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 biology 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 overfishing 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 population of biological organisms such as fisheries 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 resource 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 populations are becoming increasingly limited and caches are declining due to overexploitation [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 biomass 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 fishery 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 population size at time , is the intrinsic growth rate, is the carrying capacity 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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