Concept of exponential decay model

A quantity is subject to exponential decay, if it decreases at a rate proportional to its current value, time or space. In fish biology the most useful way to express decay through the time of a group of fish born at same time is instantaneous rate of total mortality. For estimation of mortality rate, the number of survivors in a cohort as a function of time has to be determined. The number of survivors attaining age t is denoted as N(t). The mortality in a fish stock occurs due to the natural or fishing mortality.

Where,

∆Ν =Change in number, ∆Τ = Change in time, Z=Instantaneous rate of total mortality=Total mortality rate= Total mortality coefficient and S =Survival rate,

S= e-Z or ln S = - Z (survival rate) or Z= - ln S (total mortality rate)

The annual rate of total mortality (A) can be calculated as,

A=1 -S, or A= 1-e-Z

As an example, consider the number of survivors at age t = 0.5 year, N(0.5), and the number of survivors one day later, N(0.50274) (1 day = 1/365 year = 0.00274 year). The number of specimens lost during that day is: N(0.5) - N(0.50274). To designate the change in numbers during a relatively short time period (1 day) we use the symbol ∆Ν; ∆Ν(0.5) = N(0.50274) - N(0.5). Note that ∆Ν is negative because it

represents a loss from the cohort. The rate of change in numbers is written: ∆Ν/∆Τ; where ∆Τ is the length of the time period (1 day in this case). If Z remains constant throughout the life of a cohort it can be expressed mathematically as:

This above equation is called as “Exponential Decay Model” and (together with the growth equation) it is a corner-stone of the theory of exploited fish stocks Baranov, 1918; Thompson and Bell, 1934; Fry, 1949 and Beverton and Holt, 1957).

∆Ν/∆Τ = e-Z=S

Ν(t)=N(Tr)*exp[-Z(t-Tr)]

Exponential decay curves, for Z = 0.2, 0.5, 1 and 2 per year, with recruitment, N(Tr) = 1000 fish

Dynamic pool models

The concept of the dynamic pool as a model for fish populations is complicated. The population consists of all individuals alive at any time. This population is continuously reduced in size by deaths, due to natural mortality or to fishing, and is augmented by recruitment of young fish referred as “dynamic pool”. Although, these are involved a dynamic balance between production (due to recruitment and individual growth) and depletion (due to mortality). These are usually described as a function of the age of the individuals involved, and there is therefore a close correspondence between the dynamic pool concept and age-structured methods of analysis and prediction.

The primary differences from the surplus models are

1. Dynamic pool models account for variable growth, mortality and reproductive potential by age

2. Currently used to examine reproduction and recruitment potential

Dynamic pool methods seek to estimate number of the fish population at a particular time from available data, and predict its future evolution under various assumptions about natural and anthropogenic effects, especially levels of fishing. The data to be analysed are generally some estimates of catch-at-age or length at age. Estimates of the rates of mortality due to natural causes and that due to fishing,

as well as their sum, the total mortality rate, are the natural variables used to describe the fate of the fish.

Within the general class of dynamic pool methods, there tends to be a fairly clear separation between the techniques for the analysis of past data, to estimate the current state and structure of the population, and those for forecasting its future evolution. The former are exemplified by the ubiquitous technique known as Virtual Population Analysis (VPA). The latter include both short-term catch forecast methods, and long-term analyses such as that of yield-per-recruit (Beverton and Holt, 1957). These models relatively complicated considering cohort abundance, weight of fish, mortality, growth, fishing effort, recruitment and yield.

The mathematical models used in fishery science analyses the history of a fishery which can be transformed in such a way that the knowledge of the past can be used to predict future yields and biomass at different levels of fishing effort. Thus, models can be used to forecast the effects of development and management measures such as increase or reduction of fishing fleets, changes in minimum mesh sizes, closed seasons, closed areas etc. The first prediction model was developed by Thompson and Bell (1934). In the mean time the Beverton and Holt’s (1957) model based on rigorous assumptions but requiring less calculations also called the “yield-per-recruit model” is being widely used. The yield per recruit model (Beverton and Holt, 1957) is a model describing the state of stock and the yield in a situation when the fishing pattern has been the same for such a long time that all fish alive have been exposed to it since they recruited hence, called as “steady state model”.

The relationship between recruitment and spawning is still not well understood. The only point understood is (0, 0) i.e., if there are no spawners there is no recruitment. It is reasonable to assume that at low levels of the parent stock there is a direct positive linear relationship with the number of offspring or recruitment, but when it occurs we have to assume that the parental stock comes down to such a level that recruitment over fishing can occur.

Assumptions of Beverton and Holt model (1957)

1. Recruitment is constant, yet not specified. Hence, the expression “yield per recruit”

2. All fish of a cohort are hatched on the same date

3. Recruit and selection are ‘knife-edge’

4. The fishing and natural mortalities are constant from the entry to the exploited phase

5. There is a complete mixing within the stock

6. The length-weight relationship has the exponent 3 i.e.,W= q*L3 or isometric

The assumed life history of a cohort in Beverton and Holt model is depends on exponential decay model. The parameters required are:

1. At age Tr the cohort recruits to the fishing ground, all at the same time: ‘knife-edge recruitment’.

2. From Tr to Tc they suffer only from natural mortality this is assumed to remain constant throughout the lifespan of cohort.

3. At age Tc, ‘the age at first capture’, the cohort is assumed to be suddenly exposed to full fishing mortality which is assumed remain constant for the rest of the cohort’s life. The sigmoid shaped gear selection curve called “knife-edge selection”.

4. The catch from the cohort therefore assumed to be zero before the cohort has attained the age Tr.

Life history of a cohort as assumed in the Beverton and Holt model

The numbers of survivors at age Tr is the recruitment to the fishery: R = N(Tr)The number of survivors at age Tc is : N(Tc) = R exp[ -M(Tc-Tr)]The number of survivor at age t, where t > Tc is:

N(t) = N(Tc) – exp[- (M+F) (t- Tc) = R exp[-M(Tc – Tr) – (M+F)(t – Tc)]

The fraction of total recruitment N(Tr) or R surviving until age t is obtained by dividing both sides of the equation by R and becomes:

N(t)/R = exp[ - M(Tc – Tr) – (M+F) (t – Tc)

It gives the number of fish at time t per recruit, i.e. as the fraction of each fish that recruited to the fishery.Beverton and Holt Yield Per Recruit (Y/R) model (1957)

Beverton and Holt (1957) developed this model describing the state of the stock and the expected yield in a situation where a given fishing pattern has been operating for a long time, i.e. under steady state condition. The definition of the recruits may vary among the authors; here, it has been given as…………

(1). fully metamorphosed young fish,

(2). whose growth is described adequately by the VBGF,

(3). whose instantaneous rate of the natural mortality remains same as in adults

(4). which occur at (or swim into) the fishing ground

To derive the mathematical expression for the model we take a starting point in the catch equation in the form of equation.

C(t, t+∆t) = ∆t F N(t) …………………………………………………………………………………..1

This equation gives the number of fish caught in time period from t to t + ∆t from a cohort. To get the corresponding yield in weight, this number should be multiplied by the individual weight of a fish. If ∆t is small , then body weight of a fish will remain approximately constant during the time period from t to t + ∆t, and the yield becomes:

Y(t, t+ ∆t) = ∆t *F *N(t) *w(t) ……………………………………………………………………………….…...2

Where w(t) is the body weight of a t years old fish, as defined by the weight converted Von Bertalanffy equation. To get the yield per recruit for the time period from t to t+∆t, Eqn.2 is divided by the no. of recruits, R:

Y(t, t+ ∆t)/R = F*N(t)/R *w(t) *∆t……………………………………………………………………..………...3

(Eqn.3 is the Beverton and Holt model for a short time period)

Where, N(t)/R = exp[ - M(Tc – Tr) – (M+F) (t – Tc)……………..……………………………..4

To get the total yield per recruit for the entire life span of the cohort, Y/R, all the small contributions defined by Eqn.3 must be summed:

Y/R = Y( Tc, Tc +∆ t)/R + Y(Tc+∆t, Tc + 2∆t)/R +Y(Tc+2∆t, Tc +3∆t)/R ………..+Y(Tc +(n-1)∆t, Tc+ n∆t)/R ………………………………………………………………….5

Where ‘n’ is some large number, so large that the number of fish older than Tc+n∆t, i.e. N(Tc+ n∆t), is so small that it can be ignored.

The next step is to convert the eqn.5 into a form which can easily be calculated. So, eqn. 5 can be converted as:

Where,

S= exp[-k(Tc- t0)]; K= VBG parameter; t0= age at length zero; Tc= age at first capture; Tr= age at recruitment; W∞= asymptotic body weight; F= fishing mortality; M= natural mortality; Z= F+M, Total mortality.

Equation 6, is the “Beverton and Holt yield per recruit model”(1957) which is written in the form suggested by Gulland (1969).

The two parameters F and Tc are those which can be controlled by fisheries managers, because F is proportional to effort, and Tc is a function of gear selectivity. Thus, Yw/R is considered as function of F and Tc. The Yw/R has a “maximum sustainable yield” which depends on age of the first capture Tc.

Result of a stock and yield assessment with the yield per recruit model

Yield per recruit curves with different ages of first capture (Tc)

If the M is high it is difficult to point out the Fmsy and the Yw/R curve runs parallel to the X-axis (see A).

Y/R = F * exp [-M*(Tc –Tr)] *W∞ *[ - – ] …..…..6

Yield per recruit as a function of F for the parameter :K = 0.37/yr, Tc = 1.0 yr, t0 = - 0.2 yearM = 1.1/yr Tr = 0.4 year , W∞= 286 gm.

A

Yield per recruit as a function of F for the parameter:A: M = 1.1/yr, B: M = 0.2/yrK = 0.37/yr, Tc = 1.0 yr, t0 = - 0.2 yearTr = 0.4 yr, W∞ = 286 grams

B

• In graph B, the curve B has a pronounced maximum and lower value of Fmsy

and higher values of MSY/R as compared to curve A. • Graph A, does not have maximum level, in such cases one must not conclude

that effort should be increased or decreased. Rather, it is the biomass per recruit which should be examined or another version of Y/R model should be used.

Beverton and Holt’s Relative Yield per Recruit (Y’/R) Model (1966)

For fisheries management purposes, it is important to be able to determine changes in the Y/R for different values of F, for example if F is increased by 20% then the yield may decrease by 15%. The absolute values of Y/R expressed in grams per recruit are not important for this purpose. Therefore, Beverton and Holt (1966) also developed a "relative yield per recruit model" which can provide the kind of information needed for management. This model has the great advantage of requiring fewer parameters, while it is especially suitable for assessing the effect of mesh size regulations. It belongs to the category of length-based models, because it is based on lengths rather than ages. The formula given is…

(Y’/R) = E * U M/K * [1- + – ]

The relative yield per recruit (Y’/R) is a function of U and E and the only parameter required here is M/K.

M = = K/Z

U =1–Lc/L∞, the fraction of growth to be completed after entry into exploited phase.E = F/Z, the exploitation rate. (U= F/Z*(1-e-Z) exploitation ratio)

Beverton and Holt’s relative yield per recruit (Y/R)’ curve corresponding to the Y/R-curve

• The relative yield per recruit (Y’/R) can be transformed into yield per recruit (Y/R) as

Y’/R = Y/R*exp[M*(Tr-to)/ W∞]

.

Beverton and Holt’s Biomass Per Recruit (B/R) model

Beverton and Holt's biomass per recruit model expresses the annual average biomass of survivors as a function of fishing mortality (or effort). The average biomass is related to the catch per unit of effort. The relationship between CPUE and numbers caught, which multiplied by the body weight on both sides gives following equation.

CPUE (t) = q * N (t) • CPUE is the weight of catch per unit of effort.

CPUE (t) * W(t) = q * W (t) • multiplied by the body weight on both sides

CPUE W(t) = q * B (t) • If N (t) * W (t) is replaced by B (t), the symbol of biomass

The catch in number / year can be expressed as C = F * and yield per year as Y = F * B’ ; Where, B’ is the average biomass in sea during a year. Where average biomass ,B’ follows that, B/R = Y/R*1/F. Because of the assumption of a constant parameter system the yield from a stock during one year is equal to the yield from a single cohort during its life span.

Therefore we have the following simple relationship between Y /R, and average biomass per recruit (B’/R) as: Y/R = F* B’/R

B’/R can be calculated as:

(B’/R) = exp[-M(Tc-Tr)] * W∞ * [ ]

Biomass curve of (CPUE in weight) as a function of effort

Advantages of Y/R models

Both F and M are explicit in the model

Increased biological realism

Avoid having to address year-to-year variation in recruitment

Can see effects of F and Age of Entry on age and size in the catch

Disadvantages/ Limitations of Y/R

• assume constant recruitment

• This assumes age-structure remains stable

• Ignore any temporal variation in F and M

• Stable environment

• No density-dependence in growth and mortality.

• Yield-per-Recruit is good for determining if ‘growth overfishing’ is occurring.

But, since the models assume constant recruitment, they can’t detect ‘recruitment overfishing’. One of the serious drawbacks of the yield per recruit model is that it disregards ‘recruitment overfishing’. With an ever decreasing

biomass for increasing effort a point may be reached when the stock is no longer big enough to sustain the constant recruitment assumed.

Length based Thompson and Bell Model (1934)

The first predictive model was developed much earlier than the Beverton and Holt model by Thompson and Bell (1934). The Thompson and Bell model is the exact opposite of the VPA or cohort analysis and inverse of length structured VPA. It is used to predict the effects of changes in the fishing effort on future yields, while VPA or cohort analysis are used to determine the numbers of fish that must have been present in the sea, to account for a known sustained catch, and the fishing effort that must have been expended on each age or length group to obtain the numbers caught. Therefore, VPA or cohort analysis is called historic or retrospective models, while the Thompson and Bell model is predictive. The Thompson and Bell method consists of two main stages:

• An analysis based on fishing mortality per size (age) group (so called F-array),

size (age) group-specific catches, death, yield, biomass and value

• A prediction of the effect of change in F-array on the catches, death, yield,

biomass and value in future.

The first of these two parts can be achieved through VPA or slight modification of the catch curve routine, where the fishing mortality are estimated for each age or length group.

Input parameters: The "length-based Thompson and Bell model" takes its inputs from a length-based cohort analysis. The inputs consist of the fishing mortalities by length group (the so-called F-at-length array), the number of fish entering the smallest length group, and the natural mortality factor ‘H’ by length group, which must be the same as the ones used in the cohort analysis. Additional inputs are the parameters of a length-weight relationship (or the average weight of a single fish by length group) and the average price per kg by length group.

Output parameters: The outputs for each length group the number at the lower limit of the length group, N(L1), the catch in numbers, the yield in weight, the biomass multiplied by ∆t, i.e. the time required to grow from the lower limit to the upper limit of the length group and the value.

• Thompson and Bell formula is based on Jones length based cohort analysis

which is given as follows:

It can be rewritten as

Where,

Which, is the same factor as used in Jones' length-based cohort analysis

• In order to calculate the yield (catch in weight) by length group the catch C (in

numbers) has to be multiplied by the mean weight of the length group, (L1,L2), which is obtained from

Where, q and b are the parameters of the length-weight relationship

The yield of this length group will be,

The value of the yield is given by:

Where, (L1,L2) is the average price per kg of fish between lengths L1 and L2

The corresponding mean biomass * ∆t is:

The annual yield is simply the sum of the yield of all length groups for each month:

Y = YiƩ

The annual value is likewise the sum of the value of all length groups for each month:

V = ViƩ

Basic features of the length-based Thompson and Bell analysis

Since the length-based Thompson and Bell analysis is derived from Jones' length-based cohort analysis which in turn is based on Pope's age-based cohort analysis, the length-based Thompson and Bell method has the same limitations as Pope's age-based cohort analysis. The approximation to VPA in the predictive mode is valid for values of F*∆t up to 1.2 and of M*∆t up to 0.3 (Pope, 1972). If the F's are high, nonsensical results will come out of the analysis, such as negative stock numbers.

Yield, biomass and value of yield per 1000 shrimps calculated by the age-based Thompson and Bell model (1934), data collected from Kuwait shrimp fishery (Garcia & van Zalinge 1982)

REFERENCES

1. Sparre, P. and Venema, S.C.,1989. “Prediction models”, Introduction to Tropical Fish Stock Assessment. FAO Fisheries Technical Paper No. 306/1. 231-234

2. Gayanilo F.C. and Pauly D., 1997. “Yeild per recruit and prediction”, FAO-Iclarm Fish Stock Assessment Tools. 178-207

3. Beverton, R.J.H. and S.J. Holt, 1957. On the dynamics of exploited fish populations. Fish. Invest. Minist. Agric. Fish. Food G.B.(2 Sea fish)- 553.

4. Vivekanandan, E.(2005). “Analytical models of stock assessment”, Stock Assessment of Tropical Fishes. ICAR. 70-75

5. Chakraborty, S. K., 2010. “Prediction models” in CAFT programme on Fisheries Resources management (course manual). CIFE Mumbai, p208-215