- Email: [email protected]
Qualitative behavior of a fishery system
Qualitative Behavior of a Fishery System* GUR
HUBERMAN
Yale lJniversi@, School of Organization and Management, 56 Hillhouse Avenue, New Haven, Connecticut 06520
Received 15 January 1978; revised 16 March 1978
ABSTRACT A global portrait
of the phase plane for a fishery
model is obtained
for any acceptable
values of the parameters. Three different structures of the phase plane are recovered. The first predicts an eventual collapse of the fishery. The second predicts an unstable limit cycle and an eventual stability of solutions which start inside the limit cycle. The last structure predicts two possible stable equilibria, one with high catch rate, and the other with no catch. Each structure corresponds to a different domain in the parameter space. The boundaries of these domains are found by solving the relevant differential equation for a saddle-to-saddle separatrix in the phase plane by perturbation methods.
INTRODUCTION We consider a model of the combined dynamics of animal population and human fishing effort. We assume that logistic growth perturbed by predation governs the increase of the population. Then we substract the harvest from the natural growth, and obtain the differential equation for the dynamics of the population. For the human effort we assume that the rate of change in the effort is proportional to the net income. This reflects the fact that the hunted population is a common property. A phase-plane analysis is performed in order to study the equilibria of the system. The main problem is to determine for which parameters a saddle-to-saddle separatrix is obtained. We obtain approximate solutions to the problem by using perturbation methods. Consequently we can draw three distinct phase-plane portraits and eliminate other possibilities. In all these structurally different portraits we see that the fish population can be driven to a low undesirable equilibrium. In certain cases, a control of the parameters can prevent a collapse of the population (and the harvest). *This research
MATHEMATICAL OElsevier
was partially
supported
BIOSCIENCES
North-Holland,
Inc., 1978
by NSF Grant
ENG-7615599.
42, 1-14 (1978) 0025-5564/78/090001+
1402.25
GUR HUBERMAN
2 I.
THE MODEL The basic equation
for the dynamics
of the population
is
where u is the population density, r is time, E is the fishing effort and r, K,P,a are parameters. The term ru(1 - u/K) represents logistic growth. The predation term - @‘/(a’+ u2) re p resents a type-III S-shaped functional response of the prey (Holling [4]). Following Holling, we assume that the effect of predation saturates at relatively low population density, and therefore a/K is small. The term - /?/u2(az+ u2) is not the only way to represent a type-111 S-shaped functional response. It was chosen for mathematical convenience. Following Clark and Munro [l], the term yEu represents the human harvest. The rate of change in the fishing effort is governed by the equation dE = ySEu - ypE. dr
(1.2)
The term y15Eu represents the total income, and the term ypE represents the total costs of the fishery. Equations (1.1),(1.2) along with an initial condition on II and E, say at r =O, determine u and E for r > 0. We shall study the solutions of this system (called trajectories in the u - E plane) for different sets of initial conditions and parameters. The formulation of this model was inspired by a similar model of Ludwig et al. [6]. This model and others are surveyed by May [2]. The first model of this type was introduced by Smith [8]. In order to reduce the number of parameters we introduce the following parameters and quantities: “~5, xc!!
Qcf,
a)
a= -ya2, ;
y=??E, P
b=-$
P t= -r. cl
In terms of these we have dx -& = q(x) dv ;?i =au(x-b),
--v,
(1.3)
3
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
where g(x)=R(l-$)-j-$ The function g(x) has either one or three zeroes. The former case yields a straightforward phase-plane portrait, and we shall discuss only the latter case. The relevant domain in the Q-R plane is illustrated in Fig. 1. Let the zeros of g(x) be denoted by x,,xz,xa, and let x4 denote the maximum of g(x). As we have already indicated, Q = K/a is large. Therefore, for 0 < E, G R < i we have
x,=;(;+{F)+O(Q-i)
xz=;(
as
-;+@)+O(Q-‘)
x,=Q-;+O(Q-‘)
Q+co,
as as
Q-+co,
Q-+co,
x4=$--;&+ O(Q-')
as
Q+cO.
The system (1.3) is a nonlinear autonomous system of ordinary differential equations. We refer the reader to Hirsch and Smale [3] for a treatment of this type of system. The equilibria of the system which occur in the first quadrant are
Eo= (0, ‘$3 Ei=(x,,O),
i= 1,2,3,
&= (kg(b)). E4 occurs if x2 < b < x3. If b is outside this interval, the situation is neither realistic nor mathematically interesting, and we consider only the case x,< b
and repels in the
GUR HUBERMAN
4
(3) E, is a saddle point which repels in the x-direction. (4) E3 is a saddle point which attracts in the x-direction.
(5) E4 is asymptotically
stable [unstable]
if x=b.
at The motions II.
in the phase plane are illustrated
A SADDLE-TO-SADDLE
in Fig. 2.
SEPARATRIX
The x axis is a separatrix that joins E, and E,. The motions in the phase plane suggest that there may be another separatrix that goes from EJ to E2. The existence of such a separatrix will convey information about the global portrait of the phase plane. Let T, be the (unique) orbit in the first quadrant that converges to E2 as t+cc, and let TJ be the (unique) orbit in the first quadrant that converges to E, as t+cc (see Fig. 2). If there is a saddle-to-saddle separatrix, then T2 = T3. In particular, T2= T3 for x= 6. Hence the motivation to find the value of b for which such a separatrix is obtained. From here on we shall refer to this as the solution for 6, and denote it by 6. We shall concentrate on the equation a_v(x-b)
dy _=
dx
xg(x;Q)-xY
(11.1)
.
We shall find the b for which there exists a solution, y that satisfies
Y(X;) =Q
i=2,3,
(11.2)
i.e., there is a saddle-to-saddle separatrix. Equation (11.1) is too complicated to integrate exactly, and we use a perturbation method. is where structural changes occur,’ we scale: Observing that x0 = m
~ .*
(11.3)
B= &
‘In fact, a Hopf bifurcation
takes place near b = xw
*Following
,j=O,
this, uj=xj/a
1,2,3;
G=d/vo
.
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
s 2 3
---
---
~--------
<
GUR HUBERMAN
h
/ /P
/
/ /
QUALITATIVE The
7
BEHAVIOR OF A FISHERY SYSTEM
new form of (11.1.1) is dY
_= du
d?j
(u--).I’
u(G-Y)
(11.4)
’
where
Now we try a parameter of the form P= u_Q”’and the following expansions for the saddle-to-saddle separatrix and B: For small P we have
For large P we have Y(‘)(u;Q)=
Y#)(u;Q)+P-‘Y~‘)(u;Q)+P-~Y$‘)(u;Q)+..., .
~(‘)=B~‘)(Q)+P-‘B(‘)(Q)+P-~B~‘)(Q)+...
(11.6)
B$)(Q) appears to be O(1) as Q+co, and m is chosen so that also By)=O(l) as Q-cc. It appears that m=3/2 is the appropriate m. To obtain asymptotic solutions for small [large] P we substitute (11.5) [(11.6)] into (11.4). Then we expand in powers of P and equate the coefficients of equal powers of P. For the rest of the paper we shall be interested only in the expansions of 8. We present them, omitting the details of the manipulations and the perturbation-theoretic considerations. For small P we have
BP = uo(Q ), B,c”,=
where ’ = d/du.
(11.7)
G(uo;Q)[2G"(uo;Q>+uoG"'(uo~Q>l 2~4 G”(uo)f
Q,,,-,
3
(11.8)
Therefore i@(Q)
Q 3/2-m = 8R
+
O(l)
as
Q+cc.
For large P we have u3-“2
lnu,-
lnu,
’
(11.9)
.
0.0389 0.0389 0.0389
“Large P” is good for P=5.59
-
50 50 50
0.108 0.108 0.108
Both are good for P= 1.77
-
50 50 50
0.3 0.3 0.3
Noneisgood for P= 1.77
55.90 5.59 0.56
17.67 1.77 0.117
17.67 1.77 0.117
17.67 1.77 0.177
P=aQ312
._ ._ ._ -
500 500 500
50 50 50
0.5 0.5 0.5
Q
“Small P” is good for P= 1.77
R
-
155.80 150.78 131.21
22.57 22.35 21.74
15.87 14.03 13.045
12.57 11.06 10.04
b^
Computed
Results
2308.6 332.91 135.35
41.58 23.48 21.67
37.08 15.22 13.035
21.37 11.6 10.02
ii
“Small P”
TABLE 1 A Sample of the Computer
1381. 120.8 3.2
155.99 155.99 155.99
22.58 22.58 22.58
84. 5. 0.3
0.1 3.5 18.9
0.04 1. 3.9
2.3 15. 15.
16.25 16.25 16.25
m
133. 8.5 0.08
P”
1.8 15. 27.
d
“Large
“Large P” relative error
12.8 12.8 12.8
117. 4.8 0.1
m
“Small P” relative error
QUALITATIVE
BEHAVIOR OF A FISHERY SYSTEM
Therefore
h&”=
a ln(Q/xz)
+
o(1)
as Q-J.
In both cases we see that for large Q, B > U, (or g>x,). This observation is the main tool in the next section. The analytic work was accompanied by numerical integration. The program was run for R-values ranging from 0.014 to 0.5, Q-values from 50 to 5000, and u-values from 10d5 to 103. For P > 10, the numerical solution for 6 was not more than 5% away from the asymptotic “large P” approximation. For P< 1 the numerical solution was not more than 7% away from the asymptotic “small P” approximation. For P between 1 and 10 the results varied. Sometimes they were close to the “large P” approximation, sometimes to the “small P” approximation, sometimes to both and sometimes to neither. A sample of the numerical results for intermediate P is given in the Table 1. III.
THE GLOBAL
PORTRAIT
OF THE PHASE PLANE
Now we are ready to draw the phase-plane portrait for the various parameter values. The result of the previous chapter (6> x,,) plays a major role in our analysis. We note that when b < 6, T3 lies above T,. Hence, T3 (which starts at E3) must go to E,. As a result we have a domain bounded by solution trajectories. The boundaries are the x-axis and T3, and any solution which starts inside this domain cannot leave its boundaries. For x2 < b < x,, there is only one asymptotically stable equilibrium in the phase plane, namely E,. It is on the boundary of K, the compact set bounded by T3 and the x-axis. Therefore every solution (except those starting exactly at the equilibria or on T2) will converge to E,. This conclusion holds for solutions which start inside K or outside it in the first quadrant. For x,, < b < 6, E4 is asymptotically stable and T, still lies above T,. One can use the Poincare-Bendixon theorem [6] to show than an unstable limit cycle must occur in this case. This limit cycle is around E4 and inside K. The existence of a limit cycle for b near x,, can also be shown by noticing that at b = x,, a Hopf bifurcation takes place. We refer the reader to [5] for a statement of the theorem. For x0 < b < 6, every solution which starts in the domain bounded by the limit cycle will remain there, and converge to Ed. Outside the closed set bounded by the limit cycle, all the solutions converge to E, (except those starting at E,, E3 or on T,).
10
GUR HUBERMAN
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
11
12
GUR HUBERh4AN
---
/
_ /--/
mu
>
QUALITATIVE
BEHAVIOR
OF A FISHERY
13
SYSTEM
SUMMARY
We have indicated three structurally different portraits of the phase plane in the interior of the first quadrant. The first portrait was obtained for b 6, i.e., when T, > T,. Then solutions which start at an equilibrium or on T2 remain there. Solutions which start under T2 converge to E4, and all the other solutions converge to E,. A forth portrait that might have occurred would have E4 as an asymptotically unstable equilibrium and an asymptotically stable limit cycle surrounding E4. It was ruled out because d > x,,. V.
INTERPRETATION
As we could see, there is a danger of driving the harvested population to E,. Clearly, this is an undesirable equilibrium, the equilibrium of a graveyard: low animal density and no harvesting activity. Another feature is the possibility of fluctuations both in the effort and in the population. With b (x,,d) these fluctuations may be undesirable but not fatal if they occur inside the limit cycle. But if they are outside the limit cycle they end up at E,-a disaster. If the situation is not too bad, a collapse can be avoided by regulating the fishery, i.e., by controlling either a or b. The higher b is, the smaller the attraction domain of E, is. Given (x (O),y (0)), we can determine b such that (x(O),y(O)) are in the interior of the limit cycle (in the second portrait) or under T, (in the third portrait), or we need b > x3 to achieve one of those. Then a collapse is inevitable. The main conclusion is that even from such a simple model it can be seen that nature is not always forgiving; there is a danger of depleting this type of resource.
l
Thanks are due to Dr. D. Ludwig for persistent support and many useful discussions. Thanks are also due to Dr. l? Wan for helpful comments regarding an early draft of this paper. REFERENCES 1 C. W. Clark and G. R. Munro, The economics of fishing and modem capital theory: simplified approach, J. Environmental Economics and Management 2:92-106 (1975). 2 E. A. Coddington and N. Levinson, McGraw-Hill, New York, 1955.
Theory
of Ordinaty
Differential
a
Equations,
14
GUR HUBERMAN
3 M. W. Hirsch and Smale, D$erential Equations, Dynamical System, and Linear Algebra, Academic, New York, 1974. 4 C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomological Society of Can& 45 (1965). 5 N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Appl. Math. 52: (4) (1973). 6 D. Ludwig, D. Jones and C. S. Holling, An analytical model for the interaction between
budworm and balsam, in preparation. 7 R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature 269: 471-477 (1977). 8 V. L. Smith, On models of commercial fishing, J. Political Economy 77: 181-198 (1969).
HUBERMAN
Yale lJniversi@, School of Organization and Management, 56 Hillhouse Avenue, New Haven, Connecticut 06520
Received 15 January 1978; revised 16 March 1978
ABSTRACT A global portrait
of the phase plane for a fishery
model is obtained
for any acceptable
values of the parameters. Three different structures of the phase plane are recovered. The first predicts an eventual collapse of the fishery. The second predicts an unstable limit cycle and an eventual stability of solutions which start inside the limit cycle. The last structure predicts two possible stable equilibria, one with high catch rate, and the other with no catch. Each structure corresponds to a different domain in the parameter space. The boundaries of these domains are found by solving the relevant differential equation for a saddle-to-saddle separatrix in the phase plane by perturbation methods.
INTRODUCTION We consider a model of the combined dynamics of animal population and human fishing effort. We assume that logistic growth perturbed by predation governs the increase of the population. Then we substract the harvest from the natural growth, and obtain the differential equation for the dynamics of the population. For the human effort we assume that the rate of change in the effort is proportional to the net income. This reflects the fact that the hunted population is a common property. A phase-plane analysis is performed in order to study the equilibria of the system. The main problem is to determine for which parameters a saddle-to-saddle separatrix is obtained. We obtain approximate solutions to the problem by using perturbation methods. Consequently we can draw three distinct phase-plane portraits and eliminate other possibilities. In all these structurally different portraits we see that the fish population can be driven to a low undesirable equilibrium. In certain cases, a control of the parameters can prevent a collapse of the population (and the harvest). *This research
MATHEMATICAL OElsevier
was partially
supported
BIOSCIENCES
North-Holland,
Inc., 1978
by NSF Grant
ENG-7615599.
42, 1-14 (1978) 0025-5564/78/090001+
1402.25
GUR HUBERMAN
2 I.
THE MODEL The basic equation
for the dynamics
of the population
is
where u is the population density, r is time, E is the fishing effort and r, K,P,a are parameters. The term ru(1 - u/K) represents logistic growth. The predation term - @‘/(a’+ u2) re p resents a type-III S-shaped functional response of the prey (Holling [4]). Following Holling, we assume that the effect of predation saturates at relatively low population density, and therefore a/K is small. The term - /?/u2(az+ u2) is not the only way to represent a type-111 S-shaped functional response. It was chosen for mathematical convenience. Following Clark and Munro [l], the term yEu represents the human harvest. The rate of change in the fishing effort is governed by the equation dE = ySEu - ypE. dr
(1.2)
The term y15Eu represents the total income, and the term ypE represents the total costs of the fishery. Equations (1.1),(1.2) along with an initial condition on II and E, say at r =O, determine u and E for r > 0. We shall study the solutions of this system (called trajectories in the u - E plane) for different sets of initial conditions and parameters. The formulation of this model was inspired by a similar model of Ludwig et al. [6]. This model and others are surveyed by May [2]. The first model of this type was introduced by Smith [8]. In order to reduce the number of parameters we introduce the following parameters and quantities: “~5, xc!!
Qcf,
a)
a= -ya2, ;
y=??E, P
b=-$
P t= -r. cl
In terms of these we have dx -& = q(x) dv ;?i =au(x-b),
--v,
(1.3)
3
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
where g(x)=R(l-$)-j-$ The function g(x) has either one or three zeroes. The former case yields a straightforward phase-plane portrait, and we shall discuss only the latter case. The relevant domain in the Q-R plane is illustrated in Fig. 1. Let the zeros of g(x) be denoted by x,,xz,xa, and let x4 denote the maximum of g(x). As we have already indicated, Q = K/a is large. Therefore, for 0 < E, G R < i we have
x,=;(;+{F)+O(Q-i)
xz=;(
as
-;+@)+O(Q-‘)
x,=Q-;+O(Q-‘)
Q+co,
as as
Q-+co,
Q-+co,
x4=$--;&+ O(Q-')
as
Q+cO.
The system (1.3) is a nonlinear autonomous system of ordinary differential equations. We refer the reader to Hirsch and Smale [3] for a treatment of this type of system. The equilibria of the system which occur in the first quadrant are
Eo= (0, ‘$3 Ei=(x,,O),
i= 1,2,3,
&= (kg(b)). E4 occurs if x2 < b < x3. If b is outside this interval, the situation is neither realistic nor mathematically interesting, and we consider only the case x,< b
and repels in the
GUR HUBERMAN
4
(3) E, is a saddle point which repels in the x-direction. (4) E3 is a saddle point which attracts in the x-direction.
(5) E4 is asymptotically
stable [unstable]
if x=b.
at The motions II.
in the phase plane are illustrated
A SADDLE-TO-SADDLE
in Fig. 2.
SEPARATRIX
The x axis is a separatrix that joins E, and E,. The motions in the phase plane suggest that there may be another separatrix that goes from EJ to E2. The existence of such a separatrix will convey information about the global portrait of the phase plane. Let T, be the (unique) orbit in the first quadrant that converges to E2 as t+cc, and let TJ be the (unique) orbit in the first quadrant that converges to E, as t+cc (see Fig. 2). If there is a saddle-to-saddle separatrix, then T2 = T3. In particular, T2= T3 for x= 6. Hence the motivation to find the value of b for which such a separatrix is obtained. From here on we shall refer to this as the solution for 6, and denote it by 6. We shall concentrate on the equation a_v(x-b)
dy _=
dx
xg(x;Q)-xY
(11.1)
.
We shall find the b for which there exists a solution, y that satisfies
Y(X;) =Q
i=2,3,
(11.2)
i.e., there is a saddle-to-saddle separatrix. Equation (11.1) is too complicated to integrate exactly, and we use a perturbation method. is where structural changes occur,’ we scale: Observing that x0 = m
~ .*
(11.3)
B= &
‘In fact, a Hopf bifurcation
takes place near b = xw
*Following
,j=O,
this, uj=xj/a
1,2,3;
G=d/vo
.
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
s 2 3
---
---
~--------
<
GUR HUBERMAN
h
/ /P
/
/ /
QUALITATIVE The
7
BEHAVIOR OF A FISHERY SYSTEM
new form of (11.1.1) is dY
_= du
d?j
(u--).I’
u(G-Y)
(11.4)
’
where
Now we try a parameter of the form P= u_Q”’and the following expansions for the saddle-to-saddle separatrix and B: For small P we have
For large P we have Y(‘)(u;Q)=
Y#)(u;Q)+P-‘Y~‘)(u;Q)+P-~Y$‘)(u;Q)+..., .
~(‘)=B~‘)(Q)+P-‘B(‘)(Q)+P-~B~‘)(Q)+...
(11.6)
B$)(Q) appears to be O(1) as Q+co, and m is chosen so that also By)=O(l) as Q-cc. It appears that m=3/2 is the appropriate m. To obtain asymptotic solutions for small [large] P we substitute (11.5) [(11.6)] into (11.4). Then we expand in powers of P and equate the coefficients of equal powers of P. For the rest of the paper we shall be interested only in the expansions of 8. We present them, omitting the details of the manipulations and the perturbation-theoretic considerations. For small P we have
BP = uo(Q ), B,c”,=
where ’ = d/du.
(11.7)
G(uo;Q)[2G"(uo;Q>+uoG"'(uo~Q>l 2~4 G”(uo)f
Q,,,-,
3
(11.8)
Therefore i@(Q)
Q 3/2-m = 8R
+
O(l)
as
Q+cc.
For large P we have u3-“2
lnu,-
lnu,
’
(11.9)
.
0.0389 0.0389 0.0389
“Large P” is good for P=5.59
-
50 50 50
0.108 0.108 0.108
Both are good for P= 1.77
-
50 50 50
0.3 0.3 0.3
Noneisgood for P= 1.77
55.90 5.59 0.56
17.67 1.77 0.117
17.67 1.77 0.117
17.67 1.77 0.177
P=aQ312
._ ._ ._ -
500 500 500
50 50 50
0.5 0.5 0.5
Q
“Small P” is good for P= 1.77
R
-
155.80 150.78 131.21
22.57 22.35 21.74
15.87 14.03 13.045
12.57 11.06 10.04
b^
Computed
Results
2308.6 332.91 135.35
41.58 23.48 21.67
37.08 15.22 13.035
21.37 11.6 10.02
ii
“Small P”
TABLE 1 A Sample of the Computer
1381. 120.8 3.2
155.99 155.99 155.99
22.58 22.58 22.58
84. 5. 0.3
0.1 3.5 18.9
0.04 1. 3.9
2.3 15. 15.
16.25 16.25 16.25
m
133. 8.5 0.08
P”
1.8 15. 27.
d
“Large
“Large P” relative error
12.8 12.8 12.8
117. 4.8 0.1
m
“Small P” relative error
QUALITATIVE
BEHAVIOR OF A FISHERY SYSTEM
Therefore
h&”=
a ln(Q/xz)
+
o(1)
as Q-J.
In both cases we see that for large Q, B > U, (or g>x,). This observation is the main tool in the next section. The analytic work was accompanied by numerical integration. The program was run for R-values ranging from 0.014 to 0.5, Q-values from 50 to 5000, and u-values from 10d5 to 103. For P > 10, the numerical solution for 6 was not more than 5% away from the asymptotic “large P” approximation. For P< 1 the numerical solution was not more than 7% away from the asymptotic “small P” approximation. For P between 1 and 10 the results varied. Sometimes they were close to the “large P” approximation, sometimes to the “small P” approximation, sometimes to both and sometimes to neither. A sample of the numerical results for intermediate P is given in the Table 1. III.
THE GLOBAL
PORTRAIT
OF THE PHASE PLANE
Now we are ready to draw the phase-plane portrait for the various parameter values. The result of the previous chapter (6> x,,) plays a major role in our analysis. We note that when b < 6, T3 lies above T,. Hence, T3 (which starts at E3) must go to E,. As a result we have a domain bounded by solution trajectories. The boundaries are the x-axis and T3, and any solution which starts inside this domain cannot leave its boundaries. For x2 < b < x,, there is only one asymptotically stable equilibrium in the phase plane, namely E,. It is on the boundary of K, the compact set bounded by T3 and the x-axis. Therefore every solution (except those starting exactly at the equilibria or on T2) will converge to E,. This conclusion holds for solutions which start inside K or outside it in the first quadrant. For x,, < b < 6, E4 is asymptotically stable and T, still lies above T,. One can use the Poincare-Bendixon theorem [6] to show than an unstable limit cycle must occur in this case. This limit cycle is around E4 and inside K. The existence of a limit cycle for b near x,, can also be shown by noticing that at b = x,, a Hopf bifurcation takes place. We refer the reader to [5] for a statement of the theorem. For x0 < b < 6, every solution which starts in the domain bounded by the limit cycle will remain there, and converge to Ed. Outside the closed set bounded by the limit cycle, all the solutions converge to E, (except those starting at E,, E3 or on T,).
10
GUR HUBERMAN
QUALITATIVE BEHAVIOR OF A FISHERY SYSTEM
11
12
GUR HUBERh4AN
---
/
_ /--/
mu
>
QUALITATIVE
BEHAVIOR
OF A FISHERY
13
SYSTEM
SUMMARY
We have indicated three structurally different portraits of the phase plane in the interior of the first quadrant. The first portrait was obtained for b
INTERPRETATION
As we could see, there is a danger of driving the harvested population to E,. Clearly, this is an undesirable equilibrium, the equilibrium of a graveyard: low animal density and no harvesting activity. Another feature is the possibility of fluctuations both in the effort and in the population. With b (x,,d) these fluctuations may be undesirable but not fatal if they occur inside the limit cycle. But if they are outside the limit cycle they end up at E,-a disaster. If the situation is not too bad, a collapse can be avoided by regulating the fishery, i.e., by controlling either a or b. The higher b is, the smaller the attraction domain of E, is. Given (x (O),y (0)), we can determine b such that (x(O),y(O)) are in the interior of the limit cycle (in the second portrait) or under T, (in the third portrait), or we need b > x3 to achieve one of those. Then a collapse is inevitable. The main conclusion is that even from such a simple model it can be seen that nature is not always forgiving; there is a danger of depleting this type of resource.
l
Thanks are due to Dr. D. Ludwig for persistent support and many useful discussions. Thanks are also due to Dr. l? Wan for helpful comments regarding an early draft of this paper. REFERENCES 1 C. W. Clark and G. R. Munro, The economics of fishing and modem capital theory: simplified approach, J. Environmental Economics and Management 2:92-106 (1975). 2 E. A. Coddington and N. Levinson, McGraw-Hill, New York, 1955.
Theory
of Ordinaty
Differential
a
Equations,
14
GUR HUBERMAN
3 M. W. Hirsch and Smale, D$erential Equations, Dynamical System, and Linear Algebra, Academic, New York, 1974. 4 C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomological Society of Can& 45 (1965). 5 N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Appl. Math. 52: (4) (1973). 6 D. Ludwig, D. Jones and C. S. Holling, An analytical model for the interaction between
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