A Model for Bio-Economics of Fisheries

A Model for Bio-Economics of Fisheries

G. Shanmugam 1, K. B. Naidu 2

1Associate Professor, Dept of Mathematics, Jeppiaar Engineering College, Chennai,

2Professor, Department of Mathematics, Sathyabama University, Chennai

Abstract. In this paper a model for growth of fish, a model for fishing economics and delay model for fishing are considered. The maximum sustainable yield for fishing is obtained. In the delay model the three cases of equilibrium population being equal to (or) greater then (or) less then the ratio of carrying capacity and rate of growth are considered.

1. Introduction

   The World population is growing at enormous rate, creating increasing demand for food. Food comes from renewable resources. Agricultural products are renewable resources, since every season new crops are produced in farms. Fisheries are a renewable resource since fish are reproduced in lakes and seas. Forests are renewable resources since they reproduce periodically. As these resources are renewable, the quality of the resources will certainly degrade, leading to shortage. Over fishing will lead to decline in fisheries. Global warming again has an impact on the growth of agriculture, fisheries and forests. It is imperative that we should manage these resources economically to prevent a catastrophic bust in our global economy.

   Mathematical bio-economics is the mathematical study of the management of renewable bio resources. It takes into consideration not only economic factors like revenue, cost etc., but also the impact of this demand on the resources.

   One of the mathematical tools used in bio economics is differential equations. To model the growth or decline of fish population, we use differential equations.

2. Model for Growth of Fish

As a first step in the model of growth of fisheries, we assume that Fish reproduce at a rate that is linearly dependent on the number of fish present at time t.

We take r b d 0 where b is the birth rate and d is the death rate, both being constant. Then the model

for the growth of fish population is

dN rN dt

(1)

where N (t) is the population size of the fish at time t and r is the reproduction rate of fish, taken to be con-

stant. Let the initial population of the fish be

N0 , that is

N(0) N0

(2)

rt

rt

Solving (1) with the initial condition (2) we have

N (t)

N0 e

(3)

The model (1) and (2) or its solution (3) is called Malthusian growth model. We can interpret this as follows.

rt

rt

Initially there is a population of size N0

0 , however small it may be; then the population grows exponentially

to the size

N (t)

N0e in time t. (Malthus predicted a dooms day for the human population using the above

model). This model is not realistic as for as the growth of fisheries is concerned. To add realism to this model we have to consider the effect of crowding of fish, limitation of space and resources. We may call it carrying capac-

rN 2

ity. Such an effect is called negative density dependence denoted by the region (Lake).

Then the model for the growth of the fish can be written as

where K is the carrying capacity of

K

dN rN dt

rN 1

rN 2 K N K

(4)

The first term in the RHS of (4) is a linear (Malthusian) term and the second term

rN 2

K

is a nonlinear term

which represents crowding effect. We can solve the equation (4) by separating the variable using partial fraction. But this solution does not give any; insight in the growth process of fish population. We have to undertake stabil- ity analysis of the model (4). Equate the RHS of (4) to zero, that is

rN 1 N 0

K

Then N

1. and N

   K are two equilibrium states.

   Consider the zero equilibrium state N 0

   Put N 0 then (4) becomes

   2

   2

   d r r

   dt K

   d r

   dt

   This gives

   (neglecting 2) (5)

   cert as t (6)

   Therefore N 0 is an unstable equilibrium state, which means that the fish population grows exponentially.

   Now consider the non-zero equilibrium state N

   K . Put N

   K where 1 then (4) becomes

   d r K dt

   d r

   dt

   ce rt

   1 K r r

   2

   2

   K K

   0 as t .

   (7)

   Thus N K is a stable equilibrium state. Hence we conclude that after a long time the fish population will

   tend to the size K, the carrying capacity of the Lake.

   3 A Model for Fishing Economics

   In the previous section we discussed the model for the bio growth of fish. Now we consider the economics of fish growth. We know that the profit in any business is governed by the law:

   Profit = Revenue Costs

   The total revenue is determined by

   Total revenue = (Price of each resource harvested) (Total number of resources

   (Yield))

   Yield = qEN (8)

   where q is called catchability coefficient which represents environmental factors, the ability to use location

   equipment to catch fish ( q

   used etc.,)

2. ), E is the effort exerted in harvesting the fish (depending on the number of boats

If the market price for the resources is p then the total revenue is given by Total revenue = Price Yield

= pqEN (9)

The total cost of harvesting the resources is proportional to the effort exerted. That is Total cost = cE ,

where c is the constant which represents external control on cost ( such as price of gasoline used in the power motor).

Thus the profit of harvesting the resources is given by Economic rent = pqEN cE

Harvesting the resource will reduce its abundance hence we have to subtract the yield qEN from the RHS of the equation (4) and we get

dN rN 1 N dt K

qEN

where qEN is the harvesting term.

Thus the bio economic model that governs the abundance of resource (fish) and the profit made from the re- source is given by

dN rN 1 N dt K

and

r rN qE 0

K

qEN

(10)

Economic rent = pqEN cE (11)

Equilibrium states of (10) are given by

rN 1 N K

N r rN K

qEN 0

qE 0

Thus we have to solve the equation.

Therefore N

K qEK

r

(12)

The economic rent is the difference between total revenue and total cost. If costs exceed revenue, then the people will leave the resource because their efforts are not successful. Similarly, if the revenue is greater than the cost, the effort of expanded harvesting the resource will increase.

To calculate the yield of the resources put the expression N

Yield Y(E) = qEN

K qEK

r

in yield.

= qE K

= qK rE r

qEK r

qE 2

(13)

Note that this function is quadratic in E. The maximum sustainable yield (MSY) is found by maximizing Y(E). This occurs at the value of E where

E r

MSY 2q

(14)

We solve the system of equations when both economic rent and

dN are equal to zero.

dt

pqE* N * CE * 0

Then from (10) and (11)

*

*

rN * 1 N K

qE* N * 0

We have

N * c ,

pq

or

E* r t q

N * 0,

c pqK

E* 0

(15)

or N * K, E* 0

We can combine (15) and write

x* , E*

c , r

pq q

1

0,0

K ,0

c pqK

(16)

Therefore maximum sustainable yield is

Y (EMSY )

qk r 2

r 2q

qr2

4qz

= kr (17)

4

Figure 2 Bio-conomic Model of Fisheries in Equilibrium

[The arched curve represents total revenue from fishing effort in equilibrium. At a certain point, the number of fish being taken actually diminishes, even though fishing effort increases. It would technically be possible to take every fish out of the Lake. The straight line (Cost of Effort) represents total cost per fishing effort. The vertical distance between the total revenue and the total cost curve is economic profit. The maximum quantity of fish that could be removed annually is represented by the maximum sustainable yield.]

4 A Delay Model

We consider the model equation for fishing when there is time delay of T to reach maturity, the finite gesta- tion period.

dN rN t 1

dt

N t T K

qEN t

18)

where r is the rate of growth of the fish population, K is the carrying capacity of the region, q is the catchabili-

ty coefficient, E is the effort exerted in harvesting the resource.

Substituting N t N *

(t)

(19)

where N * is an equilibrium state in (18), we get

d r N dt

*

rN *

(t) 1 N t T K

*

*

t T

K

d

rN *

qE N *

t T

(t)

Thus the approximation equation is

We take solution of (20) in the form

dt

(t)

K

ce t

(20)

(21)

from which we get

rN *e T

T 0

K

(22)

Analytic solution for (22) is difficult to be obtained. We will examine whether there are any solutions of

(22) with real part Re >0 corresponding to which there is instability;

That is t as t

For this put i

There exists a real number 0 such that all solutions of equation (22) satisfy Re < 0

To examine this consider

e

e

rN * i T

K

t

t

rN *

e If

K

, e t

(23)

Which requires that . Thus there must be a number 0 such that Re < 0

Let us introduces 1 z

(24)

Then

w(z)

T

1 ze Z

(25)

Thus w(z) has an essential singularity at z = 0, Then by Picards theorem in complex analysis, in the neigh-

borhood of z = 0, w(z) has infinitely many roots of (22).

rN * i T

Now substituting i in (22) we have i e

K

T

T

rN *

e

K

os T

sin T

(26)

The real and imaginary parts of (26) are

rN *e

rN *

e

K

T cos T

T sin T

(27)

(28)

We will now determine the range of T such that 0 . That is, we will find conditions such that the upper

limit

(of

) is negative.

We will first consider the simple case where is real. That is then equation (22) becomes

T

T

rN *

e

K

(29)

This has no positive root, since e T 0 for all T

Next we consider 0 . Then from equation (27) and (28), if is a solution of (27) and (28), is also a solution.

Suppose > 0 (with out loss of generality); then from equation (27), 0 requires T

Multiplying (28) by T

for all T .

2

we get

TrN *

e

K

T sin T

T (30)

2

Case(i) If N *

K then Te T sin T 0 T

r 2 2

(31)

Case(ii) If N *

K which is not possible.

r

Case (iii) If N * K then

r

TrN *

e

K

T sin T

Te T

sin T

2

0 T , (32)

2

since sin T

1 and e t

1 if 0

5 Conclusion

A model for growth of fish is given in section-II. A model for fishing economics giving the maximum sustain- able yield is given in section-III. A delay model for fishing is given in section-IV and interpreted.

References

[1]. Murray. J. D , Mathematical Biology, Springer Verlag, Berlin, Heidelberg, NweYork London (1989).

1. Clark. Colin W. Mathematical Bio economics: The Optimal Management of Renewable Resources. 1990. John Wiley & Sons.

2. Elizabeth S. Allman, Jone A Rhodes, Mathematical Models in Biology an Introduction, Cambridge Uni- versity Press, Cambridge New York (2004).