Fishharvestingexperiencedbyde

Global Journal of Science Frontier Research: F

Mathematics and Decision Sciences Volume 17 Issue 2 Version 1.0 Year 2017 Type : Double Blind Peer Reviewed International Research Journal

Publisher: Global Journals Inc. (USA)

Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Fish Harvesting Experienced by Depensation Growth Function By Getachew Abiye Salilew Madda Walabu Universty Abstract- Getachew [1] introduced there is overfishing in Lake Tana, Amhara, Ethiopia using critical depensation model. In this paper, we applied this technique for depensation model. We investigated that if we consider depensation model we found that there is overfishing. Keywords: mathematical bio-economics, overfishing. GJSFR-F Classification: MSC 2010: 91B84

FishHarvestingExperiencedbyDepensationGrowthFunction

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Fish Harvesting Experienced by Depensation Ref Growth Function 201 r

Getachew Abiye Salilew ea Y

371 Abstract - Getachew [1] introduced there is overfishing in Lake Tana, Amhara, Ethiopia using critical depensation model. In this paper, we applied this technique for depensation model. We investigated that if we consider depensation model we found that there is overfishing. Keywords : mathematical bio-economics, overfishing.

I. Introduction II 14. ue ersion I -

s The economics of renewable resource use is essentially a multi-disciplinary s I undertaking, integrating both biological and economic aspects. When modeling the dynamics of the resource, one has to choose a level of analysis. An intuitive entity is the XVII organism itself (a ﬁsh, a tree, or a cow) that experiences growth and mortality. While growth and mortality determine the dynamics of the existing number of individuals, it

Economics of Fish Harvesting with of Fish Harvesting Economics with is the potential to reproduce which characterizes renewable resources. Resources whose ) - F

reproduction is completely outside of the control of the resource users can perhaps best ( be analyzed in the framework of “eating a cake of unknown size”. We will meet resources whose reproduction can be completely controlled when discussing forestry Research Volume

TJPRC: JMCAR TJPRC: 3(2016) 1 issues, but aquaculture could be another example. For most resource management problems however, it will be useful to model a reproduction function which depends in some (possibly highly nonlinear, possibly very stochastic) way on the existing number of Frontier MathematicalBio individuals, which in turn are inﬂuenced by the current exploitation regime. In the

, , absence of regulation control over harvesting behavior, the resource stocks are subject to open access [1]. In addition to the viewpoint of an organism, one could also focus on the dynamics of the underlying processes. Most models will take an aggregated view Science of

Salilew means analyzing a ﬁshery, a forest, an ecosystem as a whole. e In different renewable resource management, it is important to balance ecological Journal

Abiy and economic needs. For example if we consider one of the renewable resource (fish), the fishery management is the consideration of the ecological effects of harvesting.

Fisherman work to provide fish for a growing human population but because of this Global some fish populations have been dangerously declining. A major focus in fishery management is how best to ensure harvesting sustainability [2, 4, 5, 6]. The object of Getachew CriticalDepensation Lake in Tana, the management is to devise harvesting strategies that will not drive species to

1. extinction. Therefore, the notion of persistence, extinction times of the populations and precautionary harvesting policy, is always critical. A control variable of every fishery

Author: Department of Mathematics, College of Natural and Computational Science, Madda Walabu Universty, Bale Robe, Ethiopia. e-mail: [email protected]

management is the fishing effort [3, 8], which is defined as a measure of the intensity of fishing operations. As fishery management is the balance between harvesting 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. Mathematical bio economics is the study of the management of renewable resources. It takes into consideration not only economic questions like 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. In this paper we consider depensation (weak Alee effect) deterministic model with a constant harvest Ref rate as well as time dependent. Optimization and numerical calculations were used to determine the harvest rate that produces maximum yield under different population 201 r

density scenarios. The dynamic mathematical models set on the background of biology F 3. ea

Y and economics knowledge. The integration of these seemingly different subjects namely Epidemiology,

mathematics, biology and economics creates the source of interesting results and give . B 381 valuable applications for the peoples living with fishing activities and those policy rauer C. and Castillo makers who involved control of overfishing.

II. Ma thematical Biology of Dep ensation Model V II Deterministic models of fishery populations can be classified into three types Springe

ue ersion I namely compensation, depensation and critical depensation. Compensation model is a s s growth type where population declination is compensated by increased growth rate. I

Depensation model is the opposite case to composition growth model. The critical r - Verlag, 2001. Verlag, XVII depensation modelis the generalized logistic model which is extremely in opposite of the - depensation model. A population’s dynamics are depensatory or depensation is said to Chavez, occur if the per- capita rate of growth decreases as the density decreases to low levels.

) Component of the life-history such as fecundity or survival during a particular stage or

F Math

( the mechanisms that affect these components (such as group defense or mate-finding difficulty) are called depensatory if they decrease the per-capita growth rate as

abundance declines to low levels. Depensation model is the label most often used in and Biology ematical Population in Models fisheries. The strong depensation model is called critical depensation model. By the Research Volume work done [9] some populations experience reduced rates of survival and reproduction when reduced to very low densities. Mathematical biology expression of depensation is

Frontier given by growth model as: ( / ) = [1 ( / )] (1) 𝑎𝑎 Science Here in the growth model𝑑𝑑𝑑𝑑 (1),𝑑𝑑𝑑𝑑 ( )𝑟𝑟rep𝑥𝑥 resents− 𝑥𝑥 fish𝑘𝑘 biomass, represents intrinsic of growth rate of fish, is ecological carrying capacity, 0 and 1, and ( / ) 𝑥𝑥 𝑡𝑡 𝑟𝑟 growth rate of fish without harvest. Model (1) has the property that for > 1 there is,

Journal at low stock levels, depensation,𝑘𝑘 which is a situation where𝑎𝑎 ≥ the proportionate𝑎𝑎 ≠ growth𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 rate is an increasing function of the stock size, as opposed to being𝑎𝑎 a decreasing

Global function (compensation) in the simple logistic case where = 1. The biological growth

model (1) exhibiting depensation at the stock level below, 0, and compensation thereafter as shown the figure below. 𝑎𝑎 𝑦𝑦

𝑑𝑑𝑑𝑑

0 Notes 𝑑𝑑 𝑦𝑦 𝑘𝑘 Figure 1: Growth curve of depensation model for = 1.4, = 2 = 3.2, 0 = 1

In the figure 1 we have shown the rate change 𝑟𝑟 curve of𝑎𝑎 the 𝑘𝑘depensation𝑦𝑦 model 201 for some positive particular values of the parameters as shown. The curve is plotted for r ea the population size function ( ) versus the population rate change function = Y ( / ). The maximum rate change of population, ( ), occurs when the′ 391 𝑥𝑥 𝑡𝑡 𝑥𝑥 population size be ( ) = ( /( + 1)) for > 0 and the corresponding′ maximum rate 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡 𝑀𝑀𝑀𝑀𝑀𝑀 𝑥𝑥 change of population is given by ( ) = ( /( + 1)) 1 ( /( + 1)) . 𝑥𝑥 𝑡𝑡 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑎𝑎

′ 𝑎𝑎 V

III. Solution𝑀𝑀of𝑀𝑀𝑀𝑀 t𝑥𝑥he D�ep𝑟𝑟 en𝑎𝑎𝑎𝑎sation𝑎𝑎 Mo�de−l 𝑎𝑎 𝑎𝑎 �� II ue ersion I s The solution of depensation model (1) with initial condition (0) = 0 is s obtained as follows. Using techniques of separable variables the model (1) can be I

rewritten as [ / ( )] = ( / ) and integrating both sides we get 𝑥𝑥 [ 1/ 𝑥𝑥( XVII )] = ( / ) . T𝑎𝑎 o integrate the left hand side, we have to consider the following𝑎𝑎 𝑑𝑑𝑑𝑑 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑟𝑟 𝑘𝑘 𝑑𝑑𝑑𝑑 ∫ 𝑥𝑥 𝑘𝑘 − two cases by assuming that the initial value (0) = 0; is an integer and > 1 and

𝑥𝑥 𝑑𝑑𝑑𝑑 ∫ 𝑟𝑟 𝑘𝑘 𝑑𝑑𝑑𝑑 then applying integration by partial fraction. )

𝑥𝑥 𝑥𝑥 𝑎𝑎 𝑎𝑎 ( Case1: Let is an even integer so that = 2 , Z+.

[1/ ( ) ] = [1/ 2 ( )] = [1/( 2) ( )] = ( / ) + , where 𝑎𝑎 𝑎𝑎 𝑛𝑛 𝑛𝑛 ∈

𝑎𝑎 𝑛𝑛 𝑛𝑛 Research Volume ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑟𝑟 𝑘𝑘 𝑡𝑡 ℓ If = 1 then = 2 and thus we do have [1/ ( )] = [1/ 2( )] . ℓ ∈ ℛ 𝑎𝑎 The solution is obtained using integration by partial fraction and is = Frontier 𝑛𝑛 𝑎𝑎 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 ∫ ∫ 𝑥𝑥 𝑘𝑘 0 4 + ln . If = 2 then = 4 and thus we do have [1/ ( 𝑘𝑘−𝑥𝑥 )] 𝑥𝑥 = 0 0 𝑙𝑙𝑙𝑙 � � − 𝑥𝑥 𝑘𝑘 Science ( / ) . The𝑘𝑘− 𝑥𝑥solution𝑥𝑥 is obtained using integration by partial fraction and is:

𝑟𝑟𝑟𝑟𝑟𝑟 � � − 𝑛𝑛 𝑎𝑎 ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 of 2 3 2 3 𝑟𝑟 𝑘𝑘 𝑡𝑡 3 0 ln 2 3 = + ln 2 3 2 3 0 0 2( 0) 3( 0) Journal 𝑥𝑥 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑘𝑘 𝑘𝑘 � � − − − 𝑟𝑟𝑘𝑘 𝑡𝑡 � � − − − In general𝑘𝑘 − 𝑥𝑥 by 𝑥𝑥mathematical𝑥𝑥 𝑥𝑥 induction for𝑘𝑘 − 𝑥𝑥= 2 using𝑥𝑥 partial𝑥𝑥 fraction𝑥𝑥 we get the solution: Global 𝑎𝑎 𝑛𝑛 + + + = ( 1 1 + 2 2 + + + ) = ( / ) + ( 2) ( ) 2 ( 2)2 ( 2) 𝑑𝑑𝑑𝑑 𝐴𝐴 𝑥𝑥 𝐵𝐵 𝐴𝐴 𝑥𝑥 𝐵𝐵 𝐴𝐴𝑛𝑛 𝑥𝑥 𝐵𝐵𝑛𝑛 𝑐𝑐 � 𝑛𝑛 � ∙∙∙ 𝑛𝑛 𝑑𝑑𝑑𝑑 𝑟𝑟 𝑘𝑘 𝑡𝑡 ℓ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 1 𝑥𝑥 1 𝑘𝑘 − 𝑥𝑥 1 1 This is the general solution. Here 1 = 2 , 2 = 2 2 , , 1 = 4 , = 2; 1 1 1 1 1 𝑛𝑛 𝑛𝑛 − 𝑛𝑛− 𝑛𝑛 1 = 2 1 , 2 = 2 3 , … , 1 = 3 , =𝐴𝐴; =𝑘𝑘 𝐴𝐴= 2 𝑘𝑘and ∙∙∙ 𝐴𝐴. 𝑘𝑘 𝐴𝐴 𝑘𝑘 𝑛𝑛 − 𝑛𝑛 − 𝑘𝑘 𝑘𝑘 𝑛𝑛− 𝑘𝑘 𝑛𝑛 𝑘𝑘 𝑛𝑛 𝑘𝑘 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝑐𝑐 𝐴𝐴 ℓ ∈ ℛ ©2017 Global Journals Inc. (US) Fish Harvesting Experienced by Depensation Growth Function

Case 2: Let is an odd integer so that = 2 + 1, Z+. [1/ ( )] = 1/ (2 +1)( ) = ( / ) + , where 𝑎𝑎 𝑎𝑎 𝑛𝑛 𝑛𝑛 ∈ 𝑎𝑎 𝑛𝑛 3 ∫ 𝑑𝑑 If𝑘𝑘= −𝑑𝑑 1𝑑𝑑𝑑𝑑 then ∫� =𝑑𝑑 3 and 𝑘𝑘thus −𝑑𝑑we�𝑑𝑑𝑑𝑑 have,𝑟𝑟 𝑘𝑘[1/𝑑𝑑 (ℓ )] ℓ ∈= ℛ( / ) . The solution is obtained using integration by partial fraction and is: 𝑛𝑛 𝑎𝑎 ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑟𝑟 𝑘𝑘 𝑡𝑡 2 2 2 0 ln 2 = + ln 2 2 0 0 2( 0) 𝑥𝑥 𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑘𝑘 Notes � � − − 𝑟𝑟𝑘𝑘 𝑡𝑡 � � − − If = 2 the𝑘𝑘n − 𝑥𝑥= 5𝑥𝑥and 𝑥𝑥thus we do have𝑘𝑘 − 𝑥𝑥 [1/ 𝑥𝑥5( 𝑥𝑥)] = ( / ) . The

201 solution is obtained using integration by partial fraction and is:

r 𝑛𝑛 𝑎𝑎 ∫ 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑟𝑟 𝑘𝑘 𝑡𝑡 ea

Y 2 3 4 2 3 4 4 0 ln 2 3 4 = + ln 2 3 4 401 2 3 4 0 0 2( 0) 3( 0) 4( 0) 𝑥𝑥 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 � � − − − − 𝑟𝑟𝑘𝑘 𝑡𝑡 � � − − − − 𝑘𝑘In − 𝑥𝑥 general𝑥𝑥 by𝑥𝑥 mathematical𝑥𝑥 𝑥𝑥 induction for𝑘𝑘 − 𝑥𝑥= 2 +𝑥𝑥 1 and𝑥𝑥 using partial𝑥𝑥 fraction𝑥𝑥 we get the solution: V 𝑎𝑎 𝑛𝑛 II 1 2 3 2 2 +1 ue ersion I

s = ( + + + + + + ) = ( / ) +

s 2 +1 2 3 2 2 +1 ( ) 𝑛𝑛 𝑛𝑛 I 𝑑𝑑𝑑𝑑 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐵𝐵 � 𝑛𝑛 � ∙∙∙∙ 𝑛𝑛 𝑛𝑛 𝑑𝑑𝑑𝑑 𝑟𝑟 𝑘𝑘 𝑡𝑡 ℓ 1 1 1 1 XVII 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑘𝑘 − 𝑥𝑥 This is the general solution. Where 1 = 2 +1 , 2 = 2 , 3 = 2 1 , 4 = 2 2 ,

1 1 1 𝑛𝑛 𝑛𝑛 𝑛𝑛 − 𝑛𝑛 − , 2 = 2 , 2 +1 = , = 2 +1 and 𝐴𝐴 . 𝑘𝑘 𝐴𝐴 𝑘𝑘 𝐴𝐴 𝑘𝑘 𝐴𝐴 𝑘𝑘

𝑛𝑛 ) ∙∙∙ 𝐴𝐴 𝑛𝑛 Whe𝑘𝑘 n w𝐴𝐴e𝑛𝑛 combine𝑘𝑘 𝐵𝐵the 𝑘𝑘above twoℓ ∈cases ℛ we found that + and 2 the F ( general implicit solution of the depensation model is: ∀𝑎𝑎𝑍𝑍 ∈ 𝑎𝑎 ≥ 1 1 = = ( 1) + (2) ( ) =1

Research Volume 𝑛𝑛 𝑑𝑑𝑑𝑑 𝑥𝑥 𝑎𝑎− 𝑘𝑘 𝑎𝑎− 𝑎𝑎 𝑥𝑥 𝑘𝑘−𝑥𝑥 𝑘𝑘−𝑥𝑥 𝑛𝑛 𝑛𝑛 𝑥𝑥 Result (3) is the∫ required𝑙𝑙 𝑙𝑙particular� � − ∑ implicit� �solution𝑟𝑟 𝑘𝑘 of the𝑡𝑡 depensationℓ model (1).

Frontier 1 1 1 0 1 1 =1 = ( ) + =1 (3) 𝑛𝑛 0 0 𝑛𝑛 𝑥𝑥 𝑎𝑎− 𝑘𝑘 𝑎𝑎− 𝑥𝑥 𝑎𝑎− 𝑘𝑘

Science 𝑙𝑙𝑙𝑙 �𝑘𝑘−𝑥𝑥� − ∑𝑛𝑛 𝑛𝑛 �𝑥𝑥 � 𝑟𝑟 𝑘𝑘 𝑡𝑡 𝑙𝑙𝑙𝑙 �𝑘𝑘−𝑥𝑥 � − ∑𝑛𝑛 𝑛𝑛 �𝑥𝑥 �

of The following graph represents the stock level of the depensation model.

( ) Journal 𝑥𝑥 𝑡𝑡

Global 𝑘𝑘

𝑡𝑡 Figure 2: Typical solution curve for depensation model for = 1.6, = 2.0, = 3.0

In figure 2 we have time series plot for depensation model which verifying local stability of the three equilibrium point in model (1). The depensation model has two equilibrium points namely the trivial and non-trivial equilibrium points = 0 or = dx x respectively which are obtained by making = rxa 1 = 0. The equilibrium point dt k 𝑥𝑥 𝑥𝑥 𝑘𝑘 = 0 is semi stable since f (0) = 0. The non-trivial equilibrium point = is stable � − � for > 0 and unstable for < 0[7, 11]. 𝑥𝑥 ′ 𝑥𝑥 𝑘𝑘 Notes 𝑟𝑟 IV. Mathematical𝑟𝑟 Bio-Economics of Depensation Model A fishery is an area with an associated fish or aquatic population which is harvested for its commercial or recreational value. Population dynamics describes the 201 r

ways in which a given population grows and shrinks over time, as controlled by birth, ea death, and emigration or immigration. It is the basis for understanding changing fishery Y patterns and issues such as habitat destruction, predation and optimal harvesting rates. 411 With the natural positive population growth the population size can be brought down whenever harvesting is introduced. Schaefer catch equation is a bilinear short-term harvest function and it assumes that effort always removes a constant proportion of the V

stock. Depensation Mathematical Bio-Economics model is given by = ( ) ( , ) II 𝑑𝑑𝑑𝑑

a ue ersion I

where f(x) = rx [1 (x/k)] is the growth function of fish and ( , ) = is the s 𝑑𝑑𝑑𝑑 𝑓𝑓 𝑥𝑥 − ℎ 𝐸𝐸 𝑥𝑥 s

harvest function of fish. And thus we do have I − ℎ 𝐸𝐸 𝑥𝑥 𝑞𝑞𝑞𝑞𝑞𝑞 ( / ) = [1 ( / )] (4) XVII 𝑎𝑎 Here in the model (4),𝑑𝑑𝑑𝑑 (𝑑𝑑𝑑𝑑) rep𝑟𝑟resents𝑥𝑥 − fish𝑥𝑥 𝑘𝑘biomass− 𝑞𝑞𝑞𝑞𝑞𝑞, represents intrinsic growth rate of fish, is ecological carrying capacity, 0 and 1, is time, ( / ) is

𝑥𝑥 𝑡𝑡 𝑟𝑟 )

growth rate of the fish with harvest function ( , ). F 𝑘𝑘 𝑎𝑎 ≥ 𝑎𝑎 ≠ 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ( a) Equilibrium Points of Bio-Economics of Depensation Model ℎ 𝐸𝐸 𝑥𝑥 The equilibrium points of model (4) are obtained by making ( / ) = 0

[1 ( / )] = 0. This implies that the trivial equilibrium point = 0 or the Research Volume 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ⟺ non-𝑎𝑎 trivial equilibrium point: 1 1 ( / ) = 0 for > 1 are the equilibrium 𝑟𝑟𝑥𝑥 − 𝑥𝑥 𝑘𝑘 − 𝑞𝑞𝑞𝑞𝑞𝑞 𝑥𝑥 𝑎𝑎− 2 points. If we take = 2, we get, + = 0. So that the non-trivial Frontier equilibrium points are 𝑟𝑟𝑥𝑥 � − 𝑥𝑥 𝑘𝑘 � − 𝑞𝑞𝑞𝑞 𝑎𝑎 𝑎𝑎 𝑟𝑟𝑥𝑥 − 𝑟𝑟𝑘𝑘𝑘𝑘 𝑞𝑞𝑞𝑞𝑞𝑞 2 2

+ ( ) 4 ( ) 4 Science = = (5) 1 2 2 2 𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟−� 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 of Provided that𝑥𝑥 > 4 𝑟𝑟 and since𝑜𝑜𝑜𝑜 𝑥𝑥 > ( )2𝑟𝑟 4 both equilibrium

points are positive for positive parameters , , and . Journal The stability analysis𝑟𝑟𝑟𝑟 of𝑞𝑞 𝑞𝑞the equilibrium𝑟𝑟𝑟𝑟 points� 𝑟𝑟𝑟𝑟 is −obtained𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 by identifying the 𝑟𝑟 𝑞𝑞 𝐸𝐸 𝑘𝑘

algebraic sign of the first derivative of the function at each equilibrium points. That is, Global ( / ) = ( ) = [1 ( / )] . So, its first derivative for = 2 is, ( ) = 3 2 2 . Since𝑎𝑎 (0) = < 0 implies equilibrium point = 0 is stable. Next 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑔𝑔 𝑥𝑥 𝑟𝑟𝑥𝑥 − 𝑥𝑥 𝑘𝑘 − 𝑞𝑞𝑞𝑞𝑞𝑞 𝑎𝑎 𝑔𝑔′ 𝑥𝑥 𝑟𝑟 1 we have, ( ) = ( /2) + 2 ( )2 4 < 0 this implies that in (5) is 𝑟𝑟𝑟𝑟 − 𝑘𝑘 𝑥𝑥 −1 𝑞𝑞𝑞𝑞 𝑔𝑔′ −𝑞𝑞𝑞𝑞2 𝑥𝑥 1 ′ 1 stable. Further we have, ( ) = + 2 + ( )2 4 this implies that, 𝑔𝑔 𝑥𝑥 − 𝑟𝑟𝑟𝑟 𝑞𝑞𝑞𝑞2 − � 2𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 𝑥𝑥 𝑟𝑟𝑟𝑟 ( )2 4 ′ = is stable if ( )2 4 < 4 and unstable otherwise. 2 2 𝑔𝑔 𝑥𝑥 − 𝑞𝑞𝑞𝑞 � 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟−� 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑥𝑥 𝑟𝑟 � 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 − 𝑞𝑞𝑞𝑞 ©2017 Global Journals Inc. (US) Fish Harvesting Experienced by Depensation Growth Function

b) Maxim um Sustainable Yield (MSY) Of The 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 measured𝐻𝐻 in𝐸𝐸 𝑥𝑥terms𝑞𝑞 𝑞𝑞𝑞𝑞of biomass; fishing effort and is a constant catchability of coefficient. And substituting the non-trivial bio-economic 𝐻𝐻 𝐸𝐸 𝑞𝑞 equilibrium points of (5) in (6) gives the harvesting function as a function of effort . Ref Let ( 1, ) = 1( ) and ( 2, ) = 2( ) then we got 𝐸𝐸 201 𝐻𝐻 𝑥𝑥 𝐸𝐸 𝐻𝐻 𝐸𝐸 𝐻𝐻( 𝑥𝑥) =𝐸𝐸 ( /2𝐻𝐻 )𝐸𝐸 + ( )2 4

r (7) 1 10.

ea Y

𝐻𝐻 𝐸𝐸 𝑞𝑞𝑞𝑞 𝑟𝑟 �𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘� Resources. G. Munro, 421 ( )2 4 2( ) = ( /2 ) (8)

𝐻𝐻 𝐸𝐸 𝑞𝑞𝑞𝑞 𝑟𝑟 �𝑘𝑘𝑘𝑘 − � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝐸𝐸�

The effort at the maximum sustainable yield denoted by is obtained by The Canadian Journal of Economics of Journal Canadian The Jurisdict Extended Fisheries,

V making the first derivative of with respect to the effort equal to zero. That is 𝑀𝑀𝑀𝑀𝑀𝑀 II ( ) 2 2 𝐸𝐸 1 = 0, gives = 0 or = . Thus, = . We have the same result for, ue ersion I 9 9 s 𝐻𝐻 𝐸𝐸 s 𝑑𝑑 𝐻𝐻 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 ( 2)

I 𝑀𝑀𝑀𝑀𝑀𝑀 𝑑𝑑𝑑𝑑 = 0. 𝐸𝐸 𝐸𝐸 𝑞𝑞 𝐸𝐸 𝑞𝑞 𝑑𝑑 𝐻𝐻 XVII An𝑑𝑑𝑑𝑑d thus the corresponding Maximum Sustainable Yield in (7) to be

4 2 1 = 1( ) = 27 )

F 𝑀𝑀𝑀𝑀𝑀𝑀 𝑘𝑘 𝑟𝑟 ( Again the corresponding Maximum𝑀𝑀𝑀𝑀𝑀𝑀 Sustainable𝐻𝐻 𝐸𝐸 Yield in (8) to be

2 2 Property Economics Common and the of ion 2 = 2( ) =

Research Volume 27 𝑀𝑀𝑀𝑀𝑀𝑀 𝑘𝑘 𝑟𝑟

From this we conclude that, 𝑀𝑀𝑀𝑀𝑀𝑀1 = 2 𝐻𝐻 2𝐸𝐸. 15(1982).

Frontier c) The Open Access Yield (OAY) For The Depensation Model 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 A work done in [10] shows that economic models of fishery are underlined by biological models and it is impossible to formulate any useful economic model of fishery

Science without specifying the underlining biological dynamics of the fishery. Based on constant

of price and unit cost of effort the total revenue denoted by will be calculated using

the formula ( ) = . ( ), where is the average price per kilogram of fish. The 𝑇𝑇𝑇𝑇

Journal relationship between cost and effort is assumed to be linear and then the total cost of fishing effort𝑇𝑇𝑇𝑇 denoted𝐸𝐸 𝑝𝑝by𝐻𝐻 𝐸𝐸 is defined𝑝𝑝 as ( ) = . , where is the unit cost of

Global effort that includes cost of labor and capital and is the unit of effort and thus the 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸 𝑐𝑐 𝐸𝐸 𝑐𝑐 total economic rent of fishery denoted by defined as 𝐸𝐸 ( ) =𝑇𝑇𝑇𝑇𝑇𝑇 ( ) ( ) (9) At the open access point, total fishing costs are equal to total revenues from the 𝑇𝑇𝑇𝑇𝑇𝑇 𝐸𝐸 𝑇𝑇𝑇𝑇 𝐸𝐸 − 𝑇𝑇𝑇𝑇 𝐸𝐸 fishery. Then the open access effort is obtained by equating ( ) = ( ). Where ( ) = and ( ) = which yields = . To calculate the effort for the Open Access Yield we used two non-trivial equilibria in (5).𝑇𝑇𝑇𝑇 And𝐸𝐸 thus𝑇𝑇𝑇𝑇 we𝐸𝐸 have two 𝑇𝑇𝑇𝑇 𝐸𝐸 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑇𝑇𝑇𝑇 𝐸𝐸 𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐𝑐𝑐 © 2017 Global Journals Inc. (US) Fish Harvesting Experienced by Depensation Growth Function

equations namely 1 = and 2 = . And substituting their corresponding values respectively gives 𝑝𝑝𝑝𝑝𝑥𝑥 𝐸𝐸 𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑥𝑥 𝐸𝐸 𝑐𝑐𝑐𝑐 + ( )2 4 = (10) 2 𝑝𝑝𝑝𝑝 𝑟𝑟 𝐸𝐸�𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘� 𝑐𝑐𝑐𝑐 ( )2 4 = (11) 2 𝑝𝑝𝑝𝑝 From (10), we have 𝑟𝑟 𝐸𝐸=� 0𝑘𝑘 𝑘𝑘 −=� 𝑘𝑘𝑘𝑘 1 − 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 that� is𝑐𝑐𝑐𝑐 = 1 provided, Notes 2 2 < . 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑝𝑝𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝 𝑂𝑂𝑂𝑂𝑂𝑂 𝑝𝑝𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝 And thus the corresponding𝐸𝐸 Open𝐸𝐸 Access �Yield− to� be 𝐸𝐸 � − � 𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝 201 r

= ( ) = . + ( )2 4 ea 1 1 2 Y 𝑞𝑞 𝑂𝑂𝑂𝑂𝑂𝑂 𝐻𝐻 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 �𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 � 431 𝑟𝑟 4 2 = (1 ) + ( )2 (1 ) 1 2 𝑐𝑐 𝑐𝑐 𝑘𝑘𝑟𝑟 𝑐𝑐 𝑐𝑐

𝑂𝑂𝑂𝑂𝑂𝑂 − �𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − − � V

𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 II Equation (11) gives = 0 or = 2 1 that is = 2 1 ue ersion I

𝑐𝑐𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 s provided, < . s 𝑝𝑝𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝 𝑂𝑂𝑂𝑂𝑂𝑂 𝑝𝑝𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝

𝐸𝐸 𝐸𝐸 � − � 𝐸𝐸 � − � I And thus the corresponding Open Access Yield to be

𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝 XVII = ( ) = ( )2 4 2 2 2 𝑞𝑞 𝑂𝑂𝑂𝑂𝑂𝑂 𝐻𝐻 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 �𝑘𝑘𝑘𝑘 − � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 �

2 4 ) 𝑟𝑟 2 = 1 ( ) (1 ) F 2 2 ( 𝑐𝑐 𝑐𝑐 𝑘𝑘𝑟𝑟 𝑐𝑐 𝑐𝑐 𝑂𝑂𝑂𝑂𝑂𝑂 � − � �𝑘𝑘𝑘𝑘 − � 𝑘𝑘𝑘𝑘 − − � d) The Maximum Economic𝑝𝑝𝑝𝑝 Yield𝑝𝑝 𝑝𝑝𝑝𝑝(MEY) of Depensation𝑝𝑝 𝑝𝑝Model 𝑝𝑝𝑝𝑝𝑝𝑝

The maximum economic yield is attained at the profit maximizing level of effort Research Volume

which is obtained using equation (9). So, [ ( ) / ] = 0 implies [ ( ) /

] = [ ( ) / ]. To calculate the effort for the Maximum Economic Yield we Frontier 𝑑𝑑�𝑇𝑇𝑇𝑇𝑇𝑇 𝐸𝐸 � 𝑑𝑑𝑑𝑑 𝑑𝑑�𝑇𝑇𝑇𝑇 𝐸𝐸 � used two non-trivial equilibria 1 and 2 in (5). And thus we have two equations 𝑑𝑑𝑑𝑑 𝑑𝑑�𝑇𝑇𝐶𝐶 𝐸𝐸 � 𝑑𝑑𝑑𝑑 namely [ ( 1 )/ ] = [ ( )/ ] and [ ( 2 )/ ] = [ ( )/ ]. And 𝑥𝑥 𝑥𝑥 Science substituting the corresponding values of and in these equations give respectively

1 2 of 𝑑𝑑 𝑝𝑝𝑝𝑝𝑥𝑥 𝐸𝐸 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑝𝑝𝑝𝑝𝑥𝑥 𝐸𝐸 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑 ( ) + ( 𝑥𝑥)2 4 𝑥𝑥 = (12)

2 Journal 𝑑𝑑 𝑝𝑝𝑝𝑝 𝑑𝑑 𝑐𝑐𝑐𝑐

𝑑𝑑𝑑𝑑 � 𝑟𝑟 𝐸𝐸�𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘�� 𝑑𝑑(𝑑𝑑 ) ( )2 4 = (13) 2 Global 𝑑𝑑 𝑝𝑝𝑝𝑝 𝑑𝑑 𝑐𝑐𝑐𝑐 From equation (12), we𝑑𝑑𝑑𝑑 do� 𝑟𝑟have𝐸𝐸�𝑘𝑘 𝑘𝑘the− � following𝑘𝑘𝑘𝑘 − 𝑘𝑘 𝑘𝑘𝑘𝑘𝑘𝑘�� 𝑑𝑑𝑑𝑑

2 2 2 2 2 2 + + 1 = 0 9 2 2 9 3 𝑟𝑟 𝑐𝑐 𝑐𝑐 𝑟𝑟 𝑐𝑐𝑐𝑐 𝑐𝑐 𝐸𝐸 − �2𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � 𝐸𝐸 � − � 2 𝑘𝑘𝑞𝑞 2 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑞𝑞 𝑝𝑝𝑝𝑝𝑝𝑝 Setting = + 2 and = 1 , we obtain 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝐴𝐴 𝑘𝑘𝑘𝑘 �𝑘𝑘 𝑝𝑝 � − 𝑝𝑝 𝑞𝑞 𝐵𝐵 − 𝑝𝑝𝑝𝑝𝑝𝑝 ©2017 Global Journals Inc. (US) Fish Harvesting Experienced by Depensation Growth Function

2 = ± , provided that ( /3 )2 ( / ) and, > 0. 3 3 3 𝑟𝑟 𝐴𝐴 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐

T𝐸𝐸hus efforts𝑞𝑞 � 𝑘𝑘𝑘𝑘 at� maximum� 𝑘𝑘𝑘𝑘 � − 𝑝𝑝 𝑝𝑝economic� yield are: 𝐴𝐴 𝑘𝑘𝑘𝑘 ≥ 𝑐𝑐𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝 𝐴𝐴

2 2 2 2 + 2 1 2 2 2 2 = + + 1 1 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 Notes 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 � �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� 𝑞𝑞 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 2 201 2 2 + 2 2 2 r 2 1 2 2 ea + 1 Y 2= 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 441 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 � − �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� 𝑞𝑞 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 And thus the corresponding Maximum Economic Yields in (7) to be

V 1 = 1 1

II 2 𝑀𝑀𝑀𝑀𝑀𝑀 ue ersion I 𝑀𝑀𝑀𝑀𝑀𝑀 𝐻𝐻 �𝐸𝐸 �

s 2 2

s 2 1 + 2 1 2 2 2 2 I = + + 1 × 1 6 3 𝑐𝑐 𝑐𝑐 3 2

XVII �𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀𝑀𝑀 � �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝

) 2 2 + 2 2 F 4 2 2 1 2 2 2 ( × + ( )2 + + 1 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 ⎛ 𝑘𝑘𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 ⎞ 𝑘𝑘𝑘𝑘� 𝑘𝑘𝑘𝑘 − � �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� ⎜ 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 ⎟ Research Volume = ⎝ 2 1 2 ⎠ 2 2 2 𝑀𝑀𝑀𝑀𝑀𝑀 Frontier 𝑀𝑀𝑀𝑀𝑀𝑀 𝐻𝐻 �𝐸𝐸 � 2 1 + 2 1 2 2 2 2 = + 1 × 2 𝑐𝑐 𝑐𝑐 2 6 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3

Science 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀𝑀𝑀 � − �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − ��

of 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 2 2 2 2 4 2 + 2 1 2 2 2 2 Journal × + ( )2 + 1 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 ⎛ 𝑘𝑘𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 ⎞ 𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − � −�� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� Global ⎜ 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 ⎟ ⎝ Similarly from equation (13), we have the same effort as the above. And thus the⎠ corresponding Maximum Economic Yields in (8) to be

2 2 2 2 1 + 2 1 2 2 2 2 = + + 1 × 3 𝑐𝑐 𝑐𝑐 2 6 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀𝑀𝑀 � �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 © 2017 Global Journals Inc. (US) Fish Harvesting Experienced by Depensation Growth Function

2 2 2 2 4 2 + 2 1 2 2 2 2 × ( )2 + + 1 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 ⎛ 𝑘𝑘𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 ⎞ 𝑘𝑘𝑘𝑘−� 𝑘𝑘𝑘𝑘 − � �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� ⎜ 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 ⎟ 2 2 2 2 ⎝ 1 + 2 1 2 2 2 2 ⎠ = + 1 × 4 𝑐𝑐 𝑐𝑐 2 Ref 6 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀𝑀𝑀 � − �� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − ��

𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 201 2 2 2 r

2 ea 2 + 2 2 2 4 1 2 2 Y × ( )2 + 1 𝑐𝑐 𝑐𝑐 2 3 �𝑘𝑘 �𝑘𝑘𝑘𝑘 3 � − � 3 451 ⎛ 𝑘𝑘𝑟𝑟 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 𝑐𝑐 ⎞ 𝑘𝑘𝑘𝑘−� 𝑘𝑘𝑘𝑘 − � −�� 𝑘𝑘 �𝑘𝑘𝑘𝑘 � − � − � − �� ⎜ 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑝𝑝 𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 ⎟

⎝ V. Parameter Estimation ⎠ V a) Basic Parameters Estimation II

14. ue ersion I

- Using the time series data [1], we have the following parameters estimation. s s Table 1 I

XVII Parameters Symbol Value

Carrying capacity 2.57 × 1013

Economics of Fish Harvesting with of Fish Harvesting Economics with 11 ) - Cat ch ability constant 2.197 × 10 𝑘𝑘 𝑘𝑘𝑘𝑘 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓ℎ F ( Cost of effort − 𝑞𝑞 182.50 𝑝𝑝𝑝𝑝/ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

Price of effort 11 / 𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑘𝑘𝑘𝑘 Research Volume TJPRC: JMCAR TJPRC: 3(2016) 1 Intrinsic growth rate 0.5 𝑝𝑝 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑘𝑘𝑘𝑘 𝑟𝑟 Table 2: Parameter estimation for depensation model Frontier

MathematicalBio

, , Description Formula Value [ kg per day ]

Science 2 /9 1.3 × 1023 of Salilew 2 22 e 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 (1 (𝑟𝑟𝑟𝑟/ 𝑞𝑞) )/ 1.7 × 10

23 𝑂𝑂𝑂𝑂𝑂𝑂 2 1.29 × 10 Journal Abiy 𝐸𝐸 𝑐𝑐𝑐𝑐( +− 𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝 )/𝑝𝑝3𝑞𝑞 𝑀𝑀𝑀𝑀𝑀𝑀 𝐸𝐸 𝑟𝑟 𝐴𝐴 4�𝐴𝐴2 /−27𝐵𝐵 𝑞𝑞 4.9 × 1025 Global 23 𝑀𝑀𝑀𝑀𝑀𝑀 (1 ( / )) 𝑘𝑘+𝑟𝑟 ( )2 4 2 /2 91.4 × 10

Getachew CriticalDepensation Lake in Tana, 𝑂𝑂𝑂𝑂𝑂𝑂 𝑐𝑐 − 𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝 �𝑘𝑘𝑘𝑘 � 2𝑘𝑘𝑘𝑘 − 2𝑟𝑟 𝐵𝐵� 𝑝𝑝𝑝𝑝 24

1 = + ( ) (4 /3) /6 48.9 × 10 1. 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 𝜑𝜑 �𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑟𝑟 𝜑𝜑 � 2 2 2 + 2 2 Where; A = 𝑐𝑐 𝑐𝑐 , = 1 , and = + 3 𝑘𝑘�𝑘𝑘𝑘𝑘 𝑝𝑝 �−𝑝𝑝 𝑞𝑞 𝑐𝑐𝑐𝑐 𝑐𝑐 � 𝑘𝑘𝑘𝑘 � 𝐵𝐵 𝑝𝑝𝑝𝑝 � − 𝑝𝑝𝑝𝑝𝑝𝑝 � 𝜑𝜑 𝐴𝐴 √𝐴𝐴 − 𝐵𝐵

Table 3: Parameter estimation to depensation model for different values of .

r=0.5 R=1 R=2 𝑟𝑟 = 1.3 × 1023 = 2.6 × 1023 = 5.20 × 1023 25 25 2 5 1 = 4.89 × 10 1 = 9.795 × 10 1 = 19.59 × 10 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 25 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 25 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 25 2 = 2.45 × 10 2 = 4.898 × 10 2 = 9.795 × 10 𝑀𝑀𝑀𝑀𝑀𝑀 = 1.67 × 1022 𝑀𝑀𝑀𝑀𝑀𝑀 = 3.336 × 1022 𝑀𝑀𝑀𝑀𝑀𝑀 = 6.67 × 1022 23 2 3 2 3 𝑀𝑀𝑀𝑀𝑀𝑀1 = 91.47 × 10 𝑀𝑀𝑀𝑀𝑀𝑀1 = 182.94 × 10 𝑀𝑀𝑀𝑀𝑀𝑀1 = 365.885 × 10 𝑂𝑂𝑂𝑂𝑂𝑂 23 𝑂𝑂𝑂𝑂𝑂𝑂 23 𝑂𝑂𝑂𝑂𝑂𝑂 23 𝐸𝐸 2 = 2.77 × 10 𝐸𝐸 2 = 5.53 × 10 𝐸𝐸 2 = 11.07 × 10 𝑂𝑂𝑂𝑂𝑂𝑂 23 𝑂𝑂𝑂𝑂𝑂𝑂 23 𝑂𝑂𝑂𝑂𝑂𝑂 23 Ref 1 = 1.29 × 10 1 = 2.58 × 10 1 = 5.162 × 10 𝑂𝑂𝑂𝑂𝑂𝑂 = 8.4 × 1021 𝑂𝑂𝑂𝑂𝑂𝑂 = 16.81 × 1021 𝑂𝑂𝑂𝑂𝑂𝑂 = 33.62 × 1021 𝑀𝑀𝑀𝑀𝑀𝑀2 𝑀𝑀𝑀𝑀𝑀𝑀2 𝑀𝑀𝑀𝑀𝑀𝑀2

201 𝐸𝐸 24 𝐸𝐸 24 𝐸𝐸 2 4 1 = 48.97 × 10 1 = 97.94 × 10 1 = 195.885 × 10 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 1. r 𝐸𝐸 23 𝐸𝐸 23 𝐸𝐸 23

2 = 46.79 × 10 2 = 93.58 × 10 2 = 187.156 × 10 ea Tana, in Lake Depensation Critical Getachew Y 24 24 24 𝑀𝑀𝑀𝑀𝑀𝑀3 = 23.94 × 10 𝑀𝑀𝑀𝑀𝑀𝑀3 = 47.88 × 10 𝑀𝑀𝑀𝑀𝑀𝑀3 = 95.75 × 10 𝑀𝑀𝑀𝑀𝑀𝑀 22 𝑀𝑀𝑀𝑀𝑀𝑀 22 𝑀𝑀𝑀𝑀𝑀𝑀 22 461 4 = 6.92 × 10 4 = 13.84 × 10 4 = 27.68 × 10 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 b) The Economic𝑀𝑀𝑀𝑀𝑀𝑀 Model Estimation𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 In this section we calculated the profit for depensation models by using real data

Abiy

V [1]. In a commercial fishery, the appropriate measure of gross benefits is the total II

revenue that accrues to firms. Assuming that fish are sold in a competitive market, each e

ue ersion I

Salilew

s firm takes the market price as given and so the revenue obtained from a harvest is s

I given by ( ) = ( ). And finally the economic rent or profit denoted by is 𝑝𝑝 𝐻𝐻

XVII defined in terms of total cost and total revenue by = . , 𝑇𝑇𝑇𝑇 𝐸𝐸 𝑝𝑝𝑝𝑝 𝐸𝐸 𝑃𝑃 Bio Mathematical c) The depensation economic model parameter estimations 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑃𝑃 𝑇𝑇𝑇𝑇 −𝑇𝑇𝑇𝑇 In this case we do have the following parameter estimation with the harvest at

2 ) the given type of Effort is obtained by: ( ) = + ( ) 4 .

TJPRC: JMCAR 3(2016) 1 3(2016) TJPRC: JMCAR

F 2

( 𝑞𝑞𝑞𝑞

Table 4: The depensation economic model𝑟𝑟 parameter estimations 𝐻𝐻 𝐸𝐸 �𝑘𝑘𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘� ( ) ( ) ( )

0.5 182.5 11 2.1 2.5 1.3 1.6 1.2 4.8 91.4 48.9

Research Volume 𝑀𝑀𝑀𝑀𝑀𝑀 𝑂𝑂𝑂𝑂𝑂𝑂 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀 𝑂𝑂𝑂𝑂𝑂𝑂 𝑀𝑀𝑀𝑀𝑀𝑀 -

𝑟𝑟 𝑐𝑐 𝑝𝑝 × 10𝑞𝑞 11 × 1𝑘𝑘013 ×𝐸𝐸1023 ×𝐸𝐸1022 ×𝐸𝐸1023 𝐻𝐻×𝐸𝐸1025 𝐻𝐻× 𝐸𝐸1023 𝐻𝐻 ×𝐸𝐸1024 withEconomics Harvesting Fish of And thus the profit with− different type of harvest function is given by:

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

( ) = ( ) ( ) = . ( ) .

Science ( 𝑀𝑀𝑀𝑀𝑀𝑀) = 53.877549𝑀𝑀𝑀𝑀𝑀𝑀 × 1025𝑀𝑀𝑀𝑀𝑀𝑀237.330848𝑀𝑀𝑀𝑀𝑀𝑀× 1023 =𝑀𝑀𝑀𝑀𝑀𝑀51.50424 × 1025

of 𝑃𝑃 𝐸𝐸 𝑇𝑇𝑇𝑇 𝐸𝐸 − 𝑇𝑇𝑇𝑇 𝐸𝐸 𝑝𝑝 𝐻𝐻 𝐸𝐸 − 𝑐𝑐 𝐸𝐸

- 14. The depensation economic model for Open Access Yield (OAY) 𝑃𝑃 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

Journal ( ) = ( ) ( ) = . ( ) . 23 23 23 𝑃𝑃(𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 ) = 𝑇𝑇𝑇𝑇1006𝐸𝐸𝑂𝑂.1903𝑂𝑂𝑂𝑂 −×𝑇𝑇𝑇𝑇10𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 30.44𝑝𝑝 𝐻𝐻× 10𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 =−975𝑐𝑐 𝐸𝐸.𝑂𝑂7503𝑂𝑂𝑂𝑂 × 10 Global The depensation economic model for Maximum Economic Yield (MEY) 𝑃𝑃 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ( ) = ( ) ( ) = . ( ) . 24 23 24 𝑃𝑃(𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 ) = 𝑇𝑇𝑇𝑇538.𝐸𝐸684𝑀𝑀𝑀𝑀𝑀𝑀×−10𝑇𝑇𝑇𝑇 𝐸𝐸𝑀𝑀235𝑀𝑀𝑀𝑀 .5227436𝑝𝑝 𝐻𝐻 𝐸𝐸𝑀𝑀×𝑀𝑀𝑀𝑀10− 𝑐𝑐=𝐸𝐸515𝑀𝑀𝑀𝑀𝑀𝑀.13173 × 10 𝑃𝑃 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 VI. Resu−lts and Concl usions 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 Biologically 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 harvest, which means the additional effort will have less return. Using data [1] there is over fishing for different cases of the natural growth rates = 0.5, = 1 and = 2 of fish as in table 3. Without loss of any generality we prefer to analyze the tabular approximate value for = 0.5 as our choice of the parameter is similar𝑟𝑟 to𝑟𝑟 that of =𝑟𝑟 1 and = 2. When the natural growth rate = 0.5, carrying 𝑟𝑟 capacity = 2.57 × 1013 , effort for maximum sustainable yield, = 𝑟𝑟 𝑟𝑟 𝑟𝑟 Ref 1.300443 × 1023 , effort for open access yield = 0.1668069 × 𝑘𝑘 𝑘𝑘𝑘𝑘 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓ℎ 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 1023 , effort for maximum economic yield = 1.2905355813 × 𝑘𝑘𝑘𝑘 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓ℎ 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 23 201 10 . And thus we observed from these values that𝑀𝑀 𝑀𝑀𝑀𝑀all efforts are greater 𝑘𝑘𝑘𝑘 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓ℎ 𝐸𝐸 r

than the carrying capacity therefore there is overfishing if we consider the depensation ea model.𝑘𝑘𝑘𝑘 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓ℎ Y In the economic point of view we have the approximate price of total population 471 of fish in [1] is, 28.2 × 1013 . In the depensation economic model parameter estimations we have: ( ) = 51.50424 × 1025 , ( ) = 975.7503 × 1023 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and ( ) = 515.13173 × 1024 . And thus in the depensation model the

V 𝑃𝑃 𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑃𝑃 𝐸𝐸𝑂𝑂𝑂𝑂𝑂𝑂 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 economic rent or the profit obtained by all kinds of effort are greater than the price of II 𝑀𝑀𝑀𝑀𝑀𝑀

𝑃𝑃 𝐸𝐸 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ue ersion I 14.

the total population of fish and therefore there is overfishing. To keep the sustainability s - s

of fish we must reduce the effort levels. I

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