CHAPTER 3 - COHORT

3.1 COHORT - INTRODUCTION

A cohort or annual class or a generation, is a group of individuals born in the same spawning season. The following scheme illustrates the different phases of the life cycle of a cohort:

Figure 3.1 Cycle of life of a cohort

Let us start, for example, with the egg phase. The phases that follow will be larvae, juvenile and adult.

The number of individuals that arrive in the fishing area for the first time is called recruitment to the exploitable phase. These individuals grow, spawn (once or several times) and die.

After the first spawning the individuals of the cohort are called adults and in general, they will spawn again every year, generating new cohorts.

The phases of life of each cohort which precede the recruitment to the fishing area (egg, larvae, pre-recruits), are important phases of its life cycle but, during this time they are not usually subjected to exploitation. The variations in their abundances are mainly due to predation and environmental factors (winds, currents, temperature, salinity,…). In these non exploitable phases mortality is usually very high, particularly at the end of the larvae phase (Cushing, 1996). This results in a small percentage of survivors until the recruitment. Notice that this mortality is not directly caused by fishing.

The recruitment of a cohort during the exploitable phase, may occur during several months in the following schematic ways:

Figure 3.2 Types of annual recruitment to the exploitable phase

With some exceptions, the forms of recruitment can be simplified by considering that all the individuals are recruited at a certain instant, t_r called age of recruitment to the exploitable phase. It was established that recruitments will occur on 1 January (beginning of the year in many countries). These two considerations do not usually change the results of the analyses, but simplify them and agree with the periods of time to which commercial statistics are referred.

It should be mentioned that not all the individuals of the cohort spawn for the first time at the same age. The proportion of individuals which spawn increases with age, from 0 to 100 percent. After the age at which 100 percent of the individuals spawned for the 1^st time, all the individuals will be adult. The histogram or curve that represents these proportions is called maturity ogive.

In certain cases, the maturity ogive can also be simplified supposing that the 1^st spawning occurs at the age t_mat designated as age of 1^st maturity. This simplification means that the individuals with an age inferior to t_mat are considered juveniles and those with the same age or older, are considered adults.

Figure 3.3 represents a maturity ogive with the shape of a histogram or curve:

Figure 3.3 Maturity ogive

3.2 EVOLUTION OF THE NUMBER OF A COHORT, IN AN INTERVAL OF TIME

Consider the interval (t_i, t_i+1) with the size T_i = t_i+1 - t_i of the evolution of a cohort with time and Nt the number of survivors of the cohort at the instant t in the interval T_i (see Figure 3.4).

The available information suggests that the mean rates of percentual variation of N_t can be considered approximately constant, that is, rmr (N_t) (constant.

Basic assumption

The relative instantaneous rate of variation of N_t, in the interval T_i is:

rir (N_t) = constant negative = - Z_i

Figure 3.4 Evolution of N in the interval T_i

The model of the evolution of N_t, in the interval T_i, is an exponential model (because rir(N_t is constant). This model has the following properties:

Properties

1. General expression. From the basic assumption

rir(N_t) = -Z_i

with the initial condition that, for t = t_i it will be N_t = N_i then:

2. Number of survivors, N_i+1, at the end of interval T_i

3. Number of deaths, D_i, during the interval T_i

(notice that D_i is positive but the variation DN_i = N_i+1- N_i is negative)

4. Cumulative number of survivors, N_cumi, during the interval T_i

5. Approximate central value, N_centrali, in the interval T_i

6. Mean number, , of survivors during the interval T_i

when is small (Z_i.T_i < 1)

Comments

1. The basic assumption is sometimes presented in terms of absolute instantaneous rates, that is:

air (N_t) = - Z_i.N_t { air (N_t) proportional to N_t} or
air (lnNt) = -Z_i

Z_i = mortality total coefficient, assumed constant at the interval T_i

Notice that:

+Z_i = rir of total mortality of N_t
- Z_i = rir of variation of N_t

2. Unit of Z_i

From the definition, it can be deduced that Z_i is expressed in units of [time]-1_. By agreement, the unit year^-1 has been adopted, even when the interval of time is smaller or bigger than a year.

The following expressions show, in a simplified way, the calculation of the unit of Zi, with the rules and usual symbols [..(of dimension in the determination of physical units.

3. Annual survival rate, S_i

When T_i =1 year it will be:

and also

S_i = Annual survival rate in the year i

(or percentage of the initial number of individuals that survived at the end of the year).

1 - S_i = Annual mortality rate in the year i

The percentage of the initial number of individuals that die during the year is, by definition, the relative mean rate rmr (N_t) of mortality of N_t, over one year, in relation to the initial number, N_I

4. Absolute mean rate

5. Relative mean rate in relation to

6. Notice that S_i takes values between 0 and 1, that is,

0 ≤ S_i ≤ 1 but Z_i can be > 1

7. If the limits of the interval T_i were (t_i, ∞) then it would be:

T_i = ∞
N_i+1 = 0
D_i = N_i
and

3.3 CATCH, IN NUMBER, OVER AN INTERVAL OF TIME

The causes of death of the individuals of the cohort due to fishing will be separated from all other causes of death. These other causes are grouped together as one cause designated as natural mortality. So, from the properties of the exponential model, the result will be

rir(Nt) _total = rir(Nt) _natural + rir(Nt) _fishing

Supposing that, in the interval T_i, the instantaneous rates of mortality due to natural causes and due to fishing are constant and equal to M_i and to F_i, respectively, then

Z_i = F_i + M_i

Multiplying both factors of the previous equation (equality) by, then:

is the number, D_i, of deaths due to total mortality,

In the same way will be the number of dead individuals due to fishing, that is, the Catch, C_i, in number, and then:

Notice too that, will be the number of dead individuals due to "natural" causes.

The exploitation rate, E_i, during the interval T_i was defined by Beverton and Holt (1956) as:

and then, or,

The capture in number, C_i, in the interval T_i, can be expressed in the following different ways:

Comments

1. Ricker (1975) defines the exploitation rate, Ei*, as the percentage of the initial number that is captured in the interval Ti, that is: E_i* = C_i/N_i.

a) Ricker’s definition may be more natural, but mathematically Beverton & Holt’s definitions are more useful.

b) It is easy to verify that

2. The exploitation rate, E_i, does not have any unit, it is an abstract number.

3. The possible values of E_i are between 0 e 1, being 0 when there is no exploitation and 1 when the capture C_i is equal to the number of total deaths D_i, that is, when M_i = 0.

3.4 INDIVIDUAL GROWTH

In order to study the evolution of the biomass of a cohort, one can use the model of the evolution of a cohort, in number, and combine it with a model of the evolution of the mean weight of an individual of the cohort. In effect, the biomass B_t is equal to N_t.W_t where W_t is the individual mean weight at the instant t.

To define a model for the individual growth weight W_t, there are then two possibilities:

ALTERNATIVE 1:

A) To define a model for the mean individual growth in length, L_t
B) To define the relation Weight-Length.
C) To combine A) with B) and obtain a mode for the mean individual growth in weight, W_t

ALTERNATIVE 2:

D) To define directly growth models for W_t and L_t.

ALTERNATIVE 1

A) Model for the individual growth by length

The models that are used in fisheries biology are valid for the exploitable phase of the resource. The most well known is the von Bertalanffy Model (1938) adapted by Beverton and Holt (1957). The existing observations suggest that there is an asymptotical length, that is, there is a limit to which the individual length tends.

Figure 3.5 von Bertalanffy Model

t - age
L_t - individual mean length at the age t
L_∞ - asymptotical length
t_r - beginning of the exploitable phase

So, L_t presents an evolution where:

air(L_t) is not constant (because growth is not linear)
rir(L_t) is not constant (because growth is not exponential)

However, it can be observed that the variation of the quantity (L_∞ - L_t) (which we could call "what is left to grow"), presents a constant relative rate and can be described by an exponential model. So, we can adopt the:

Basic assumption

rir (L_∞ - L_t) = - K = negative constant during all the exploited life

where K is the growth coefficient (attention: the growth coefficient K is not the velocity of growth but the relative velocity of what "is left to grow"!!).

The properties of this model can be obtained directly from the general properties of the exponential model. The initial condition:

where t_a (and L_a), that corresponds to an instant within the exploitable phase, will be adopted.

The properties of the model of individual growth by length by Beverton & Holt (1957) deduced from the exponential model of are summarized as:

B) Relation Weight-Length

It is common to use the power function to relate the individual weight to the total (or any other) length. Then:

where constant a is designated as condition factor and constant b as allometric constant. This relation can be justified accepting that the percentage of growth in weight is proportional to the percentage of growth in length, otherwise, the individuals would become disproportionate. Thus, the basic assumption is:

where b is the constant of proportionality.

ALTERNATIVE 2

D) Model directly for W_t and L_t

In alternative 1, a model was developed for growth in length, and then a model was built for growth in weight, using the relation between weight and length.

The basic assumption adopted by alternative 1 used the characteristic instead of weight in order to be able to have a characteristic with a rate rir constant, that is, an exponential model. The relation W=a.L^b was adopted to obtain the model of growth in weight. Notice that it can be said that L was considered as a function of W, that is, L= (W/a)^1/b. It will then be possible to adopt, instead of that function of W, another function of the weight, in order to be able to formulate directly the basic assumption:

with the initial condition

Properties

The properties of this model (once it is an exponential model) can be obtained directly from the general properties of the exponential model. It is particularly interesting to derive the general expression W_t resulting from different choices of function H.

Comments

1. Gompertz equation, in weight, is similar to Gompertz equation, in length, but, in their simplified forms, t* represents different ages, because they will correspond, respectively, to W_a = 1 and to L_a = 1.

2. Gompertz equation, in length, is similar to von Bertalanffy if L_t is substituted by ln L_t. In practice, this fact allows the utilization of the same particular methods to estimate the parameters in both equations, using L in the von Bertalanffy expression and lnL in the Gompertz expression. (See Section 7.4)

3. It is important to notice, once again that, in practice, L_centrali and W_centrali are used instead of the mean values, and

4. Gompertz growth curve in length, has an inflection point, (t_infl, L_infl), with:

t_infl = t_a+(1/K).ln(ln(L_∞/L_a)) L_infl = L_∞ / e

5. Gompertz growth curve in weight has an inflection point, (t_infl, W_infl) with:

t_infl = t_a+(1/K).ln(ln(W_∞/W_a)) W_infl = W_∞ / e

6. Richards growth curve in length does not have an inflection point but the growth curve in weight has an inflection point, (t_infl, W_infl).

In the particular case of the von Bertalanffy equation it will be:

W_infl = (8/27).W_∞ and t_infl = t₀+(1/k).ln3

7. Some authors refer Gompertz equation in other ways, for example, using the inflection point t_infl and the asymptotic weight W_∞, instead of the parameters t_a and W_a.

It will then be or

Sometimes the length expression is presented in its general form:

The parameters of the length model will then be: L_∞ = a; k = -c et t_infl = (1/c).ln(-b)

8. The growth in length presents an inflection in fish farming, where the study of growth covers very young ages and it is common to use the Gompertz equation. In fisheries, the tradition is to use the von Bertalanffy equation.

9. A model that can sometimes be useful, is the Ricker model (1975). This model is valid for a certain interval of time T_i and not necessarily for all the exploitable life of the fishery resource. In fact, the model is based on the basic assumption that the individual growth is exponential in the interval T_i.

It will be, for example, where K_i can be different from one interval to the next.

3.5 BIOMASS AND YIELD, DURING THE INTERVAL T_i

1. Biomass

Theoretically, it could be said that the biomass at the instant t of the interval T_i is given by:

B_t = N_t. W_t

Thus, the cumulative biomass during the interval T_i would be:

and the mean biomass in the interval T_i would be:

In the same way, the mean weight of the cohort, , in the interval T_i would be:

The biomass can be obtained by dividing both terms of the fraction by T_i, as

2. Yield

The yield, Y_i, during the interval T_i will be expressed as the product of the catch in numbers, times the individual mean weight:

Comments

In practice is considered approximately equal to at the interval T_i.

Other expressions of Y_i will also be:

3.6 COHORT DURING THE EXPLOITABLE LIFE

Consider the evolution of a cohort during the exploitable life, beginning at age t_r, and intervals of time, T_i, covering all the exploitable phase (frequently the intervals are of 1 year...).

Figure 3.6 illustrates the evolution of the number of survivors of the cohort, N, and the evolution of the catches in number, C, which are obtained during the successive intervals of time T_i.

Figure 3.6 Evolution of the number of survivors of the cohort, N, and the catches in number, C

Figure 3.7 illustrates the evolution of the biomass of the cohort, B, and the evolution of the catches in weight, Y, which can be obtained during the successive intervals of time T_i.

Figure 3.7 Evolution of the biomass of the cohort, B, and the catches in weight, Y

Values of the most important characteristics of the cohort, during all the exploitable phase

Comments

1. At first sight it may seem that the values of the characteristics of a cohort during all the exploitable phase are of little interest, because, very rarely is fishing applied to an isolated cohort. At each moment, the survivors of several cohorts are simultaneously present and available for fishing.

2. Despite this fact and for reasons which will be mentioned later, it is important to analyse the characteristics of a cohort during all its exploitable life. Knowledge of the evolution of a cohort, in number and in biomass, and particularly the critical age, is important for the success of the activities in fish farming. As B_t = N_t. W_t, the critical age, t_crítical, will be the age t in the interval T_i where

rir (W_t) = - rir(N_t)=M because, the critical age being the maximum biomass, the derivative of B will be equal to zero.

Notice that N_cum can be expressed in function of the recruitment as

N_cum = R.{...}

where {...} represents a function of biological parameters and annual fishing mortality coefficients F_i during the life of the cohort. B_cum can also be expressed as:

B_cum = R.{...}

where the function {...} also includes growth parameters.

3.7 SIMPLIFICATION OF BEVERTON AND HOLT

Beverton and Holt (1957) deduced algebraic expressions for the characteristics of a cohort during the exploited life, adopting the simple assumptions:

1. The exploited phase of the cohort is initiated at age _tc and is extended to the infinite.
2. The natural mortality coefficient, M, is constant during all the exploitable phase.

Figure 3.8 Life of a cohort during the exploitable phase

3. The fishing mortality coefficient, F, is constant during all the exploited phase.
4. Growth follows the von Bertalanffy equation with for L_a = 0 for t_a = t₀

Comments

1. The simplification of Beverton and Holt allows the calculation of any characteristic of the cohort during its life with algebric expressions, avoiding the addition of the values of the characteristic in the successive intervals T_i. This was useful for calculations in the 60-70's when computers were not available. It is also useful when the only available data is natural mortality, M, and growth parameters.

2. At present, the simplified expressions are also useful to study the effects on the biomass, on yield and on the mean weight of the catch due to changes in the fishing mortality coefficient, F, and/or on the age of first capture, t_c. These analyses are usually illustrated with figures. For example, Figure 3.9 exemplifies the analyses of the biomasses and of the catches in weight, obtained from a cohort during the exploited life, subjected to different fishing mortality coefficients, assuming a fixed age t_c.

Figure 3.9 Evolution of the biomass and of the catch in weight from a cohort subject to different fishing mortality coefficients and fixed t_c (notice that the Figure illustrates only the analysis and does not take into consideration the scales of the axis).

3. Notice that the forms of the previous curves, Y and B_cum against F, do not depend on the value of the recruitment and so, they are usually designated as curves of biomass and yield per recruit, B/R and Y/R, respectively. The calculations are usually made with R=1000.

4. The mean weight, the mean age and the mean length of the catch do not depend on the value of the recruitment. The curves of the characteristics of a cohort during its life against the fishing level, F, or against the age of first catch, t_c, deserve a careful study, for reasons which will be stressed in the chapter concerning the long-term projections of the stock.

5. B_cum was calculated as:

where

The calculations can also be made using other values of the constant b, from the relation W-L, different from the isomorphic coefficient, b=3, using the incomplete mathematical function Beta (Jones,1957).

6. The means and ¯t can be calculated from the cumulative expression

These means are designated as weighted means, being the weighting factors, the number of survivals, N_t, at each age t.