Journal of Mathematics and Computer Applications Research (JMCAR) ISSN(P): 2250 - 2408; ISSN(E): Applied Vol. 3 , Issue 1 , Dec 201 6 , 1 - 14 TJPRC Pvt . Ltd.

MATHEMATICAL BIO ECONOMICS OF FISH HARVESTING WITH

CRITICAL DEPENSATION IN LAKE TANA

GETACHEW ABIYE SALILEW Madda Walabu Universty, Department of Mathematics, Bale Robe, Ethiopia ABSTRACT

Fishing management in Lake Tana, Ethiopia is the main concern of this paper. To study the fishing activities and fishing management in Lake Tana we have considered ordinary differential equations which represent the dynamics of fishing activities. We class ify this model into mathematical and mathematical bio - economics model. In the mathematical biology stream we have study the stability analysis of the equilibrium points with the behavior of their respective solutions. In the mathematical bio - econom ics stream we investigate the efforts for maximum sustainable yield (MSY), open access yield (OAY) and the maximum economic yield (MEY). These efforts are calculated based on the real data collected from the stake holders around Lake Tana. We investigate t hat if we consider the critical depensation model we found that there is if all efforts are less than the thresh hold value = × kg of fish . The same results found in the analysis of economic models that overfishing exists in the critical depensation model. O r

KEYWORDS: Mathematical Bio - Economics, MSY , OAY, MEY, Overfishing, Catch Per Unit Effort i g i n

a l

Received: Aug 27, 2016 ; Accepted: Sep 13, 2016 ; Published: Sep 21, 2016 ; Paper Id.: JMCARDEC20161 A r t i c

1. I NTRODUCTION l e

Environmental resources are described as renewable and non - renewable. If they are renewable, they have a capacity for reproduction and growth otherwise not. Renewable resources include of biological organisms such as and forests which have a natural capacity for growth. There is one similarity between renewable and non - renewable resources that both are capable of being fully exhausted: that is they become to zero if excessive and prolonged harvesting or extraction activity is carried ou t. In the absence of regulation control over harvesting behavior, the stocks are subject to open access.

Mathematical bio economics is the study of the management of renewable resources. It takes into consideration not only economic questions lik e revenue, cost, price, effort etc., but also the impact of this demand on the resource. The aim of fish harvesting management is to gain a sustainable development of activity so that, future generation can also benefit from the resources. It is observed i n many countries that, fish are becoming increasingly limited and caches are declining due to [1]. Overfishing and waste of resource rent in fisheries are caused by free and open access to the resource exploitation [5]. During the course of time high levels of fishing effort cause a serious reduction in the size of the fish stock and consequently the rate of catch per unit of effort is reduced [3]. Biological overfishing occurs when fishing mortality has reached a level where the stock has negative marginal growth or slowing down biomass growth. Economic or bio - economic overfishing additionally considers the cost of fishing and defines overfishing as a situation of negative marginal growth of resource rent .

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The object of the management is to devise harvesting strategies that will not drive species to extinction. Therefore, the notion of persistence, extinction times of the populati ons and precautionary harvesting policy, is always critical. A control variable of every management is the fishing effort [2, 4], which is defined as a measure of the intensity of fishing operations. As fishery management is the balance between har vesting and its ecological implications, it is important to fish in such a way that a species is sustainable and not in danger of becoming extinct. In this paper we considercritical depensation (strong Alee effect) deterministic models with a constant harv est rate as well as time dependent harvesting in the case of Lake Tana, Amara region, Ethiopia which is experienced by open access fishery.The dynamic mathematical models set on the background of biology and economics knowledge. The integration of these se emingly different subjects namely mathematics, biology and economics creates the source of interesting results and give valuable applications for the peoples living with fishing activities and those policy makers who involved control of overfishing. This st udy tries to show whether or not overfishing indeed exists.

2. MATHEMATICAL BIOLOGY OF CRITICAL DEPENSATION MODEL

In order to investigate the economics of a renewable resource, it is first necessary to describe the pattern of biological growth of the reso urce. First we consider the growth function for a population of some species of fish which by convention is called a fishery. The deterministic models of fishery populations can be classified into three types namely Compensation, Depensation and Critical d epensation. Compensation growth is a growth type where population declination is compensated by increased growth rate and has equation of the form / = [ 1 − ( / ) ] . Depensation ( week Alee effect ) growth is the opposite case to composition growth model and has equation of the form / = [ 1 − ( / ) ] . In both of these models ( ) denotes the at time , is the intrinsic growth rate, is the and ≠ 1 is any real number. The critical depensation model (strong Alee effect) is the generalized logistic model which is extremely in opposite of the depensation model. By the work done [6] some populations experience reduced r ates of survival and reproduction when reduced to very low densities. This reduced per – capita growth rate at low densities is called depensation. The strong Depensation is called critical depensation. Mathematical biology expression of t he critical depensation is given by growth model as,

( / ) = ( ) [ 1 − ( / ) ] [ ( / ) − 1 ] (1)

Here in the growth model (1), ( ) represents fish biomass , represents intrinsic growth rate of fish, is ecological carrying capacity , is the critical mass quantity and ( / ) growth rate of the fish without harvest. In case of critical depensation model (1) the following observations can be made.

 The growth rate of fish population is negative as long as the population size of fish li es below the critical mass

quantity and above the carrying capacity .

 The growth rate of fish population is positive when the population size of fish lies on ( , ) .

 The per capita growth rate, [ ( / ) / ] , is always a positive quantity.

 Both and are positive quantities such that 0 < < .

 Rate of growth of fish population increases with increasing ( ) over some range of population size [ 0 , ] .

 The critical depensation model (1) has three equilibria points = 0 , = and = . The eq uilibrium point

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= is unstable and the remaining two equilibria points = 0 and = are asymptotically stable.

 If the initial population size is assumed to start with some value above then the population will grow and reach the carrying capacity over time otherwise it will die out over time.

Solution of the Critical Depensation Model

The solution of critical depensation model (1) with initial condition ( 0 ) = is obtained as foll ows. Using techniques of separable variables the model (1) can be rewritten as [ / ( − ) ( − ) ] = ( / ) and integrating both sides we get ∫ [ 1 / ( − ) ( − ) ] = ∫ ( / ) . The integrand on the left hand side can be simplified using partial fraction method. Let [ 1 / ( − ) ( − ) ] = [ / ] + [ / ( − ) ] + [ / ( − ) ] . Here , and are unknown constants and they determine as follows. After simplification and comparison of the coefficients we obtain = [ − 1 / ] , = [ 1 / ( − ) ] and = [ 1 / ( − ) ] and thus the integral equation takes the form [ − 1 / ] ∫ + [ 1 / ( − ) ] ∫ ( − ) + [ 1 / ( − ) ] ∫ ( − ) = ∫ [ / ] . On evaluating the integrals we get [ − 1 / ] − [ 1 / ( − ) ] ln ( − ) + [ 1 / ( − ) ] ln ( − ) = [ / ] + ( ( − ) ) ln . Applying antilogarithmic function we got the implicit solution which can be expressed as

( ) ( ) ( − ) ( − ) = (2)

For model (1) the result (2) is the required general implicit solutions of the critical depensation model.

Substituting the initial condition ( 0 ) = at ( 0 ) = the general solution (1) is:

( ) ( ) ( ) ( / ) [ ( − ) / ( − ) ] [ ( − ) / ( − ) ] = (3)

The result (3) is the required particular implicit solution of the critical depensation model (1)

Figure 1 : Growth Curve of Critical Depensation Model for = . , = . , =

In the figure 1 we have shown the rate change curve of the critical depensation model for some particular values of the parameters as shown. The curve is plotted for the population size function ( ) versus the population rate change ′ function = ( / ) . Clearly the rate change of population is negative in the interval ( 0 , ) and it is positive in the interval ( , ) . The rate change of population increases up from 0 to maximum and then falls down to 0 in the interval ′ ( , ) . The maximum rate ch ange of population, ( ) , occurs when the population size be ( ) = ( / 3 ) and the ′ corresponding maximum rate change of population is given by ( ) = { ( / 3 ) [ 1 − ( 1 / 3 ) ] [ ( 1 / 3 ) − 1 ] } >

/ 0 where the parameter is defined as = ( + ) + − + . The following graph represents the stock level of the critical depensation model

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Figure 2 : Typical Solution Curve for Critical Depensation Model for = . , = . , = .

In figure 2 we have time series plot for critical depensation model which verifying local stability of the three equilibrium point in model (1). (i) The equilibrium point = is unstable; (ii) The equilibrium points = 0 and = are stable since all the solution curves that start with any initial value around 0 and are running towards and approaching = 0 and = respectively as t tends to infinity.

3. MATHEMATICAL BIO - ECONOMICS OF CRITICAL DEPENSATION MODEL

Exploitation of biological resources and the harvest of population species are commonly practiced in fisheries, forestry and wild life management. With the natural positive population growth the population size can be brought down whenever harvesting is introduced. As lo ng as the population size is maintained above the population size continues growing and reaches the carrying capacity then harvested population quantity satisfies the following inequality:

< ( ) − . Where ( ) be the present population size and be the critical mass quantity. If the population size is brought below due to harvesting then the population size naturally decreases and dies down eventually. That is, the population size continues decreasing and reaches extinction whenever the harvested population quantity satisfies the following inequality: > ( ) − . Schaefer catch equation is a bilinear short - term harvest function and it assumes that effort always removes a constant prop ortion of the stock. The Critical Depensation

Mathematical Bio - Economics model is given by = ( ) − ℎ ( , ) where ( ) = ( ) [ 1 − ( / ) ] [ ( / ) − 1 ] is the growth function of fish and ℎ ( , ) = is the harvest function of fish. And thus we do have

( / ) = ( ) [ 1 − ( / ) ] [ ( / ) − 1 ] − (4)

Here in the model (4), ( ) represents fish biomass , represents intrinsic growth rate of fish, is ecological carrying capacity , is the critical mass quantity, is time, ( / ) is growth rate of the fish with harvest function ℎ ( , ) .

3.1 Equilibrium Points of Bio - Economics of Critical Depensation Model

The equilibrium points of model (4) are obtained by making ( / ) = 0 ⟺ [ 1 − ( / ) ] [ ( / ) − 1 ] − = 0 . Then we get the cube equation [ − / ] x + [ r ( + ) / ] x − − = 0 . This implies that the model has trivial equilibrium point = 0 or the non - trivial bio - economic equilibrium points are the roots of x − ( + ) x +

( 1 + ( / ) ) = 0 which are

= ( 1 / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) )

= ( 1 / 2 ) ( + ) − ( + ) − 4 ( 1 + ( / ) ) (5) www.tjprc.org [email protected] Mathe matical Bio Economics of Fish Harvesting with Critical Depensation in Lake Tana 5

Provided that ( + ) > 4 ( 1 + ( / ) ) .

The stability analysis of the equilibrium points is obtained by identifying the algebraic sign of the first derivative of the function at each equilibrium points. That is let ( / ) = ( ) = [ 1 − ( / ) ] [ ( / ) − 1 ] − implies ( ) = ( − / ) + ( / ) ( + ) − ( + ) . So that its firstderivative is ′ ( ) = ( − 3 / ) +

( 2 / ) ( + ) − ( + ) . Since ′ ( 0 ) = − ( + ) < 0 which implies that the trivial equilibrium point = 0 is ′ ′ stable. We have also ( ) = ( − 3 / ) + ( 2 / ) ( + ) − ( + ) where is in (5). Thus ( ) = +

2 − , this implies that is stable when ( + 2 ) < and is unstable when ( + 2 ) > where =

′ ( ( + ) / 2 ) + [ ( + ) / 2 ] ( + ) − 4 ( 1 + ( / ) ) . Similarly ( ) = ( − 3 / ) + ′ ( 2 / ) ( + ) − ( + ) where is in (5). Thus ( ) = + 2 − Ω , this implies that is stable when

( + 2 ) < Ω and is unstable when ( + 2 ) > Ω

where Ω = ( ( + ) / 2 ) − [ ( + ) / 2 ] ( + ) − 4 ( 1 + ( / ) ) .

3.2 Maximum Sustainable Yield (M SY ) of the Critical Depensation Model

Schaefer catch equation is a bilinear short - term harvest function and it assumes that effort always removes a constant proportion of the stock.

( , ) = (6)

Where = catch measured in terms of biomass; fishing effort and is a constant catchability of coefficient. And substituting the non - trivial bio - economic equilibrium points of (5) in (6) gives the harvesting function as a function of effort . Let ( , ) = ( ) and ( , ) = ( ) then we got

( ) = ( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) ) (7)

( ) = ( / 2 ) ( + ) − ( + ) − 4 ( 1 + ( / ) ) (8)

The effort at the maximum sustainable yield denoted by is obtained by making the first derivative of with respect to the effort equal to zero. That is

( ) ( ) = [ + + + − 4 1 + ] + [ ] = 0 (9) ( ) ( ( / ) )

( ) ( ) = [ + − + − 4 1 + ] − [ ] = 0 (10) ( ) ( ( / ) )

From equation (9) we do have the following

= ( / 9 ) η ± + + + where η = + − 4 . Then we have

= ( / 9 ) η + + + + > 0

= ( / 9 ) η − + + + < 0

And therefore we have www.tjprc.org [email protected] 6 Getachew Abiye Salilew

= = ( / 9 ) + − 4 + + + +

And thus the corresponding Maximum Sustainable Yield in (7) to be

= ( ) = ( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) )

= ( / 18 ) ( + ) + ( + ) − 4 [ 1 + ( 1 / 9 ) ]

/ Where = η + + + + .

From equation (10) we do have the following

= ( / 9 ) η ± + + + . Then we have

= ( / 9 ) η + + + + > 0

= ( / 9 ) η − + + + < 0

And therefore we have

= = ( / 9 ) + − 4 + + + +

And thus the corresponding Maximum Sustainable Yield in (8) to be

= ( ) = ( / 2 ) ( + ) − ( + ) − 4 ( 1 + ( / ) )

= ( / 18 ) ( + ) − ( + ) − 4 ( 1 + ( 1 / 9 ) )

3.3 The Open Access Yield ( OAY) for t he Critical Depensation Model

A work done in [7] shows that economic models of 0020 fishery are underlined by biological models and it is impossible to formulate any useful economic model of fishery without specifying the underlining biological dynamics of the fishery. Based on constant price and unit cost of effort the total revenue deno ted by will be calculated using the formula ( ) = . ( ) , where is the average price per kilogram of fish. The relationship between cost and effort is assumed to be linear and then the total cost of fishing effort denoted by is defined as ( ) = . , where is the unit cost of effort that includes cost of labor and capital and is the unit of effort and thus the total economic rent of fishery denoted by defined as

( ) = ( ) − ( ) (11)

At the open access point, total fishing costs are equal to total revenues from the fishery. Then the open access effort is obtained by equating ( ) = ( ) . Where ( ) = and ( ) = which yields = . To calculate the effort for the Open A ccess Yield we used the two non trivial equilibria of (5).And thus we have two equations namely = and = . And substituting their corresponding values respectively gives

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( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) ) = (12)

( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) ) = (13)

From (12) we have = 0 or = ( / ) [ ( c ( + ) / ) − ( c / ) − ] . That is = ( / ) [ ( c ( + ) / ) − ( c / ) − ] provided that ( c ( + ) / ) > ( c / ) + . And thus the corresponding Open Access Yield in (7) to be

= ( ) = ( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) )

= ( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( 1 / ) )

Where = [ ( c ( + ) / ) − ( c / ) − ] .

From (13) we have = 0 or = ( / ) [ ( c ( + ) / ) − ( c / ) − ] . And thus the corresponding Open Access Yield in (8) to be

= ( ) = ( / 2 ) ( + ) − ( + ) − 4 ( 1 + ( / ) )

= ( / 2 ) ( + ) − ( + ) − 4 ( 1 + ( 1 / ) )

The Maximum Economic Yield ( MEY) Of Critical Depensation Model

The maximum economic yield is attained at the profit maximizing level of effort which is obtained using equation

( ) ( ) ( ) (11). So that = 0 this implies that = . To ca lculate the effort for the Maximum Economic Yield we used the two non - trivial equilibria and in (5). And thus we have two equations namely ( ) = ( ) ( ) ( ) and = . And substituting the corresponding values of and in these equations give respectively

( / 2 ) [ ( + ) + ( + ) − 4 1 + ( / ) ] = [ ] (14)

( / 2 ) [ ( + ) − ( + ) − 4 1 + ( / ) ] = [ ] (15)

From equatio n (14) we do have the following

⎡ ⎤ ( − 4 / ) ⎢ ⎛ ⎞ ⎥ ( + ) + ( + ) − 4 1 + ( / ) + = 2 ⎢ ⎥ 2 ( + ) − 4 1 + ( / ) ⎣ ⎝ ⎠ ⎦

± | | Thus we have = . Where = ( − ) and = ( 2 / ) − ( + ) . ( )

Therefore, the efforts at maximum economic yield are:

= 3 − + | | + 3 Or = 3 − − | | + 3

And thus the corresponding Maximum Economic Yields in (7) to be

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( ) ( ) = = / 2 + + + − 4 1 + /

= [ 3 ( − ) + ] ( + ) + ( + ) − 4 [ 1 + [ 3 + ] ]

Where = − + | | + 3

( ) ( ) = = / 2 + + + − 4 1 + /

= ( 3 + ) ( + ) + ( + ) − 4 1 + ( 3 + )

Where = − − | | + 3

From equation (15) we do have the following

( / ) ( + ) − ( + ) − 4 1 + ( / ) − = ( ) ( / )

We have the same effort as the above = ( 3 − ) ± | | + 3 / ( 18 )

= 3 − + | | + 3 Or = 3 − − | | + 3

And thus the corresponding Maximum Economic Yields in (8) to be

( ) ( ) = = / 2 + − + − 4 1 + /

= ( 3 + ) ( + ) − ( + ) − 4 1 + ( 3 + )

( ) ( ) = = / 2 + − + − 4 1 + /

= ( 3 + ) ( + ) − ( + ) − 4 1 + ( 3 + )

4. PARAMETER ESTIMATION 4.1 Basic Parameters Estimation

Time series data from October to December in 2014 in case of Lake Tana is collected for the variables namely: the total catch, the effort, the average price of fish. These data collected from the statistics section of No.1 Lake Tana Fishing Co - operative or ganization. The variable catch was expressed in weight of biomass in kilo grams while the variable effort, which is a composite of input, was expressed in terms of number of boats, nets, traps, vessels and with regard to the variable fish price, average va lue was taken. Beyond this; informal discussion was held with fishermen and fishery officers to gather their perception on the past and present level of exploitation of the fishery.

From Lake Tana we collect data to have our parameter estimation namely: intrinsic growth rate of the fish ( ),

www.tjprc.org [email protected] Mathe matical Bio Economics of Fish Harvesting with Critical Depensation in Lake Tana 9 carrying capacity of Lake Tana ( ), the catchability constant ( ), cost of the effort ( ), price of the fish ( ) and the thresh hold value . To estimate these parameters based on Lake Tana, we have collected data for the number of fish produced in kilogram in the three months of 2014 as follows.

Table 1: Fish Production in Kilogram: Source No.1 Lake Tana Fishing Co - Operative Cat Fish in Total Fish in Month Tilapia in K gm Barbus in Kgm Kgm Kgm October 1012 30.8 60 1102.8 November 38522 1563.5 2666 42751.5 December 6060 254.8 677 6991.8 Total 45594 1849.1 3403 50846.1

Carrying Capacity ( ) and the Catchability Constant ( )

Since 3600 is the area of Lake Tana and 10 is its mean depth, thus the estimated volume of Lake Tana is: = × ℎ = 36 × 10 . And approximately one tilapia fish weigh 2 fish has a volume of 2 . 8 , and th us 36 × 10 has a capacity to carry = 2 . 57145 × 10 kg of fish. The parameter of catchability constant ( ) can be estimated by the formula:

. = = ℎ ℎ . ×

= 2 . 197054 × 10

Cost of the Effort ( )

In a normal production season a single trip by a boat with 20 gillnets of 10cm mesh size and 100m long is expected to bring 100kg of whole fish per day. The total cost needed for fuel, boat, fishing net and human resource for one trip or to produce 100kg of fish is 650 birr or 6.5 birr/kg of fish. To transport fish from harvesting sites to the processin g unit, 80 more birr is needed and the t otal expenditure reaches to 730 birr or 7.30 birr/kg for one quintal of fish. The following table shows the cost of effort for harvesting fish from Lake Tana.

Table 2: Cost Estimation of Effort to Produce 100kg of Fish in One Day Acti vities Price in Birr/day to Produce 100kg Fish Labour 200 birr Gill Net 180 birr Fuel 250 birr Boat rent 20 birr Transport 80 birr Total 730 birr

From table 2 we have the cost is = 7 . 3 / for one boot. But in one day there are 25 boats on the whole Lake Tana activate to harvest fish. And hence we can estimate the cost of fish as a whole = 25 × 7 . 3 / ℎ = 182 . 50 / ℎ in one day.

Price of Fish

The following table consists the average price of three types of fish as indicated below

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Table 3: Average Price of Fish Operative Types of Fish Species Average Price per Kilogram Tilapia 16 birr/kg Barbus 8 birr/kg Catfish 9 birr/kg

And then the price of fish calculated as the mean price as = ( 16 + 8 + 9 ) / 3 = 11 / .

The Thresh Hold Value

The effort for Open Access Yield is obtained by the formula = ( 1 − ( / ) ) ( ( / ) − 1 ) assumed to be non - zero and thus to be exist, ( 1 − ( / ) ) ( ( / ) − 1 ) > 0 . This implies that 0 < < < where = 7 . 551434371167 × 10 . That is < 7 . 551434371167 × 10 and hence let us take the thresh hold value = 7 × 10 .

Finally we summarize our parameters es timation by the following table

Table 4 Parameters Symbol Value Carrying capacity 2 . 57 × 10 ℎ Thresh hold value 7 × 10 ℎ Catchability constant 2 . 197 × 10 Cost of effort 182 . 50 / Price of effort 11 / Intrinsic growth rate 0 . 5

Parameter Estimation for the Critical Depensation Model

Table 5 : Parameter Estimation for the Critical Depensation Model Value Description Formula [ kg per day] / ( / 9 ) + − 4 + + + + 17 . 69 × 10 ( / ) [ 1 − ( / ) ] [ ( / ) − ] 1 . 74 × 10 = ( / 18 ) Ψ − Φ + | ( 2 c / ) − ( + ) | √ Φ + Ψ 6 . 21 × 10 ( / 18 ) ( + ) + ( + ) − 4 ( 1 + ( / 9 ) ) 67 . 12 × 10 ( / 2 ) ( + ) + ( + ) − 4 ( 1 + ( / ) ) 9 . 81 × 10 3 . 51 × 10 ( ξ / 36 ) ( + ) + ( + ) − 4 [ 1 + ( ξ / 18 ) ]

Where Φ = ; Ψ = 3 ; ξ = Ψ − Φ + | θ | √ Φ + Ψ

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Table 6 : Parameter Estimation for the Critical Depensation Model for Different Values of r

4.2 The Economic Model Estimation

In this section we calculated the economic rent or profit for critical depensation models by considering our real data collected from Lake Tana. The total cost of harvesting denoted by depends on the amount of effort being expended. For simplicity, harvesting costs are taken to be a linear function of effort defined by where is the cost per unit of harvesting effort, taken to be a constant and this equation imposes the assumption t hat harvesting costs are linearly related to fishing effort. Let denote the total revenue from harvesting some quantity of fish. The total revenue will depend on the quantity harvested, so we have = ( ) . In a commercial fishery, the approp riate measure of gross benefits is the total revenue that accrues to firms. Assuming that fish are sold in a competitive market, each firm takes the market price as given and so the revenue obtained from a harvest is given by ( ) = ( ) . And finally the economic rent or profit denoted by is defined in terms of total cost and total revenue by = − .

The Critical Depensation Economic Model Parameter Estimations

In this case we do have the following parameter estimation with the harvest at the given type of Effort is obtained by

Table 7 : The Critical Depensation Economic Model Parameter Estimations

( ) ( ) ( ) 2 . 1 2 . 5 17 . 7 1 . 7 6 . 2 67 . 1 351 . 0 0.5 182 . 5 11 9 . 8 × 10 11 × 10 11 × 10 13 × 10 10 × 10 9 × 10 10 × 10 12 × 10 11

And thus the profit or the economic rent with different type of harvest function is given by:

The critical depensation economic model for Maximum Sustainable Yield (MSY)

The critical depensation economic model for Open Access Yield (OAY)

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The critical depensation economic model for Maximum Economic Yield (MEY)

5. DISCUSSION S AND RESULTS

This study focuses on Lake Tana fishing activity or fishing management in which to identify whether overfishing exists or not. To do this we have collected real data from the source namely: three months catch fish, number of boats in the whole Lake. Biolog ically overfishing occurs when fish species are caught at a rate faster than they can reproduce. A continuous increase of effort might result in an increase catch but at a decreasing rate or more effort may result in proportionality a smaller harve st, which means the additional effort will have less return.

To determine whether or not over fishing occurs in Lake Tana we have considered parameter estimation for different cases of the natural growth rates = 0 . 5 , = 1 and = 2 in table 6 . Without loss of any generality we prefer to analyze the tabular approximate value for = 0 . 5 as our choice of the parameter is similar to that of = 1 and = 2 . There is overfishing in Lake Tana with the critical depensation model for values of effort less than the critical mass quantity and there is no over fishing for values of effort greater than the thresh hold value . Here we consider the critical depensation model on Lake Tana with parameter estimation: the natural growth rate = 0 . 5 , ca rrying capacity of Lake Tana = 2 . 57 × 10 kg of fish, the thresh hold value = 7 × 10 kg of fish, effort for maximum sustainable yield = 17 . 7 × 10 kg of fish, effort for open access yield = 1 . 7 × 10 kg of fish, effort for maximum economic yield = 6 . 2 × 10 kg of fish. And thus we observed from these values that all efforts are less than the thresh hold value and also less than the carrying capacity of Lake Tana therefore there is overfishing if we considered the critical depensation model.

I n the economic point of view we observe that the approximate price of total population fish in Lake Tana is 28 . 2 × 10 which is obtained by multiplying the carrying capacity of Lake Tana = 2 . 57 × 10 ℎ and the price of one kilo gram of fish 11 birr. In the critical depensation model we take the approximate price of fish at the thresh hold value 0 . 77 × 10 which is obtained by multiplying the thresh hold value of Lake Tana = 0 . 77 × 10 ℎ and the price of one kil o gram of fish 11 birr. And thus we do have parameter estimations: ( ) = 706 . 06 × 10 , ( ) = 104 . 73284 × 10 and ( ) < 0 . And thus in the critical depensation model the economic rent or the profit obtained by all kinds of effort are greater than the price of the total population of fish in Lake Tana and therefore there is overfishing.

6. CONCLUSIONS AND RECOMMENDATIONS

To keep the sustainability of fish we must increase effort levels but not exceeding the carrying capacity in the critical - depensation model. Even though the efforts are less than the carrying capacity , then there is overfishing in Lake

Tana using critical - depensation Model. Because the efforts are less than the critical mass quantity which implies that the natural growth rate of the fish is below zero. Therefore there is overfishing in Lake Tana using critical - depensation

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model. To continue the sustainability of fish in Lake Tana the effort level not be below the critical mass quantity and not be above the carrying capacity . Since the culture of stakeholders around Lake Tana to keep appropriate data is low and thus we cannot collect the data that we want. Therefore in the future work to have exact data the stake holders must be informed so th at giving real data gives their fishing management to be good.

ACKNOWLEDG E MENTS

Almighty God, thank you! Next to God I want to thank heartily my advisor Dr. Temesgen Tibebu, not only for his supervision and advice but also for his willingness to consult him at any point of time, his patience and the experience he shared me. Special gratitude goes to Bahir Dar; Fish production and Marketing Industry; Lake Tana No. 1 Fis hing Co - operative Organization and Livestock Resources Development and Production Agency for their assistance during the collection of data. I dearly thank my family for never ending love and support. Thank you very much.

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