Parameters in Fish Population Dynamics

1.Age-Growth 2. Mortality 3. Recruitment & Selection

Growth Parameters: Growth in fishes is not constant over years. The young grows at a faster rate and this growth goes on decreasing as the fish (animal) advances in age until there in no further growth. The length at which there is no further growth is called as length at infinity or “Asymptotic growth”.

Asymptotic growth (L∞): This is the theoretical length “L∞” beyond which the fish does not grow. In nature, however, fishes do not attain this size, since they die due to natural reasons like predation, disease, senility or fishing.

Similarly the arbitrary origin (t0) is also a theoretical value. The increase in length from year to year takes place at a certain rate called as the growth constant (k).

Therefore, these “L∞, t0 & k” are called as growth parameters, which can be estimated from age-length data by various methods.

I. Von Bertalanffy’s Equation (1938) II. Ford (1933)-Walford (1946) Method

I. Von Bertalanffy’s Equation

Lt = L∞ [1-e-k (t-to)] or Wt = W∞ [1-e-k (t-to)] 3

Where, Lt = length at given time “t” Wt = Weight at infinity L∞ = Length at infinity W∞ = Weight at infinityk = Growth constant or Growth coefficient i.e. rate at which fish attains the limiting size (∞)t0 = arbitrary origin of growth (time of birth) and e = natural logarithm

Application of von Bertalanffy’s Equation: For population analysis it is desirable to express the growth of fish in two aspects-

i. The growth of fish population in a mathematical form. ii. The basic requirement is an expression, which will give the

size (i.e. length & weight) at any given age which agrees with that observed data of size at the age which can be incorporated in yield models. This growth is considered for the whole population of fish, not for individual fish.

II. Ford-Walford Plot: This is a graphical method of calculating growth. This method is a combination of two methods framed by Ford (1933) and Walford (1946). In this method growth parameters are estimated by plotting length at a given age against the length a year earlier. Intercept on 450 lines is L∞ and slope is e-k.

Age (t) length (Lt) (Lt+1)

1 25.7 36.0

2 36.0 42.9

3 42.9 47.5

4 47.5 50.7

5 50.7 52.8

6 52.8 54.9

Y = a + bX (a – intercept and b – slope)

L∞ can be estimated graphically from the interception point of 450 diagonal (where Lt = Lt + 1) and the regression line, because for very old fish, which have stopped growing Lt = Lt + 1 = L∞, then “t0” can be estimated from equation-

Lt= L∞ [1-e – k (t - to)] or

e-k (t - to) = (L∞ – Lt) / L∞&

t0= t + 1/k loge (L∞ – Lt) / L∞

2. Mortality: Each year a proportion of the fish alive at the beginning of the year will die some by predation, disease or other natural causes and some by being caught, while others will survive until the beginning of the next year.

Mathematically this can be expressed as: N1 = P + D + O + C + N2

[N1- Number of fish at the beginning of the first year, N2- Number of fish at the beginning of the second year, P- Number of fish dying by predation, D- Number of fish dying by disease, O-Number of fish dying by other causes, C-Number of fish caught / fishing].

The most obvious way of expressing these deaths is as “proportions” of the numbers of the beginning of the year i.e. –

• Annual predation rate = P / N1

• Annual rate of death by other causes = D / N1

• Annual rate of death by other causes = O / N1

• Annual rate of death by exploitation = C / N1

• Annual rate of death by all causes (A) = (P+D+O+C) / N1

• Annual survival rate (S) = N2 / N1 = 1-A

More practically, it is better to consider the instantaneous rates, i.e. the rates applying over a short period of time (dt), during which the numbers in the population do not charge significantly, so that the numbers dying from anyone cause are not affected by numbers dying from any other cause. Then, combining all causes of natural (non-fishing) mortality, these deaths will be proportional to the instantaneous natural mortality coefficient (M) and the numbers caught to the instantaneous fishing mortality coefficient (F).

Accordingly:

dP + dD + dO = MNt dt [Instantaneous natural mortality]

dC = FNt - dt [Instantaneous fishing mortality] And total deaths which are equal to decrease in population

numbers can be total mortality coefficient (Z) = F + M [Where, F= Fishing mortality coefficient & M= Natural mortality coefficient]

Estimation of Total Mortality (Z) 1. Z from computation of data:

i. Heinchke method (1913) ii. Jackson method (1939) iii. Chapman & Robson method (1960)

iv. Cushing method (1968)

2. Z from C P U E [(Paloheimo method (1960)]

i. Heinchke Method (1913)-

Annual mortality rate (A) = N0 / ∑ N[N0 = Number of 0- year old fish and N = Total number of fish comprising all age group]

Annual survival rate (S) = 1-A = ∑ N - N0 / ∑ N

From which Z is estimated by S = e-z and log es = -ZHence, Z = - log es

Applicability of Heinchke Method -• This method seems practicable only if the 0-year class

constitutes the “fully recruited” group. • It is never possible that all the size classes of the 0-year

class are truly represented (in catch) as they are in nature. However, this method is most appropriate for assessment of ‘Z’

in populations which contain distinct 0-year group. [Therefore, these methods quite useful for short lived fish species, where the age determination of the fishes beyond an age of one year is rather difficult (because of the availability of insufficient samples of older fishes)].

For example the fishes such as Indian Lesser Sardine (S. fimbriala, S. gibbosa, S. albella), White Baits (Stoleferous spp.) and Silver Bellies (Leognathus spp, Secutor spp & Gazza spp.).

From the above expression it can be recognized that it is not necessary to know the number of fish in each age older than that coded as “0” but only their total (∑). Hence, the advantage in this method could be also used when age determination of older fish is unreliable.

ii. Jackson (1939) Method: ‘Z’ between two adjacent age groups (year class) in a given year is estimated from the survival rate (S) using the equation-

S = Nt + 1 / Nt and S = e-Z

[Nt = number of specific year old fish & Nt + 1 = number of 1 more year old fish]

Z = -log e Nt + 1 / Nt

Hence, Z = log e Nt / Nt + 1

iii. Chapman and Robson Method (1960): In this method survival rate is estimated from the equation-

S = T / ∑N + (T – 1)Where T = N1 + N2 + N3 + …, ∑N = No + N1 + N2 + …, T-1 = [(N1 + N2 + N3 + …) – 1]

iv. Cushing (1968) Method: In this method ‘Z’ is estimated from two stock densities of a year class separated by a number of years due to incomplete age composition data e.g. if the density of only the 1st ad 4th year old fish is available ‘Z’ is estimated from the relation:

Z = 1/3 log e N1 / N4

The ghal (Pseudosciaens dicanthus) fishery along the Northwest Coast of India is characterized by similar situation. where, younger age groups caught inshore with traditional gear (i.e. bag-net or gill-net) and the older age groups caught with mechanized trawler in offshore and one or more intermediate age groups to be absent in the fishery. In this instance, the “Cushing method” is quite ideal.

2. “Z” from C P U E [Poloheimo (1961) method]: A plot of monthly CPUE against respective months for a fishery through out year may take shape of a near normal curve, with ascending phase including the peak falling in one half the year and descending phase in other half.

From this data CPUE is derived separately for descending phase (c/f)2 and ascending phase (c/f)1 and ‘Z’ estimated using the expression:

Z = - log e(c/f)2 / (c/f)1Such a situation obtains in many pelagic fisheries such as the Indian

Oil Sardine, Mackerel and Seer fish along the South West Coast of India where ascending phase occurs in January to June and the descending phase occurs in July to December.

Estimation of Fishing (F) and Natural (M) mortalities: The estimation of total mortality (Z) gives an idea of removal of individuals from a population or stock, but does not provide useful information about the effects of fishing until we know how much of this total Z is due to fishing and how much to natural causes.

Therefore, it is necessary to estimate the fishing and natural mortality parameters separately, and this can be possibly achieved by relating changes in the total mortality to changes in the amount of fishing. Mathematically this can be expressed as: Z = F + M = q f + M

[Where, f = fishing effort, q = catchability coefficient (Fishing mortality coefficient generated by one unit of effort)].

• If there are two periods with different known levels of fishing effort f1 and f2 during which estimates are made of the total mortality Z1 and Z2, then: Z1 = q f1 + M and Z2 = q f2 + M

• This again gives a pair of equation for which two unknown quantities ‘q’ and ‘M’ can be calculated as fallows:

q = Z1 – Z2 / f1 – f2 and M = Z1 f2 – Z2 f1 / f2 – f1

Estimation of Fishing Mortality (F) by Tag Recovery Method: Fishing mortality (F) can also be derived from tag recovery data as follows= (Progressive number of tagged fish recovered / total number of Fish tagged) x (Days between release and recovery / 365 days).

Population size calculation (N) = nT / m [n= No. of individuals sampled, T= No. of individuals tagged and m= No. of tagged individuals in sample].

Example: 80 number of fish taggedDate

of releas

e

Date of recovery

Days b/w released recovery

No. of fish

recovered

Progressive No. tagged fish recovery

F

1st May 1st May 1st May 1st May

5th May 10th May 18th May 25th May

38

1623

23

101

25

1516

F1= 2/80 x 3/365 = 0.0002

F2= 5/80 x 8/365 = 0.0013

F3= 15/80 x 16/365 = 0.0082

F4= 16/80 x 23/365 = 0.0120