KANTHARAJAN GICAR-CIFE, MUMBAI

Exponential Decay Model and

Total Mortality- KANTHARAJAN G

Mortality in Fish Stock Assessment

The life history of a fish is a series of time intervals through which an individual survives several risks.

Risks rarely occur sequentially, but instead occur simultaneously And a fish can only die once (and once it’s dead, it’s dead!)

Need to understand the dynamics of mortality

Mortality

Total Mortality

(Z = F + M)

Predation

Disease

Spawning stress

Senescence

Stranding

Abiotic (Temp,

Pollution)

Fishing

Starvation

Johan Hjort described extreme mortality rates (>99%) ofmarine fish larvae within days of hatching - hydrodynamicstarvation

Ichthyophonus hoferi – Phocomycetes fungi which givesdisease induced mortality in North sea herring.In 1989 – 1, 300, 000 t 1991, outbreak in 1994, 760,000t (Patterson, 1996).

Water temperature - effect on spawning and survival of larvaeand juveniles (IPCC report, 2001).

Wild Atlantic salmon (Salmo salar) and marine Mammals

Natural Mortality (M)

• Interested in knowing the effect of fishing on fish stocks - clear distinction betweenmortalities caused by fishing and those caused by other factors needed.

• The mortality created by all other causes than fishing, e.g., predation includingcannibalism, diseases, spawning stress, starvation, and old age.

• Predation and starvation mortalities - linked to the ambient ecosystem• Same species - different natural mortality rates in different areas - density of predators

and competitors - influenced by fishing activities• K - linked to the longevity of the fish and longevity is related to mortality • High K-value have a high M-value• Low K-value have a low natural mortality• A slow growing species (low K) simply cannot bear high natural mortality - if it did it

would soon become extinct. • Natural mortality - Asymptotic length and weight - large fish have fewer predators than

small fish.

Types of Natural Mortality

• Physiological mortality – Caused solely by diseases, old age or both leading to deathwithout he intervention of predators. (K , Purely physiological mortality)

• Selective mortality – caused by disease, old age or both in a certain no of fish beingcharacterized by a lack of performance which makes them accessible to predators.(Middle aged group)

• Chance mortality – Mortality unrelated to physiological mechanisms and proportionalonly to a number of possible encounter with potential predator. (Small fish, Place in afood chain)

Fishing Mortality (F)Mortality due to fishing; death or removal of fish from a population due to fishingF value ranges - from 0 for no fishing

- to very high values 1.5 or 2 (1.5 or 2 times as many fish have been caught as were present at the beginning ofthe fishing season)

F = qfq = catchability coefficient

- The proportion of a fish population caught by one unit of fishing effort.- q = (C/E) (1/N)- q depends on,

The distribution of fishing effortEffectiveness of finding fish of the vesselSelectivity of the gear

Fishing Effort (f)

- The amount of fishing gear of a specific type used on the fishing grounds over a given unit of time

(e.g). hours trawled per day, number of hooks set per day or number of hauls of a beach seine per day.

- When two or more kinds of gear are used, they must be adjusted to some standard type.

- Increasing skills of skippers and fishing crews also contribute to the effectiveness of fishing effort and capacity

Mortality concepts

Exponential Decay Model• It is a corer-stone of the theory of exploited fish stocks, because in exploited fish stock numbers

surviving end to decrease exponentially with time and age according to Z.

• The higher the value of Z the faster the decrease in numbers and the lower the maximum age.

• It is usually easier to understand the concept of mortality and survival if the number of survivorsat a certain moment is expressed as a percentage of the original number.

• Table gives the percentages of survivors after one and two years respectively of two populationssubject to two different total mortality rates, Z = 0.5 and Z = 2.0.

Total mortality rate Z

percentage of survivors *

after 1 year100*N(Tr+1)/N(Tr)

after 2 years100*N(Tr+2)/N(Tr)

0.5 61% 37%

2.0 14% 2%

*) Percentage of survivors = 100*exp(-Z*(t-Tr)) and t-Tr = 1 and 2 respectively

N(t) – No. of survivors from a cohort attaining age tN(Tr) – No. of recruits to the fishery

The rate of change in numbers is written:

D N(t)/D t

where D t is the length of the time periodThe rate of change in numbers per year, D N(t)/D t depends on the number of survivors, N.N more – more will die.

D N(t)/D t = -Z N(t)

If Z is constant throughout the life,

N(t) = N(Tr)*exp[-Z*(t-Tr)] Exponential decay model

Exponential decay of a cohort with recruitment N(Tr) = 100000 and total (constant) mortality Z = 1.5 per year. Equation D N/D t = -Z*N is fulfilled for various ages of the cohort (t)

age of cohort

t years

number of survivors

N(Tr)*exp(-Z*(t-Tr))

change in numbers

during on day, D N

Z

Tr+0.0+D t 99590 -410 1.5

Tr+0.6+D t 40490 -167 1.5

Tr+1.5+D t 10497 -43 1.5

Tr+3.5+D t 522.60 -2.15 1.5

Tr 100000

Exponential decay curve with N(Tr) = 100,000 recruits and Z = 1.5 Z remains constant for all ages, t

REFERENCES

• Patterson K R., (1996). Modelling the impact of disease – induced mortality in an exploited population: Theoutbreak of the fungal parasite Ichthyophonus hoferi in the North Sea herring (Clupea herengus). CanadianJournal of Fisheries and Aquatic Sciences, vol. 53 pp: 2870 – 2874.

• Daniel pauly. (1980). On the interrelationships between natural mortality, growth parameters and meanenvironmental temperature in 175 fish stocks, J. Cons. Int. Explor. Mer, 39(2): 175 – 192.

• Per Sparre., Erik Ursin., Venema C. (1989). Introduction to tropical fish stock assessment Part 1 – Manual(FAO Fisheries Technical Paper 306/1).

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