (12). I will now, however, give three new production models in differential equations, each with a single time delay, based respectively on Beverton and Holt’s (1957), Ricker’s (1954) and Deriso’s (1980) recruitment functions. If

Y. Xiao / Ecological Modelling 128 (2000) 165–180

175

ag (t−v)Bg (t − v) bg (t−v) + Bg (t − v) where ag (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t), rg (a, t) =rg (t) and dg (a, t) =dg (t), then Eq. (12) becomes

Bg (a0, t)=

&

Á cg (a, t)wg (a, t) da à ag (t−v)Bg (t − v) −[mg (t) − rg (t)]Bg (t)− Í a 0 B: g (t) = bg (t−v)+ Bg (t − v) Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g= f,m.

If Bg (a0, t)=Bg (t−v)e ag (t − v)[1 − Bg (t − v)/bg (t − v)], where ag (t) and bg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t), rg (a, t)= rg (t) and dg (a, t)=dg (t), then Eq. (12) becomes B: g (t) =Bg (t−v)e ag (t − v)[1 − Bg (t − v)/bg (t − v)] −[mg (t)− rg (t)]Bg (t)

&

Á cg (a, t)wg (a, t) da à − Í a0 Ãdg (t)Bg (t) Ä

if (catch) , g =f,m. if (effort)

If Bg (a0, t)= ag (t−v)Bg (t − v)[1 − bg (t − v)gg (t−v)Bg (t−v)] − gg (t − v), where ag (t), bg (t) and gg (t) are recruitment parameters and v is the time delay, all of fish of sex g at time t, mg (a, t)=mg (t), rg (a, t)=rg (t) and dg (a, t) =dg (t), then Eq. (12) becomes B: g (t) =ag (t− v)Bg (t − v)[1 −bg (t − v)gg (t − v)Bg (t−v)] − gg (t − v) − [mg (t)− rg (t)]Bg (t)

&

Á cg (a, t)wg (a, t) da à − Í a0 Ãdg (t)Bg (t) Ä

if (catch) , g =f,m. if (effort)

I now derive a new production model in a system of two differential equations and give its explicit analytical solution. If mg (a, t) =mg (t), w; g (a, t) =ag (t)+ rg (t)wg (a, t), i.e. ag (t) a (t) exp − rg (t)(a − a0) − g , wg (a, t)= wg (a0, t)+ rg (t) rg (t) i.e. 1 rg (t) = {a (t)+ rg (t)wg (a, t)} wg (a, t) g and dg (a, t)=dg (t), then Eq. (12) becomes



n

&

Á cg (a, t)wg (a, t) da à B: g (t) =Bg (a0, t)− [mg (t) − rg (t)]Bg (t) + ag (t)Ng (t) − Í a 0 Ãdg (t)Bg (t) Ä

if (catch) , if (effort)

g=f,m; (14a)

if mg (a, t)= mg (t), wg (a, t) =1 (i.e. rg (a, t) =0), and dg (a, t)= dg (t), then Eq. (12) becomes

&

Á cg (a, t) da if (catch) Ã , N: g (t) =Ng (a0, t)− mg (t)Ng (t) − Í a 0 Ã dg (t)Ng (t) if (effort) Ä with a solution

g=f,m;

(14b)

176

Y. Xiao / Ecological Modelling 128 (2000) 165–180

 &

 &

&

n

t t Á à Bg (t0) exp − [mg (s) − rg (s)] ds + Bg (a0, s)+ ag (s)Ng (s)− cg (a, s)wg (a, s) da t0 t0 a0 à t t à Bg (t) = Í exp − [mg (j) −rg (j)] dj ds if (catch)Bg (t0) exp − [mg (s)+ dg (s)− rg (s)] ds s t0 à t t à à + [Bg (a0, s) + ag (s)Ng (s)] exp − [mg (j)+ dg (j)− rg (j)] dj dsif (effort), g=f,m, and t0 s Ä (15a)

 & &



 &  &

Á ÃNg (t0) exp − Ng (t) = Í Ã Ng (t0) exp − Ä

t

t0 t

 &

 &

mg (s) ds +

 &

t

t0

Ng (a0, s)−

 &

[mg (s) + dg (s)] ds +

t0

&





n  &  &

cg (a, s) da exp −

a0

t



t

s

t

Ng (a0, s) exp −

t0

 

mg (j) dj ds

if (catch)

[mg (j)+ dg (j)] dj ds

s

if (effort), g = f,m.

(15b)

By differentiating both sides of Eq. (14a) with respect to t, substituting Eq. (14b) into the resulting equation, and finally substituting Ng (t) solved from Eq. (14a) into the further resulting equation, that system of two differential equations (Eq. 14a,b) can also be converted to a one order higher differential equation for Bg (t) only B8 g (t) =

Á Ã Ã Ã Í Ã Ã Ã Ä



n



!

"

n

a; g (t) a; g (t) −2mg (t) +rg (t) B: g (t)+ r; g (t)− m; g (t)+ − mg (t) {mg (t) −rg (t)} Bg (t) +B: g (a0, t) ag (t) ag (t)



+

n

a; g (t) ag (t) − +mg (t) Bg (a0, t)+ wg (a0, t) ag (t)



n

& !

a0

"

n

a; g (t) d − mg (t) cg (a, t)wg (a, t)− ag (t)cg (a, t)− {cg (a, t)wg (a, t)} da if (catch) ag (t) dt



!

"

n

a; g (t) a; g (t) −2mg (t) −2dg (t)+ rg (t) B: g (t)+ r; g (t)− m; g (t)− d: g (t) + −mg (t) −dg (t) {mg (t) +dg (t) −rg (t)} Bg (t) ag (t) ag (t)

+B: g (a0, t) +



n

a; g (t) ag (t) − + mg (t)+ dg (t) Bg (a0, t) if (effort), g=f,m wg (a0, t) ag (t)

This equation can in turn be written as a difference equation in several ways. Its forward (X8 (t)= [X(t+ 2Dt) − 2X(t+ Dt)+ X(t)]/(Dt 2) and X: (t) =[X(t + Dt) −X(t)]/Dt) difference equation is

Bg (t +2Dt)=

!

g1Bg (t +Dt) +g2Bg (t) + g3 g4Bg (t +Dt) +g5Bg (t) + g6

if (catch) , if (effort)

(17)

g= f,m,

with g1 = g2 =

ag (t+ Dt) +[rg (t) − 2mg (t)]Dt +1 ag (t)

!

n

"

a (t+Dt) ag (t+Dt) −mg (t)Dt + 1 [mg (t) − rg (t)] + rg (t+Dt)− mg (t+Dt)+mg (t) Dt − g ag (t) ag (t)

! & !

g3 = Dt Bg (a0, t+ Dt) +

177

n

ag (t)Dt ag (t + Dt) − +mg (t)Dt Bg (a0,t) wg (a0, t) ag (t)

n

ag (t+Dt) −mg (t)Dt + 1 cg (a, t)wg (a, t) − ag (t)cg (a, t)Dt − cg (a, t+ Dt)wg (a, t) ag (t)



+



Y. Xiao / Ecological Modelling 128 (2000) 165–180

a0

" "

− cg (a, t)wg (a, t +Dt) da

ag (t+ Dt) +[rg (t) − 2mg (t) − 2dg (t)]Dt +1 ag (t) ag (t+Dt) g5 = −mg (t)Dt − dg (t)Dt + 1 [mg (t) + dg (t)− rg (t)]+ rg (t+Dt)−mg (t+Dt)−dg (t+Dt) ag (t) g4 =

!

"

n

ag (t + Dt) ag (t) ag (t)Dt ag (t + Dt) g6 = Dt Bg (a0, t+ Dt) + − +mg (t)Dt +dg (t)Dt Bg (a0, t) . wg (a0, t) ag (t) Now, it is abundantly clear that that system of differential equations (Eqs. 14a,b) or its solution (Eqs. 15a,b) is much easier to use in studies of fish population dynamics than that one order higher differential equation for Bg (t) (Eq. 16) or any of its difference equations (e.g. Eq. 17). Deriso’s (1980) system of difference equations and his delay difference equation can both be derived from Eq. (12), that system of differential equations (Eqs. 14a,b) or its solution (Eqs. 15a,b), all conditional on effort. Because of its assumptions (e.g. fish recruitment occurs at the end of the time period), it is easier to start from equation 15. Before proceeding, notice that Deriso’s (1980) growth model w(a+ 1)= a+ rw(a) has its solution w(a) =[w(a0) + a/(r − 1)]r a − a0 − a/(r −1) and its differential equation: a log(r) w; (a) = +log(r)w(a). r−1 Now, we need to use a more general form of this differential equation log(rg (t)) w; g (a, t)=ag (t) + log(rg (t))wg (a, t) rg (t)− 1 or 1 log(rg (t)) ag (t) rg (a, t)= + log(rg (t))wg (a, t) . wg (a, t) rg (t) − 1 Also, notice that + mg (t)+ dg (t) Dt −

!



n

!

 &

t

S(t) = exp −

"

"

[mg (s) + dg (s)] ds



t0

&



&

 &



is the so-called survival rate over the time interval [t0, t], t

exp

t0

&

log(rg (s)) ds :r tg− t0(t0),

t

t0

n  &

Bg (a0, s) exp −

t

[mg (j)+ dg (j)− log(rg (j))] dj ds :Bg (a0, t)

s

 

implies that fish recruitment occurs at the end of the time interval [t0, t], i.e. precisely at time t, and finally t t log(rg (s)) log(rg (t)) ag (s) Ng (s) exp − [mg (j) +dg (j)− log(rg (j))] dj ds: ag (t) N (t) r (s)− 1 rg (t)− 1 g t0 s g

n

: ag (t)Ng (t): ag (t0)Ng (t) = ag (t0)Sg (t0)Ng (t0).

Y. Xiao / Ecological Modelling 128 (2000) 165–180

178

Under these assumptions, dropping the subscript g for sex throughout and letting t0 = t−Dt, Eqs. 15a,b conditional on effort becomes Deriso’s (1980) system of difference equations

!

B(t)=a(t− Dt)S(t −Dt)N(t −Dt) +S(t − Dt)r(t− Dt)B(t− Dt)+ B(a0, t) N(t)= S(t− Dt)N(t − Dt) + N(a0, t)

with a solution t − t0 −1 Dt

5

N(t) =N(t0)

t − t0 −1 Dt

S(t0 +iDt) + 5

i=0 t − t0

B(t) = B(t0) t − t0 Dt

Dt

i

N(a0, t − iDt) 5 S(t− rDt)

i=0

r=1

−1

5

S(t0 +iDt)r(t0 +iDt)

i=0 −1

i

[S(t− (i+1)Dt)a(t −(i +1)Dt)N(t− (i+1)Dt)+ B(a0, t−iDt)] 5 S(t− rDt)r(t− rDt).

+ 5 i=0

r=1

This system of difference equations can be readily converted to a one order higher difference equation (Deriso, 1980)



B(t) = r(t−Dt)+

n

a(t −Dt) S(t − Dt)B(t − Dt) a(t −2Dt)

a(t−Dt) r(t− 2Dt)S(t − Dt)S(t −2Dt)B(t −2Dt) a(t−2Dt) w(a0, t− Dt) + 1− a(t − Dt)S(t −Dt)N(a0, t −Dt) a(t− 2Dt) +B(a0, t). −



n

Now, I turn to that age-dependent depletion model (Eq. (13)). Under three or four assumptions, it can be reduced to all existing depletion models, which need neither Eq. (6) nor 7 for data analysis. If Bg (a0, t)= 0, mg (a, t) = 0, and rg (a, t) =0, then Eq. (13) conditional on catch becomes Leslie depletion model (Leslie and Davis, 1939)

&& t

Bg (t) =Bg (t0)−

t0



cg (a, s)wg (a, s) dads, g = f,m.

a0

If Bg (a0, t)=0, mg (a, t) =0, rg (a, t) =0 and dg (a, t)= dg (t), then Eq. (13) conditional on effort becomes the generalized De Lury (1947) model

 &

t

Bg (t) =Bg (t0) exp −



dg (s) ds ,

g =f, m.

t0

4. Discussion The age-dependent production model in a differential/difference equation (Eq. (12)) is fairly general and is deducible, often trivially, to numerous production models by specifying appropriate functional forms of its various quantities, such as Bg (a0, t) and rg (a, t). In this sense, it is the fundamental production model. Similarly, the age-dependent depletion model (Eq. (13)) is the fundamental depletion model. Both are dual to (and complement) each other and can be derived from the other (see above); they make up a general theory of production models, and, together with the general age-dependent model (Eqs. (6)–(11)), provide a general theory of fish stock assessment models.

Y. Xiao / Ecological Modelling 128 (2000) 165–180

179

Although seemingly complicated, all the models developed above are as simple as existing ones and can be readily implemented in discrete form by replacing their integration signs with signs of summation. I have not provided applications, because this work deals with all stock assessment models, the choice of examples would be subjective, and the manuscript would be too long, even if a single example for an age-dependent production model is included. In fact, as I will demonstrate elsewhere, an application of an age-dependent production model conditional on catch would be sufficient for a paper! This work can be extended in at least three useful ways. First, this paper deals with stock assessment models essentially for a single species. I am developing a general theory of multi-species stock assessment models, which will describe multi-species age-dependent models, multi-species age-dependent production models, multi-species age-dependent depletion models, multi-species age-dependent estimation procedures and some of their useful special cases. Second, one might consider developing stochastic process models. Unfortunately, most fisheries biologists and ‘modellers’ might be unprepared mathematically for this development. A final useful extension might be to incorporate gains and losses of fish through such processes as diffusion and advection. However, this extension generally results in complicated models, many properties of which have not been investigated. Biological data may also be too limited to make their application truly successful. Nonetheless, such models are essential for understanding the population dynamics of some fishes, especially highly migratory species. Finally, there is a need to clarify terminology in stock assessment models. Because simple production models, models of Deriso type and depletion models are all deducible from an age-dependent model under three or four assumptions, their boundaries have now completely disappeared and their classification is no longer adequate. For example, what would those seven new production models be called? In their derivation above, I called them production models. Obviously, they are simple production models, for they are very similar in structure to previous simple production models. Equally obviously, the first production model can be reduced to the remaining six (see above), models of Deriso type, or depletion models, to which then they should all belong. To overcome this difficulty in classifying them properly, I propose that a production model be named after its originator, as is usual in science, so that we have the Schaefer (1954) production model, Deriso (1980) production model,… Nice and clear.

Acknowledgements I wish to thank Drs You-Gan Wang (formerly of CSIRO IPP&P Biometrics Unit), David MacDonald, and Vivienne Mawson, (CSIRO Division of Fisheries) for commenting on the manuscript. Drs John D. Stevens and Anthony D. M. Smith (CSIRO Division of Fisheries) are thanked for supervising this project. This work, completed on March 10, 1996, was the topic of three seminars, at NSW Fisheries Research Institute (July 2, 1996), Victorian Fisheries Research Institute (July 22, 1996) and the University of Queensland (July 31, 1996). A poster was also given at the Second World Fisheries Congress at Brisbane (July 28 to August 2, 1996).

References Beverton, R.J.H., Holt, S.J., 1957. On the dynamics of exploited fish populations. Fishery Investigations, London, Ser. 2, 19: 533 pp. Box, G.E.P., Jenkins, G.M., 1970. Time series analysis: forecast and control. Holden-Day, San Francisco, CA. xix +553 pp. De Lury, D.B., 1947. On the estimation of biological populations. Biometrics 3, 145 – 167. Deriso, R.B., 1980. Harvesting strategies and parameter estimation for an age-structured model. Can. J. Fish. Aquat. Sci. 37, 268 – 282. Deriso, R.B., Quinn II, T.J., Neal, P.R., 1985. Catch-age analysis with auxiliary information. Can. J. Fish. Aquat. Sci. 42: 815 – 824.

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