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International Journal of Advances in Science and Technology,
Vol. 1, No. 5, 2010
An Analysis for a Two Species Predator-Prey
System with Harvesting
T. K. Kar1, Ashim Batabyal2 and Swarnakamal Misra3
1
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103
E-mail: tkar1117@gmail.com
2
Department of Mathematics, Bally Nischinda Chittaranjan Vidyalaya, Bally Ghoshpara, Howrah
E-mail: ashim.bat@gmail.com
3
Department of Mathematics, Dhakuria Ramchandra High School, 10, Station Road, Dhakuria, Kolkata-
700031, India
Abstract
In this paper we have considered a prey-predator model with independent harvesting in either
species. The purpose of the work is to offer mathematical analysis of the model and to discuss
some significant qualitative results that are expected to arise from the interplay of biological
forces. The possibility of existence of bio-economic equilibria is discussed. The optimal harvesting
is studied and the solution is derived in the equilibrium case by using Pontryagin’s maximal
principle. The biological as well as economic aspects of the optimal equilibrium solution are
discussed.
Key words: Fisheries, bifurcation, bio-economic equilibrium, optimal harvesting
1. Introduction
In recent years, there is a spate of interest in bio-economic analysis of exploitation of
renewable resources like fisheries and forestry’s. Exploitation of natural resources has become a
matter of concern throughout the world in view of problem caused by rapid depletion of these
resources do meet the ever growing human needs. It has, therefore, become imperative to ensure
scientific management of exploitation of biological resources. Gradual improvement in the technical
efficiency of the fishing gear a vessel has radically changed the fishing scenario.
Fishery economists such as Clark (1985, 1990), Schaefer (1957) have made important
contributions to the development of bio-economic approaches to natural resource management. Clark
and Munro call attention to the importance of explicitly accounting for time in fishery economic
models. They present an introductory treatment of optimal control. Clark has been influential in
simulating the application of optimal control and capital theoretical concept to fishery management
problems. These studies deal with simple mathematical population models, which are solved
analytically and provide good insights into the nature of dynamic optimization.
December Issue Page 84 of 108 ISSN 2229 5216
The management of renewable resources has long been practiced using the MSY (maximum
sustainable yield) concept whose primary objective is to avoid over exploitation. The MSY is a
simple way to manage resources taking into consideration that over exploiting resources lead to a loss
in productivity. Therefore, the aim is to determine how much we can harvest without altering
dangerously the harvested population. But in recent times, several objections have been raised on
both biological and socioeconomic grounds. The main problem of the MSY is economical
irrelevance. It is so since it takes into consideration the benefits of resource exploitation, but
completely disregards the cost operation of resource exploitation. For example, it ignores the fact that
if a species is harvested such that its population decreases to a certain level, then the cost of
harvesting can become exorbitant because finding the desirable resource becomes more time
consuming. This might lead to a situation where the cost of harvesting is higher than the benefit.
Confronted with inadequacy of the MSY, many researchers in recent times have expressed their
serious reservations against the validity of the MSY concept as an operational objective for the
management of exploited living resource stocks and tried to replace it by the OSY, that is the
optimum sustainable yield, which is based on the standard cost benefit criterion used to maximize
revenues. Actually renewable resources management is complicated and constructing accurate
mathematical models about the effect of harvesting is even more complicated. This is so because to
have a perfect model we need to consider its size, growth rate, carrying capacity, competitors
combined with the cost of harvesting and the price obtained for the harvested species. More
information can be found about these factors in Clark (1990). The problem of combined harvesting of
two competing species was studied in details by Chaudhuri (1986). Optimal harvesting policy in
different combined harvesting models has been studied by Sutinen and Anderson (1985), Andersen
and Lee (1986), Mesterton Gibbons (1996), Pradhan and Chaudhuri (1999), Brauer and Soudak
(1979), Kar and Chaudhuri (2002) etc.
We consider the following set of equations to describe the dynamics of prey and predator
dx x axy
r x1 q1 E1 x,
dt k b x
(1.1)
d y axy x
d y 1 c y q 2 E 2 y ,
2
dt b x k
with x0 x0 0, y0 y0 0. Here x(t) and y(t) denote the prey and predator
populations respectively. r > 0 represents the intrinsic growth of the prey, k is the carrying capacity of
the prey in the absence of predator and harvesting. The term ax/(b+x) denotes the functional response
of the predator. This response function is termed as Holling type II response function (Holling, 1965).
> 0 is the conversion factor denoting the number of newly born predators for each captured prey.
q1 , q 2 are the catchability of prey and predator species respectability. E1 , E2 are harvesting effort
of prey and predator species respectability. The term dy represents a growth rate of predators due to
availability of alternative food sources. However, when prey is abundant taking of alternative food
diminishes. For this reason dy is modified by the factor dy(1-x/k) to indicate the diminishing of
predators growth rate as prey approaches its saturation value k. c is the predators intra-specific
competition coefficient.
2. Boundedness of the System
The system describing the evolution of X(t)=[x(t), y(t)], is said to be bounded if all the solution
trajectories of (1.1) are uniformly asymptotically bounded for t 0. In other words there exists a
constant M such that lim sup t || X(t) || M. The boundedness of solutions of the system (1.1) is
proved in the following theorem.
2
Theorem 1. All the solutions of system (1.1) which start in R are uniformly bounded.
December Issue Page 85 of 108 ISSN 2229 5216
1
Proof. We define the function w x y.
Therefore, the time derivative of w,
dw dx 1 dy x d y x c 2 q2 E 2 y
rx 1 q1 E1 x 1 y .
dt dt dt k k
For each v > 0, we have
dw
vw
k
r v q1 E1 2 1 d v q2 E2 2 .
dt 4r 4 c
Now if we assume both E1 and E2 are bounded quantity, then right hand side is bounded. Thus we
choose a
> 0, such that
dw
vw .
dt
Applying the theory of differential inequality (Birkoff and Rota ,1982) we obtain
0 w( x, y)
v
1 e w( x(0), y(0) ) e
vt v t
,
which upon letting t , yields 0 < w < /v.
2
So, we have, that all the solution of system (1.1) that start in R are confined to the region B, where
B { ( x, y) R : w
2
, for any 0 }.
v
3. Equilibria
In this section, we find all the equilibria admitted by the system (1.1). The equilibria of (1.1)
are the intersection points of the prey-isocline on which dx/dt = 0 and the predator isocline on which
dy/dt = 0. Whenever we have the prey harvesting effort E1 r q1 , then the system admits only
one equilibrium, given by P0(0, 0), which is the trivial equilibrium. If we have E1 r q1 , then
there exists another equilibrium on the boundary given by P k (r q1 E1 ) / r , 0. Again if E2 <
1
d/q2, there also exists the boundary equilibrium P2 0, (d q2 E2 ) / c .
The interior equilibrium P3(x*, y*) of the system (1.1) is the intersection of the two isoclines
x y 0 in the positive quadrant of the xy-plane.
The prey isocline x 0 is equivalent to
(b x) x
y r 1 k q1 E1 F ( x) (say) (3.1)
a
and the predator isocline y 0 gives
1 a x x
y b x d 1 k q2 E2 L ( x) (say) . (3.2)
c
The prey isocline F(x) is a parabola which has unique global maximum at the positive xy-plane. For
mathematical convenience we impose the following assumption on F(x).
Hypothesis ( H1 ). Let the prey isocline F(x) possesses a unique global maximum at
xm > 0 satisfying
December Issue Page 86 of 108 ISSN 2229 5216
dF
0 for 0 x x m ,
dx
dF k
and 0 for x m x
dx r
assuming E1 r / q1.
*
Interior equilibrium y* of P3 is given by (3.1) as y ((b x ) / a) r (1 x / k ) q1 E1 and
* *
x* will be determined from the cubic equation
2 abd adk a 2 k 2bq1 E1 k
b 2bk
ad q1 E1 k 2 rc rc rc r x
x 3 2b k x
rc r akq2 E 2 (3.3)
rc
b k ad
2
aq E
r q1 E1 2 2 0.
r bc bc
We see that the equation (3.3) has at least one real positive root if
ad aq E
q1 E1 r 2 2 . (3.4)
bc bc
Now for the existence of interior equilibrium of the system (1.1) we get the following result.
Proposition 1. There exists an interior equilibrium P3(x*, y*) in the positive xy-plane if and only if
r d ad aq E
E1 , E2 and q1 E1 r 2 2 .
q1 q2 bc bc
Proof. The variational matrix V(x, y) of the system (1.1) is given by
x rx aby ax
r 1 k k (b x) 2 q1 E1
b x
V ( x, y ) .
aby d ax x
y d 1 2cy q2 E2
(b x) bx k
2
k
From V(x, y) it follows that P(0, 0) is a repeller along both the x and y directions if
E1 r / q1 and E2 d / q2 . P k (r q1 E1 ) / r, 0 is locally stable along the x-direction but
1
unstable along y-direction. P2 0, (d q2 E2 ) / c is locally unstable along the x-direction since
ad aq E
q1 E1 r 2 2 . So, there exist an equilibrium P3(x*, y*) in the interior of the positive xy-
bc bc
plane. This follows from an application of the Poincare -Bendixson theorem (Conway and Smoller,
1986).
ad aq E
Remark: If q1 E1 r 2 2 , then either there is no interior equilibrium or there may
bc bc
exist two equilibria in the positive xy-plane. We are interested to study the case when the system
possesses unique interior equilibrium.
December Issue Page 87 of 108 ISSN 2229 5216
4. Dynamic Behaviour
From the previous result it is clear that all the boundary equilibria are non-saturated. Here we
shall consider the qualitative behaviour of the interior equilibrium point P3(x*, y*) from the following
theorem.
ad aq E
Theorem 2. Let (H1) and q1 E1 r 2 2 hold. In addition, (i) if P3 lies on the
bc bc
decreasing part of the F(x), then it is stable and (ii) if P3 lies on the increasing part of F(x), then either
stable or unstable according as c1 c c2 or c c1 where
* * *
b 2 x*
ax* r 1
q1 E1
c1
* k k
x *
r (1 k ) q1 E1 (b x )
* 2
a 2 x* x*
1
ad acy * aq2 E2
b x *
k
and c2
*
.
b 2 x*
r 1
q1 E1
k k
Proof. The characteristic equation for V(x*, y*) is given by 2 p q 0 ,
where
r x* b 2 x * q1 E1 x *
p c y * 1
k ,
b x* k b x*
(4.1)
ax * y * ab d r c b 2 x * c q1 E1
q 1 .
(b x * ) (b x * ) 2 k a k
k a
(i) From (3.4) it is clear that if P3 lies on the decreasing part of F(x), that is F(x) < 0, then p > 0
and q > 0. Hence P3 is stable. This completes the proof of part (i).
(ii) Let P3 lies on the increasing part of F(x). Then from the characteristic equation we see that
p 0 c c1 and q 0 c c2 .
* *
Thus, if P3 lies on the increasing part of F(x), it is stable if c1 c c2 . If c c1 , then p <
* * *
0 and the roots of the characteristic equation are positive or have positive real parts. Hence P 3 is
unstable. This completes the proof of part (ii).
A prey-predator system with constant parameters is often found to approach a steady state in
which the species coexist in equilibrium. But if parameters used in the model are changed, other
types of dynamical behavior may occur and the critical parameter values at which such transitions
happen are called bifurcation points.
When a stable steady state goes through a bifurcation will in general either lose its stability or
disappear entirely. Even if the system ends up in another steady state the transition to that state will
often involve the extinction of one or more levels of the food chain. On the other hand the entire
system may survive in a non-stationary state, but further bifurcation may lead to local extinction of
species. In order to preserve the system under consideration in its natural state, crossing bifurcation
should be avoided and in doing so it is of great importance to determine the critical parameter values
at which bifurcation occur.
December Issue Page 88 of 108 ISSN 2229 5216
Next we shall try to derive the criteria for the existence of small amplitude periodic
oscillations for variation of parameter of the system (1.1). To do this, we now state a theorem, giving
sufficient conditions for Hopf-bifurcation. To prove the theorem we shall follow the technique of
Hopf-bifurcation [Conway and Smoller, 1986] by considering c as bifurcation parameter.
Theorem 3. If c c1 , where x*
* 1
k b kq1E1 / r , then the system (1.1) undergoes
2
Hopf-bifurcating small amplitude periodic solutions near P3.
Proof. The theorem will be proved if we can show that the conditions for Hopf-bifurcation are
satisfied. At c c1 , the two roots of f() are purely imaginary, namely
*
i (q )1 / 2 .
Also we have,
d * rx* b 2 x* q E x*
cy (1 ) 1 1 * c c*
dc b x* k k b x 1
= - y* ≠0.
Thus, all the conditions for Hopf-bifurcation are satisfied. Hence at c c1 , there exist Hopf-
*
bifurcating small amplitude periodic solutions near P3. This completes the proof of the theorem.
P3 ( x* , y * ) is locally asymptotically stable for c c1 . Furthermore, according to the criterion a
*
simple Hopf bifurcation occurs at c c1 and for decreasing 0 c c1 , it approaches a periodic
* *
solution.
In order to ensure the existence of bifurcation let us consider the parameters of the system as a=0.8,
r=2.4, k=90, b=10.0, d=0.09, q1=0.01, q2=0.02, E1=3.0, E2=3.5, α=0.9.
We get the value of c*= 0.006810509.
90
Predator
80
70
60
populations
50
40
30
20
10 Prey
0
0 50 100 150 200 250 300 350 400
time
Figure 1. Variation of populations when c1 .006710509
*
December Issue Page 89 of 108 ISSN 2229 5216
100
90
80
70
60
Predator 50
40
30
20
10
0
0 20 40 60 80 100 120 140
Prey
Figure 2. Phase space trajectories corresponding to the value of c1 0.006710509
*
beginning with different initial levels.
90
80
Predator
70
populations
60
50
40
Prey
30
20
0 5 10 15 20 25 30
time
Figure 3. Variation of populations when c1 0.006910509
*
December Issue Page 90 of 108 ISSN 2229 5216
100
90
80
70
60
50
Predator
40
30
20
10
0
0 20 40 60 80 100 120 140
Prey
Figure 4. Phase space trajectories corresponding to the value of c1 0.006910509 beginning with
*
different initial levels.
Ecologically, the above two results show that there are threshold limits of the predator mortality
rate. It is observed that if the mortality rate lies between two distinct values, the system is stable and
below it is unstable. When the predator’s mortality rate attains the lower threshold value, the system
possesses small-amplitude periodic oscillation
Let us take r =2.4, k = 2.8 , b = 10.0 , d = 0.09 , c= 0.13, q 1 = 0.01 , q2 = 0.02 , E1 = 5.0 , E2 =3.5 in
appropriate units .
For the above values of the parameters it is found that the interior equilibrium point
P3(74.9, 16.74) exists and stable.
80
P re y
70
60
50
P o p u la t io n
40
30
20
P re d a t o r
10
0
0 2 4 6 8 10
t im e
Figure 5. Both the prey and predator populations converge to their equilibrium values.
December Issue Page 91 of 108 ISSN 2229 5216
35
30
25
Predator 20
15
P(74.9,16.74)
10
5
0
0 50 100 150 200 250
Prey
Figure 6. Phase plane trajectories beginning with different initial levels. It is showing
that the equilibrium point (74.9, 16.74) is globally asymptotically stable.
80
E =5.0
70 1
E =15.0
1
60 E =25.0
1
Prey 50
40
30
20
10
0
0 2 4 6 8 10
Time
Figure 7. Variation of prey population for different values of E1.
December Issue Page 92 of 108 ISSN 2229 5216
60
50
40
E = 25.0
p re d a t o r
1
30
20 E = 15.0
1
10 E = 5.0
1
0
0 1 2 3 4 5
t im e
Figure 8. Variation of predator population for different values of E2.
From figures 7 and 8 we observe that as E1 increases, prey population decreases but predator
population converges to the same value. This happens due to alternative source of food.
100
80
60
p re y
40
E = 25.0
2 E = 50.0
2
20
E = 2.0
2
0
0 2 4 6 8 10
t im e
Figure 9. Variation of prey population for different values of E2.
December Issue Page 93 of 108 ISSN 2229 5216
100
80
60
p re d a t o r
40
E = 2.0 E = 25.0 E = 50.0
20 2 2 2
0
0 2 4 6 8 10
t im e
Figure 10. Variation of predator population for different values of E2.
From figures 9 and 10 we observe that as E2 increases prey population increases but predator
population decreases as expected.
5. Bionomic Equilibrium
The term bionomic equilibrium is an amalgamation of the concepts of biological equilibrium
and economic equilibrium. As we already saw, a biological equilibrium is given by x 0 y .
The economic equilibrium is said to be achieved when TR (the total revenue obtained by selling the
harvested biomass) equals TC (the total cost for the effort devoted to harvesting).
Let c1 = fishing cost per unit effort of prey species,
c2 = fishing cost per unit effort of predator species,
p1 = price per unit biomass of the prey species,
p2 = price per unit biomass of the predator species.
The economic rent (revenue at any time) is given by
( p1q1 x c1 ) E1 ( p 2 q 2 y c2 ) E 2
1 2 (say),
where 1 ( p1q1 x c1 ) E1 , 2 ( p2 q2 y c2 ) E2 .
i.e., 1 and 2 represent the net revenues for the prey and predator respectively.
Although the harvesting costs per unit effort are not constant, we take it to be a constant for the
sake of simplicity. The bionomic equilibrium [ x , y , E1 , E2 ] is given by the following
simultaneous equations
x ay
r 1 q1 E1 0, (5.1)
k b x
ax x
d 1 c y q 2 E 2 0, (5.2)
b x k
and ( p1q1 x c1 ) E1 ( p2 q2 y c2 ) E2 0. (5.3)
In order to determine the bionomic equilibrium, we now consider the following cases.
Case I. If c2 p2 q2 y, i.e., if the total cost is greater than the total revenue for the predator, then
the predator harvesting will be stopped (E2 = 0). Only the prey fishery remains operational (i.e.
c1 p1q1 x ).
December Issue Page 94 of 108 ISSN 2229 5216
c1
We then have from (5.3), x .
p1q1
a c1 d c
Again from (5.2), y 1 1 .
c (b p1q1 c1 ) c p1q1k
c1
Now c1 p1 q1 x p1 q1 k 1 1 0. (5.4)
p1q1k
y 0 .
1 x a y
E1 r ( 1 ) .
q1 k b x
E1 0 provided
y
r
b x 1 x .
(5.5)
a k
Therefore, the bionomic equilibrium [ x , y , E1 , 0] exists if (5.5) holds.
Case II. If c1 p1q1 x, i.e., if the total cost is greater than the total revenue for the prey species,
then the prey harvesting will be stopped (E1= 0). Only the predator fishery remains operational (i.e.
c2 p2 q2 y ).
c2
We then have y .
p2 q2
Again from (5.1), we get
Px 2 Qx R 0, (5.6)
r b a c2
where P , Q 1 r , R br.
k k p2 q2
In equation (5.6), the product of the roots is R/P and the sum of the roots = -Q/P. We now have the
following possibilities, depending on the parameter values:
(i) When R< 0, we have Q2– 4PR> 0 since P> 0. Hence both the roots of equation (5.6) are real and
opposite sign. Then there can be only one bionomic equilibrium. Having obtained x from
equation (5.6), we have
1 a x x
E2 d (1 ) cy .
q 2 b x k
(ii) When R> 0, Q2– 4PR may be either positive or negative. If Q2– 4PR > 0, either both the roots of
equation (5.6) are positive or negative. Hence we may have either two bionomic equilibria or no
bionomic equilibria at all. If Q2– 4PR < 0, both the roots are complex and therefore, there can not
be any bionomic equilibrium.
Case III. If c1 p1q1 x and c2 p2 q2 y then the fishery will be closed.
Case IV. If c1 p1q1 x and c2 p2 q2 y , i.e. if the total costs be less than the total revenues for
both the species, then the fishery is in operation.
c1 c2
In this case, x and y .
p1q1 p2 q2
r c a c2 p1
From (5.1) and (5.2), we have E1 1 1
p q (b p q c ) ,
q1 p1q1k 2 2 1 1 1
December Issue Page 95 of 108 ISSN 2229 5216
1 a c1 c c c2
and E 2 d 1 1
p q k p q .
q 2 b p1 q1 c1 1 1 2 2
Now E1 0 if and only if
r c a c2 p1
1 1
p q (b p q c ) , (5.7)
q1 p1 q1k 2 2 1 1 1
and E2 0 if and only if
a c1 c c c2
d 1 1
pqk . (5.8)
b p1 q1 c1 1 1 p2 q2
Thus the nontrivial bionomic equilibrium point [ x , y , E1 , E2 ] exists if and only if the
conditions (5.7) and (5.8) hold together.
6. Optimal Harvesting Policy
Our objective is to maximize the present value J of a continuous time-stream of revenues
given by
J e t p1 q1 x c1 E1 (t ) p 2 q 2 y c2 E2 (t ) d t ,
0
where denotes the instantaneous annual rate of discount. We intend to maximize J subject to the
state equations (1.1) by invoking Pontryagin’s maximal principle (Pontryagin et. al., 1962). The
control variables Ei(t) (i = 1, 2) are subject to the constraints 0 Ei(t) (Ei)max .
The Hamiltonian for the problem is given by
H e t ( p1q1 x c1 ) E1 ( p2 q2 y c2 ) E2 1 ( F1 q1 E1 x) 2 ( F2 q2 E2 y),
where 1(t) and 2(t) are the adjoint variables and
x axy
F1 rx 1 ,
k b x
axy x
F2 d y 1 c y 2 .
b x k
The control variables E1 and E2 appear linearly in the Hamiltonian function H. Therefore, the
necessary conditions for the singular controls to be optimal are
H
0, i 1, 2 .
Ei
H c
0 1 e t p1 1 ,
q1 x
Now (6.1)
E1
H
c
and 0 2 e t p 2 2
.
(6.2)
E2
q2 y
t
Thus the shadow prices e i (t ), (i 1, 2) do not vary with time in optimal equilibrium.
Hence they satisfy the transversality condition at , i.e., they remain bounded as t .
H 1
Again 0 1q1 x e t ,
E1 E1
December Issue Page 96 of 108 ISSN 2229 5216
H 2
0 2 q 2 y e t .
E2 E2
This implies [Clark, (1985)] that, for each species, the user costs of harvest per unit effort must
equal the discounted value of the future marginal profit of effort at the steady state effort level.
Now
H
1
x
(6.3)
axy r x a b y d y
e t p1 q1 E1 1
(b x) 2 k 2 (b x) 2 k ,
and
H
2
y
(6.4)
ax
e t p 2 q 2 E 2 1 2 ( c y ) .
b x
Now substituting 1 and 2 from (6.1) and (6.2) into (6.3) and simplifying we get
r r d r d
2 p1q1q2 x 4 y p1q1q2 p1q1q2 r q2c1. c2 q1. 4 p1q1q2 . .b x3 y p2 q1q2 . .x 3 y 2
k k k k k
d r d r
2 p2 q1q2 .bx 2 y 2 2b p1q1q2 p1q1q2 r q2c1. c2 q1. q2c1 2 p1q1q2 . .b 2 x 2 y
k k k k
d
p2 q1q2 . .b 2 p1q1q2 ab p2 q1q2 ab q2c1a x y 2
k
r d
p1q1q2 p1q1q2 r q2 c1 . c2 q. b 2 2q2 c1b q1c2ab xy q2 c1b 2 y 0. (6.5)
k k
On the other hand substituting 1 and 2 into (6.4) and simplifying we get,
d 3 b
p 2 q 2 q1 x y p 2 q1 q 2 cx 2 y 2 {q1 q 2 p1 a ( p 2 q 2 c 2 ) q1c p 2 q1 q 2a p 2 q1 q 2 1 d
k k
p 2 q 2 q1 } x 2 y p 2 q1 q 2 c bx y 2 q1c 2 x 2 { ( p 2 q 2 c 2 ) q1bc q 2 c1 a}xy q1c 2 b x 0.
(6.6)
After finding the possible values of x, y from (6.5) and (6.6) we get
ˆ ˆ
ˆ 1 x ay
ˆ ˆ
E1 r 1 ˆ
, (6.7)
q1 k b x
ˆ 1 a xˆ xˆ
and E 2 d 1 c y .
ˆ (6.8)
q2 b x
ˆ k
Hence once the optimal equilibrium x, y is obtained, the optimal equilibrium effort
ˆ ˆ
ˆ ˆ
levels E1 and E2 can be determined from (6.7) and (6.8).
7. Simulation
Let us take r = 1.6, k = 100, a = 2.8, b =10.0, = 0.9, d = 0.09, c = 0.13, q1 = 0.01,
December Issue Page 97 of 108 ISSN 2229 5216
q2 = 0.02, p1 = 10, p2 =12, c1 = 1.3, c2 = 2.6, =0.005 in appropriate units. For the above values of
parameters we found that the optimal equilibrium point (12.38, 11.11) exists and the corresponding
optimal harvesting efforts are ˆ ˆ
E1 = 1.17 and E 2 = 1.42.
8. Concluding Remarks
In this paper, a mathematical model for a prey-predator fishery with harvesting has been
proposed and analyzed by considering the following aspects: (i) the functional response for predator
is Holling type II, (ii) predator species depends partially on an alternative source of food, and (iii)
intraspecific competition in the predator. It has been proved that the system is uniformly bounded
which, inturn implies that the system is biologically well behaved. The existence and stability of
different equilibrium points have been discussed. It has been observed that if the mortality rate, lies
between two distinct values, the system is stable and below it is unstable. When the predator’s
mortality rate attains the lower threshold value, the system possesses small amplitude periodic
oscillations.
We then examined the possibilities of existence of bionomic equilibria of the exploited system.
Next, the optimal harvest policy is discussed. The present value of a continuous time stream of
revenues is maximized by invoking Pontryagin’s maximal principle, subject to the state equations
and control constraints. It is found that the shadow prices remain constant over time in optimal
equilibrium when they satisfy the transverality condition. Also the user costs of harvest per unit effort
must equal the discounted value of the future marginal profit of effort at the steady state effort level.
9. References
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December Issue Page 98 of 108 ISSN 2229 5216
Authors Profile
Dr. Tapan Kumar Kar is an Associate Professor at the Department of
Mathematics, Bengal Engineering and Science University, Shibpur,
Howrah, India. His research interests include dynamical systems, stability
and bifurcation theory, population dynamics, mathematical modeling in
ecology and epidemiology, management and conservation of fisheries,
bioeconomic modeling of renewable resources. He wrote around 66
academic papers on those topics. He also supervised several students of
master and doctor degree.
Dr. Ashim Batabyal received his Ph.D degree from Bengal Engineering and
Science University, Shibpur, Howrah-711103, India in 2010 under the supervision
of Dr. Tapan Kumar kar, Associate Professor, Department of Mathematics,
BESUS, Howrah. This author is currently working as an assistant teacher of
mathematics at Bally Nischinda Chittaranjan vidyalaya. He is a life member of I.
S. I., Kolkata.
r photo
Comes
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December Issue Page 99 of 108 ISSN 2229 5216
