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Conservation of an Ecosystem through Optimal Taxation
Bulletin of Mathematical Biology (1998) 60, 569–584
Conservation of an Ecosystem through Optimal Taxation S. V. KRISHNA, P. D. N. SRINIVASU Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India B. KAYMAKCALAN Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey In this paper we study a harvesting problem in the presence of a predator and a tax. The objective is to maximize the monetary social benefit as well as prevent the predator from extinction, keeping the ecological balance. c 1998 Society for Mathematical Biology
1.
I NTRODUCTION
Economic progress and ecological balance always have conflicting interests. Catering to the necessities and comforts of human beings invariably robs the ecological structure of the nature. This, more often than not, leads to the extinction of a species of life. Often it is possible to prevent such extinction by proper planning. Such a planning has to be either by force or dissentive. For example, if a particular activity by individuals of a region is causing severe damage of the ecosystem of that region and if the activity is inevitable then the governing authority of the region should plan a regulating policy which would keep the damage to the ecosystem minimal. If a species is becoming extinct the regulating policy should prevent it. One such activity is harvesting. Every bit of the catch is not edible and harvesting harms some of the marine species which live on the other species from the sea and which are edible to human beings. Thus, the predator species are likely to become extinct with an indiscrete increase in harvesting activity. To avoid this the regulating authority levies a tax on the catch of the harvesting agency. This acts as a deterrent to the fisher and helps the predator to grow. It sometimes acts as an incentive to the fisher (when the tax takes the form of a subsidy). The aim of this paper is to find the proper taxation policy which would give the best possible benefit through harvesting to the community while preventing the extinction of the predator. This is different from the usual optimal harvesting problem whose objective is purely monetary. It differs from the problem of 0092-8240/98/030569 + 16
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maintaining the predator at a high level without caring for the harvesting activity. It also differs from the other problems in that due to the imposition tax the effort is a dynamic evolutionary variable. Hence, the problem is three-dimensional and some useful theorems such as Du Lac theorem (Glendinning, 1994) cannot be used here. Harvesting problems with tax have been studied by Clark (1976, 1979) and Choudary and Johnson (1990). In the absence of a predator these problems were just two-dimensional. The main feature of this paper is using a natural expression for the catch which allows meaningful conditions for the existence and stability of interior equilibrium. These conditions are later used as constraints on the tax policy. We have also obtained a characteristic equation for the optimal equilibrium. This paper is organized as follows. The problem is clearly stated in the next section. In Section 3, we analyse the existence and stability of equilibria. In Section 4, we describe the path of the system; particularly how a path approaches the interior equilibrium for various tax levels. Section 5 is devoted to the main results of the paper, namely the optimal tax policy and the corresponding paths. Section 6 contains numerical verification of some key results. Conclusions and discussion are presented in Section 7.
2.
T HE P ROBLEM
The ecological set-up is as follows. There is a prey which is harvested continuously. There is a predator, living on the prey. It is assumed that the harvesting does not affect the growth of the predator population directly, and the harvesting agency does not adjust the effort due to the presence of the predator. Thus, the interaction between the harvesting agency and the predator is through the third party, namely, prey. However, there is a conflict of interests between the fisherman and the predator, i.e., a competition for a common resource. Although this situation appears to be game theoretic, the predator is unable to evolve a strategy for its survival. Hence the regulating agency comes to the rescue of the predator through a suitable tax policy. The harvesting agency’s aim is to obtain as much revenue as possible through its activity, whereas the community needs the food through harvesting and is also keen on protecting the predator from extinction. Thus the benefit to the community consists of the revenue through the harvest and the retained predator population. Thus the problem of optimization of the community’s benefit is a conditional optimal control problem in the sense that the revenue is to be maximized subject to the condition that the predator population is larger than a positive quantity as t → ∞. In order to achieve this goal a regulating agency has to curb arbitrary growth of harvesting. This is done by levying a tax on the catch (which can also be a subsidy). The problem, now, is to find the optimal taxation policy which gives maximum benefit to the society, i.e., a taxation policy which gives the maximum benefit
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through the harvesting and at the same time takes the predator population to a level from which it will not become extinct. The harvesting is affected by the predation. Tax (or subsidy) makes the harvesting effort a dynamic variable. The net benefit, from the harvesting, to the society is the revenue before the deduction of tax, obtained by the harvesting agency. One would wish for a variable taxation through an initial ‘plan’ period and settle down to a ‘constant’ taxation thereafter. The initial period is utilized to iron out the initial hiccups one might encounter at the beginning. Thus if [0, A] is the initial plan period our objective can be to find a taxation policy T (t) defined on [0, A] such that the social benefit through harvesting in [0, A] is maximized subject to the constraint that the predator population p(t) reaches a level p ∗ at t = A, such that under a suitable taxation T (t) = T ∗ , p(t) cannot approach 0 as t → ∞. We now formulate this problem mathematically. We denote by x(t), p(t) and E(t), the population of prey, the predator population and the effort at time t. The prey–predator dynamics with harvesting is given by: x x 0 (t) = γ x 1 − − αx p − h(E, x) K p 0 (t) = −βp + ρpx where γ , K , α, β and ρ have the usual interpretation and h(E, x) is a harvesting term referring to the catch when the prey population is x and harvesting effort is E. It is assumed that harvesting h is not affected by the predation directly, nor is any part of the effort E devoted to control predation. Traditionally, h(E, x) is taken as q E x, where q is the catchability coefficient. We do not take h in this form, our objection being that h(E, x) = q E x leads to the fallacial conclusion that h(E, x) tends to infinity as the effort goes to infinity with the prey population finite and fixed, or as the prey population goes to infinity with finite and fixed effort. It may be argued that as the effort and the prey population remain bounded, there is no danger of letting E or x go to infinity. Mathematically this may be granted. When we combine economics, a more realistic harvesting function would lead to very important conclusions, as we see in Section 3. We therefore take the harvesting term as (Clark, 1979): h(E, x) = q
Ex a E + 1x
where q is still the catchability coefficient and a and l are constants. We will discuss the importance of a and l in Section 3. Here we point out that lim E→∞ h(E, x) = qa x, and lim X →∞ h(E, x) = ql E. Using this harvesting term, we rewrite the equation for x as Ex x x0 = γ x 1 − − αx p − q . K a E + lx
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The controlling agency, like the government, levies a tax T on the harvesting agency. The purpose of the tax (which may be a subsidy) is to regulate the harvesting effort. We assume that the tax T is bounded: Tmin ≤ T ≤ Tmax . Introduction of tax makes harvesting effort E a dynamic variable governed by the equation Ex 0 E =λ q (m − T ) − cE a E + lx where m is the price of unit catch, c is the cost of the unit effort and λ is a factor which converts savings into capital. Thus in our problem, the state variables x, p, E are governed by the equations Ex x − αx p − q x0 = γ x 1 − (1) K a E + lx p 0 = −βp + ρpx Ex 0 E =λ q (m − T ) − cE a E + lx
(2) (3)
these are supplemented by initial conditions x(0) = x0 , p(0) = p0 , E(0) = E 0
(4)
and the positivity constraints (x, p, E) ∈ R3+ , the positive octant in R3 . The control variable is T , and the constraint on T at this moment is T ∈ [Tmin , Tmax ]. We note that the behavioral response of the fisher to the introduction of tax is to retard the growth of his effort with the increasing tax rates. This is an important and deciding factor in determining the optimal tax policy. As stated earlier, the goal is to maximize the social benefit. The monetary benefit to the society due to the harvesting activity is Z ∞ f (T ) = H (t, x, p, E, T ) dt 0
Ex where H (t, x, p, E, T ) = e−δt q a E+lx m − cE where δ is the discounting factor. As the social benefit includes preservation of predator population we should maximize f (T ) such that p(t) approaches a nonzero value as t → ∞. Let X ∗ = (x ∗ , p ∗ , E ∗ ) be such that there exists an admissible tax policy T (t) such that (1)–(3), with X (A) = X ∗ has a strictly positive solution under the policy ∗ T (t) for t ≥ A. Let T be the set of all such R ∞ policies. Let T ∈ T (subject to existence) be a tax policy which maximizes A H dt with constraints (1)–(3) and X (A) = X ∗ . We now consider the problem of approaching X ∗ most rapidly (i.e., most rapid approach—see Clark, 1976). Let T¯ be such a policy and A be the shortest time obtained thereof. Then, the optimal tax policy T0 we seek consists of T¯ (t), t ∈ [0, A] T0 = ∗ T (t), t ≥ A.
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An interesting and important path is the equilibrium or steady-state solution. It is often convenient to find an optimal equilibrium, i.e., the equilibrium corresponding to the optimal tax, so that the optimal path would consist of a path leading to the optimal equilibrium level as quickly as possible and then stay at that optimal equilibrium level with that optimal tax subsequently. We remark that for such an optimal policy to be practicable, it is necessary that the optimal equilibrium point be at least locally asymptotically stable.
3.
E QUILIBRIUM A NALYSIS
We now study the existence and nature of equilibria of the system. Particularly we are interested in the interior or positive equilibrium of the system. Since tax is a system parameter, positive equilibria exists only for some restricted tax levels. Stability of the positive equilibrium adds to the constraint on the tax, assuming the other system parameters are known. Furthermore, we wish to know the bioeconomic ramification of the bounds on the tax levels. Interior equilibrium is important for the main problem of this paper because, to control optimally we obtain the optimal tax levels and drive the system to the optimal equilibrium level most efficiently and in the shortest time possible. To begin with we list all possible equilibria. (i) The trivial equilibrium (0, 0, 0). (ii) Equilibrium in the absence of effort (E = 0) x E = β/ρ,
γ pE = α
β 1− ρK
.
(iii) Equilibrium in the absence of the predator ( p = 0) xp =
K [cl + τ (aγ − q)] , aγ τ
Ep =
K [cl + τ (aγ − q)](gτ − cl) . a 2 γ cτ
(iv) The interior (positive) equilibrium x(T ¯ ) = β/ρ, β qτ − cl γ 1− − p(T ¯ )= , α ρK αaτ ¯ ) = β(qτ − cl) E(T ρaτ ¯ )) by e(T where τ = m − T . We denote (x(T ¯ ), p(T ¯ ), E(T ¯ ). To facilitate the discussion of equilibria, we need to introduce some notions.
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(1) MEC at x (maximum effort catch at x) is the limit of the catch as the effort E → ∞. Thus MEC at x = (q/a)x, MEC at x =
q x, a
q MEC = . a unit pop
(2) MPC at E (maximum population catch at E) is similarly defined. (3) PGR at x = γ x (pure growth rate at x) is the growth rate of the prey when its population is x, in the absence of predator, harvesting, and intraspecific competition (Malthusian growth). (4) PGRRC at x = γ (K − x) (pure growth rate of the remaining capacity at x) is the PGR at K − x. (5) PGRRC at Eq (PGRRC at equilibrium) is γ (K − βρ ) 4(PGR at Eq) (6) Recouping quotient (RQ) = θ = . MEC at K Equilibrium (ii) is the usual prey–predator equilibrium, whereas (iii) is the equilibrium with harvesting and without predator. Neither these nor the trivial equilibrium is of interest to us. However, the equilibrium (x p , 0, E p ) exists, that is x p and E p are positive, only if 0 < qτ − cl < aγ τ . The existence of the positive equilibrium puts restrictions on the tax level T . These and the additional bounds we obtain later, will be constraints on T in the optimal control problem discussed in Section 5. For the equilibrium (iv) to be positive we first need β < ρ K . This is independent of the tax. Next we must have qτ > cl. This leads to the upper bound on T , T < m − cl/q.
(5)
¯ ) positive. For p(T Such a restriction on T makes E(T ¯ ) to be positive we have to discuss two distinct situations. (A) MEC at K > PGRRC at Eq. In this case we have the lower bound for T , namely ρ K cl T >m− . (6) βγ a − ρ K (aγ − q) Thus the constraints on T , for the existence of the positive equilibrium are m−
ρ K cl < T < m − cl/q. βγ a − ρ K (aγ − q)
(7)
(B) MEC at K < PGRRC at Eq. Then no lower bound on T can nor needs to be imposed. Thus, the only constraint on the tax will be the upper bound given by (5). The inequality (5) can be interpreted as follows. ‘Tax’ plus the ‘cost-to-catch’ ratio at arbitrary large populations should be smaller than the price quoted by the fishing agency.
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Suppose this condition is not satisfied, then the fishing agent loses steadily and the disinvestment in the harvesting activity occurs. This leads to a decrease in the effort, and as a consequence the prey population increases. Thus a ‘positive’ equilibrium cannot be reached. This is an almost obvious interpretation. The inequality (6) gives a lower bound for T . But for this it is necessary that MEC at K > PGRRC at Eq. Let us examine this situation closely. Assume that MEC at K > PGRRC at Eq.
(8)
Suppose that T < m − βγ a−ρρ KK cl(aγ −q) , i.e., the government or the regulating agency has given a subsidy T to the fishing agency. Then the net price m − T to the fisherman exceeds βγ a−ρρ KK cl(aγ −q) > 0. This encourages the fishing agency to put in larger effort (E → ∞). By (8), the catch (loss to the population) is more than the pure growth of the remaining capacity. Hence the prey tends to extinction so losing positive equilibrium. Thus no equilibrium is possible (E ↑ ∞, x ↓ 0). If we have case (B) any amount of subsidy given by the regulating agency does not cause constant increase or decrease in the effort or population and hence we can get a positive equilibrium. Thus inequalities (7) in case (A) and (5) in case (B) are the constraints on the tax for the existence of the positive equilibrium. The effect of harvesting on the prey–predator system is clear. It lowers the predator equilibrium level. The level of tax affects the equilibrium positions of the predator and harvesting effort, but the equilibrium positions of the prey is not affected. Since the reduction due to harvesting in the equilibrium level of −cl the predator population, qταaτ , increases with τ = m − T , it is obvious that the equilibrium p(T ¯ ) increases with reduced tax and decreases with increased tax. This is sensible because reduced tax levels encourage harvesting effort, leading to depletion of prey which in turn causes starvation of predator, thus lowering their equilibrium level. The equilibrium level of the harvesting effort is clearly higher for lower tax levels. The equilibrium (iii) is not positive if aγ τ −lc. However, this and (5) or (7) are not compatible. Thus equilibrium (ii) exists (as a positive equilibrium) whenever the system has a positive (interior) equilibrium (iv). Combining all these results we have the following. T HEOREM 1. The prey–predator system with harvesting described by equations (1)–(3) has a unique interior equilibrium for any tax level T with m − βγ a−ρρ KK cl(aγ −q) < T < m − cl/q if MEC at K > PGRRC at Eq; and for any tax T with T < m − cl/q if MEC at K < PGRRC at Eq. The equilibrium is ¯ )) given in (iv). The equilibrium level of the predator, namely (x(T ¯ ), p(T ¯ ), E(T ¯ ) p(T ¯ ) increases with a raise of tax level. The equilibrium level of the effort E(T increases with a reduction of tax. Proof. Simple computations show that MEC at K > PGRRC at Eq implies that βγ a − ρ K (aγ − q) is positive. p(T ¯ ) in (iv) is only positive if (βaγ − ρaγ K + K ρq )τ < kρcl.
(9)
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Kρcl Thus m − T < βaγ −ρaγ , which gives the upper bound in (7) for T . If K +K ρq MEC at K ≤ PGRRC at Eq, then (9) is true for any non-negative value of τ and hence for any T < m. Thus, the only constraint in this case is given by (5). The second part of the theorem is almost trivial. We now discuss the stability of the equilibria (ii)–(iv).
T HEOREM 2. The boundary equilibrium points (iii) and (ii) are unstable under the hypotheses (7). The interior equilibrium (iv) is locally stable if recouping quotient > 1 or
ρ K cl ρ K cl ρ K cl ρ K cl T ∈ R\ m − − θ, m − + θ . 2βγ a 2βγ a 2βγ a 2βγ a
(10)
Proof. The Jacobian of the system (1)–(3) at e(T ) is
cl(qτ − cl) βγ + ρK aqτ 2 (qτ − cl)2 J (e(T ¯ )) = λ aqτ γ aτ (ρ K − β) − ρ K (qτ − cl) αaτ K −
c2l qτ 2 (qτ − cl) −cλ qτ −
0
−
αβ ρ 0 . 0
We employ the Routh–Hurwitz criterion to determine local stability. The Hurwitz matrix is −(A + E) 0 0 CFE −(B D − AE + C F) −(A + E) 0 0 CFE where A=−
βγ cl(qτ − cl) + , ρK aqτ 2
C =−
αβ , ρ
E = −cλ
D=λ
(qτ − cl) , qτ
B=−
c2l , qτ 2
(qτ − cl)2 , aqτ F=
γ aτ (ρ K − β) − ρ K (qτ − cl) . αaτ K
For the positivity of the principal minors of this matrix, we need: A + E < 0, (A + E)(B D − AE) + AC F > 0, and C F E{(A + E)(B D − AE) + AC F} > 0.
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All of these conditions are satisfied if A < 0. A < 0 is equivalent to: aβγ qτ 2 − ρ K clqτ + ρ K c2l 2 > 0. At τ = 0 the value of the expression is ρ K c2l 2 which is positive. Thus for the quadratic expression to remain positive it is sufficient if it has a negative discriminant. This means that the recouping quotient is larger than > 1. If this is not true, the quadratic expression aβγ qτ 2 −ρ K clqτ +ρ K c2l 2 = 0 has two positive roots τ1 , τ2 and the expression is negative in (τ1 , τ2 ) and positive outside. This leads to (9). This establishes the asymptotic stability of the interior equilibrium (iv). The instability of the equilibria (ii) and (iii) easily follows by considering the derivatives of the causal functions at the respective equilibrium points and observing that the determinant in each case is positive; which rules out the possibility of all the eigenvalues having negative real parts.
4.
T HE PATHS AND THE G EOMETRICAL C ONFIGURATION OF THE S YSTEM
We give a description of the geometry of the paths of the system (1)–(3). This brings out the difficulties to be encountered in a three-dimensional system. The three isoclines are: x—isocline is the surface
γl K
x2 +
γ a K
x E + (αa) pE + (αl) px − (γ l)x + (q − aγ )E = 0.
(11)
p—isocline is the plane: x = β/ρ. E—isocline is the plane:
x=
a qτ − cl
(12) E
(13)
where as the prey and the predator isoclines are independent of the tax levels, the E isocline is a plane through the p-axis and moves from nearer to the x pplane at the lowest tax levels to the E p-plane at higher tax levels. Geometrically, condition (7) assures that the curve of the intersection of plane (12) and surface (11) does meet plane (13) in the positive octant. The observation made about the E isocline now tells us that the interior equilibrium e(T ¯ ) moves from the px plane towards the p E plane as the tax level increases. From (iv) the interior equilibrium traces a smooth curve as τ varies continuously on τ > 0. The isoclines given by (11)–(13) divide R3+ into eight regions R1 , . . . , R8 . Each region is characterized by a combination of growth patterns of the state variables x, p and E. R1 − {(x, p, E) : x is increasing, p and E are decreasing}. We use the notation R1 = {+, −, −}. Similarly R2 = {+, −, +}, R3 = {+, +, +}, R4 = {−, +, +}, R5 = {−, −, +}, R6 = {−, −, −}R7 = {+, +, −}, R8 = {−, +, −}. Let P(x0 , p0 , E 0 ) be an initial state of the system. By Theorem 2, the path
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through P with tax T , will approach e(T ¯ ) asymptotically if P is sufficiently close to e(T ¯ ). The journey of the material point is described by: (6) (5)
(8)
(1) e (T)
(4)
(7)
(2)
(3)
If P is in R4 , it can either go directly to e(T ¯ ) as if it is a node or go through the cycle indicated in the above diagram. If the starting point is in R7 , then the path goes in to the region R3 and then joins the diagram. Thus, the path either goes through the cycle to reach e(T ¯ ) as if e(T ¯ ) is a spiral point or goes to R4 and then to e(T ¯ ) as if it is a node. A path starting in the region R8 goes to R4 or R6 and either goes to e(T ¯ ) as if it is a node or goes through the cycle to reach e(T ¯ ) asymptotically as if it were a spiral point. We now fix a T ∗ satisfying (10) and consider the paths, which reach e(T ¯ ∗ ), corresponding to all possible T satisfying (10). Through a lengthy but routine procedure, it is observed that the paths corresponding to T < T ∗ reach e(T ¯ ∗) ∗ from ‘below’ the surface (11) and the paths corresponding to T > T reach e(T ¯ ∗) from ‘above’ the surface (11).
5.
T HE O PTIMAL TAX P OLICY
In this section we prove the following. T HEOREM 3. Suppose the cubic equation [−{γρ 2 K (aq K − aβ) + δβ(δa Kρ + γβa − ρ K q)}q − δβρ K q 2 −ρ 2 K 2 q]τ 3 + [ρ K clq(βδ − ρ 2 K )]τ 2 +[{aρ 2 K γ (K q − β) + δβ(δaρ K + γβa − ρ K q)}mcl −δβρ K c2l 2 − ρ 2 K 2 q]τ − (ρ 3 m K 2 c2l 2 + βδρ K c2l 2 ) = 0 has a positive solution τ ∗ = m − T ∗ , T ∗ satisfying the conditions of Theorems 1 and 2. Then there exists a time A > 0 and optimal tax policies such that the system approaches the optimal equilibrium e(T ¯ ∗ ) in the period [0, A] most rapidly. The optimal taxation problem on the infinite horizon, namely, max f (T ), T subject to (1)–(4) and T satisfying conditions of Theorems 1 and 2, has a solution. Proof. We obtain the optimal tax policy to maximize the social benefit, in two stages. First we consider the problem: Z ∞ (P1 ) max f (T ) = H (t, x, p, E, T ) dt T
0
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(1), (2), (3), (4) and x, E, P > 0, T ∈ [Tmin , Tmax ]
and satisfies the hypotheses of Theorems 1 and 2 with Tmax and Tmin obtained in the last section. We use the maximum principle to obtain the optimal solution of this problem. The Hamiltonian H is H(t, x, p, E, T ) = H + µ1 x˙ + µ2 p˙ + µ3 E˙
(14)
with x, ˙ p˙ and E˙ are given by (1)–(3) respectively and µi , i = 1, 2, 3 are adjoint variables corresponding to the variables x, p, E, respectively. Using maximum Ex principle, ∂∂H = 0 gives λµ3 q a E+lx = 0. We use a singular control and find T the singular path. For this we take µ3 = 0. Continuing the use of maximum principle, 0 = µ˙ 3 = − ∂∂H , we obtain E (q E + lx)2 µ1 (t) = e−δt m − c . (15) qlx 2 Comparing this with the µ˙ 1 derived from maximum principles, we obtain qma E 2 1 µ2 (t) = −e−δt ρp (a E + lx)2 2γ x qa E 2 − µ1 (t) γ − δ − . (16) − αp − K (a E + lx)2 Finally, we obtain µ˙ 2 from the maximum principles and compare it with the µ˙ 2 obtained from (16) and (15), we are left with qma E 2 1 −δt αx(t)µ1 (t) − µ2 (t)(ρx(t) − β) = δe ρp (a E + lx)2 2γ x qa E 2 +µ1 (t) γ − δ − − αp − K (a E + lx)2 (17) x, p, E in (17) are positive solutions of (1)–(3) with given initial conditions. Suppose T ∗ is the (singular) optimal control. Then the corresponding equilibrium (x ∗ , p ∗ , E ∗ ) is a solution of (1)–(3) (with initial values (x ∗ , p ∗ , E ∗ )) (x ∗ , p ∗ , E ∗ ) is given by (iv) in Section 3. Using the solution in (17) leads to the following equation for T ∗ : [−{γρ 2 K (aq K − aβ) + δβ(δa Kρ + γβa − ρ K q)}q − δβρ K q 2 − ρ 2 K 2 q]τ 3 +[ρ K clq(βδ − ρ 2 K )]τ 2 + [{aρ 2 K γ (K q − β) + δβ(δaρ K +γβa − ρ K q)}mcl − δβρ K c2l 2 − ρ 2 K 2 q]τ − (ρ 3 m K 2 c2l 2 + βδρ K c2l 2 ) = 0, (18)
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with τ = m − T . Let T ∗ (= m − τ ∗ ) be a solution of (18) which is in the admissible set of controls obtained in Section 2. Then the optimal path for all t subsequent to an instant A would be x(t) = x ∗ (T ∗ ) for all t subsequent to an initial time A. Thus, we would have an optimal policy and an optimal path, if we could arrive at X ∗ = (x ∗ , p ∗ , E ∗ ) in a finite time A with maximum social benefit in [0, A] starting from any initial state. The period [0, A] might be dictated by several necessities; we may envisage this period as a plan period or as the shortest period of approach to X ∗ . Solution of this problem constitutes the second stage. We take A to be the shortest time to reach X ∗ . Let (x0 , p0 , E 0 ) ∈ R3+ \{0}. (x ∗ , p ∗ , E ∗ ) ∈ R3+ \{0} be as obtained in the last paragraph. We now seek the solution of the problem: (P2 ) min A(T ) T
Subject to: (1)–(3) X (0) = x0 ,
p(0) = p0 ,
E(0) = E 0 ,
x(A) = x ∗ ,
p(A) = p ∗ ,
E(A) = E ∗ , (x, p, E) ∈ R3+ \{0}
t ∈ [0, A].
This time we would be looking for nonsingular (bang–bang) control. Using maximum principle (Pontriagin et al., 1964; Cesari, 1983) we obtain the equations for the adjoint variables µ1 , µ2 , µ3 as µ˙ 1 = −
∂H ; ∂x
µ˙ 2 = −
∂H ; ∂p
µ˙ 3 = −
∂H . ∂E
(19)
The side conditions on µ1 , µ2 , µ3 are given by the equations m(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = 0;
t ∈ [0, A]
(20)
where M(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = [ Sup ]H(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t), T ). Tmin ,Tmax
˙ Equation (20) prescribes a set H being the Hamiltonian 1 + µ1 x˙ + µ2 p˙ + µ3 E. of initial conditions µ10 , µ20 , µ30 for µ1 , µ2 , µ3 . µ1 (0) = µ10 etc.
(21)
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Since (20) is an identity on [0, A] it determines a set of terminal conditions for µ1 , µ2 , µ3 . It is easy to observe that the unique solution determined by (19) and (21) has to satisfy the terminal condition determined by (20). It is now clear that the optimal tax policy is Tmax for all t ∈ [0, A] such that µ3 (t) > 0 ¯ T (t) = (22) Tmin for all t ∈ [0, A] such that µ3 (t) < 0. The optimal path is [0, A] are solutions 0¯ of (1)–(3) with the given initial states. We now combine the two stages to obtain the optimal tax policy and the optimal paths in the infinite horizon as: T (t) = T¯ (t), T (t) = T ∗ ,
t ∈ [0, A], ¯ t > A0(t) = 0(t),
0(t) = (x ∗ , p ∗ , E ∗ ),
6.
t ∈ [0, A],
t > A.
N UMERICAL V ERIFICATION OF S OME K EY R ESULTS
From Theorem 3, it is clear that once the state reaches the optimal equilibrium point X ∗ and tax is then maintained at T ∗ , the social benefit will be maximized for subsequent times. Hence, the numerical analysis will concentrate on supporting our theory about reaching X ∗ optimally, i.e., most rapidly. We have used the Runge–Kutta method of order four to compute solutions of the differential equations. From Theorem 3 of the last section, the optimal control policy is a bang–bang policy with the switches affected at those times when µ3 changes its sign. Through the numerical simulation we wish to exhibit the control policy which will take a given initial state to X ∗ in the shortest time. Due to several factors such as truncation and propagation of errors etc, the numerical result will not show the path reaching X ∗ . Thus, we proceed along the numerical path (of Theorem 3) until we are closest to the optimal equilibrium level and then switch to T ∗ . This proximity is very much dependent on the step length chosen for the nu∗ ∗| ∗| merical computation. We define the distance d(X, X ∗ ) = |x−x + | p−p∗p | + |E−E . x∗ E∗ ∗ With step length 0.01, the closest we could come to X was 0.01. We take the approximating neighborhood as {X : d(X, X ∗ ) < 0.01} which is the sphere with the center as X ∗ and radius 0.01. The values of the parameters involved are α = 0.02, λ = 0.002, δ = 0.01, ρ = 0.01, K = 5000, c = 0.1, γ = 0.1, m = 16, l = 2, a = 40, q = 1, β = 2. The sufficiency conditions for the existence of the equilibrium are verified by this set. We obtain T ∗ from the cubic equation (18). There is only one root T ∗ (= 15.4) satisfying conditions of Theorems 1 and 2. The corresponding optimal equilibrium X ∗ is (200, 66.7, 3.71). The next task is to take different sets of initial conditions. Whereas x(0), p(0)
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and E(0) can be arbitrarily chosen, µ1 (0), µ2 (0) and µ3 (0) are constrained by the relation M(x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = 0 for all t. We discuss five sets of initial conditions and tabulate the initial values, number of switches, time taken to reach the optimal equilibrium (optimal time): Table 1.
Initial values x(0) p(0) E(0) 314 3.50 51.8 117 3.34 44.9 203 22.5 47.1 196 0.06 48.2 258 1.09 124 248 13.6 11.5
No. of switches 7 7 7 7 7 27
Starting tax policy Tmin Tmin Tmin Tmin Tmin Tmax
Optimal time 4250 4307 4289 4278 3655 4719
6.1. Comparison with the suboptimal policies. Let T0 be the optimal policy obtained through Theorem 3. Let X¯ be any state vector X¯ 6= X ∗ . t1 is smaller than the optimal time, be any time such that X (t1 ) = X¯ under the control function T (t) = T0 (t), 0 ≤ t ≤ t1 . Then the tax policy Ts given by: Ts (t) =
T0 (t), T ∗,
t ∈ [0, t1 ] t > t1
is a suboptimal policy, i.e., we switch to T ∗ a bit earlier than in the case of optimal policy. A test that the optimal policy given by Theorem 3 really is optimal is that the time taken for the path under the suboptimal policy Ts , to reach the same proximity as with T0 is larger than with T0 . We remark that when the tax policy is switched to T ∗ far away from the optimal equilibrium level, as in the suboptimal policy, the ensuing path may never reach the optimal equilibrium level X ∗ . This is because X ∗ may not be globally stable. In this case the suboptimal policy is obviously not viable. Thus, the interest lies in finding out: even if we switch to T ∗ when the state is quite close to X ∗ (within the region of attraction of X ∗ , as verified by numerical procedure) whether the suboptimal policy takes longer to come as close to optimal equilibrium level as in the case of optimal policy. For this we consider the initial-state level (x(0), p(0), E(0)) as (314, 3.5, 51.8). The switch to T ∗ is made at 4243 time units when the state level is (202, 3.63, 66.4) which is separated from the optimal equilibrium level by 0.04. We find that this suboptimal policy takes 73.9 units of time since the switch to T ∗ to reach the same degree of proximity as under the optimal policy. Thus, the total time under this suboptimal policy is 4317 units of time, whereas under the optimal policy the time is 4250 units of time. This validates our claim of optimality of the policy proposed in Theorem 3.
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S UMMARY AND C ONCLUSIONS
The objective of this work is mixed, namely economic and ecological. The economic goal is to maximize the monetary benefit to the community and ecologically we wish to keep the predator and prey from extinction; or better keep the ecological balance. The instrument used to drive the system towards such a state is tax. We do not wish to forgo the food provided by the fishing (harvesting) or allow the predator to become extinct. High predator levels may be obtained at the cost of food and the other way too. If the objective was only to maintain the predator at a high level we would face an entirely different problem. The basic theme of this problem takes into cognizance the inherent competition between the predator and fisher for the common prey. As an impartial referee, the regulating agency tries to do the best for both the fisher and predator by implementing the optimal tax policy. For, under any other policy the risk exists of the predator becoming extinct or the harvest not being the best. Thus, we observe that the basic element of this problem is the inherent competition between the predator and the fisher. As we bunch all the fishing activity, ignoring the internal competition between them, into one unit, we can view this competition between two agencies, which cannot be divided into smaller units. Bio-economically we have looked for an optimal tax policy and an interior equilibrium corresponding to this tax policy. Next, we drove this system to this interior equilibrium in the shortest possible time and most beneficially as in Section 5. The optimal tax policy is a combination of bang–bang and singular control policies. Because we reach the interior equilibrium in the shortest time possible, the number of switches in the bang–bang control are decreased and thus we have a smooth tax policy. This and the optimality combine to make the policy obtained in Theorem 3, the most efficient tax policy. Mathematically, a multigoal optimization problem is reduced to a sequence of optimization problems. One on the infinite horizon, one on a finite horizon and finally a time optimal control problem. Theorem 3, and the ensuing optimal policy are obtainable only if an interior equilibrium exists and is sufficiently rough. Constraints on tax policies have to be imposed to ensure the existence and stability of the interior equilibrium. These mathematical constraints allow sensible bio-economic conclusions and thus are validated (cf. Section 3). One interesting conclusion (Theorem 2) is that the existence of interior equilibrium strongly depends on the tax level, but its stability may not depend on tax level if the prey has sufficiently large recouping quotient. Use of a nonconventional harvest function helps in obtaining such conclusions. From the numerical results, we find that the switches occur with some regularity, to start with, and then there is a long period of time during which no switch (sign change of µ3 ) takes place. Thus, the tax policy remains unchanged for a very long time. This, as we pointed out earlier, is a sign of a good and a practicable policy. We find that this situation repeats for each set of initial values considered.
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Thus, the optimal tax policy given by Theorem 3 is not only optimal theoretically, but seems to be sound from a administrative angle also. A referee suggested that we comment on the predator population to be maintained at ‘adequately large values’. In this sequel we make the following remarks. Let us assume that Tmax is smaller than m. Then the equilibrium as a function ¯ ) of T is continuous on [Tmax , Tmin ]. p(T ¯ ) is an increasing function of T and E(T ¯ ¯ is a decreasing function of T . Let PM and Pm be the maximum and minimum values of p(T ¯ ). Then for any p˜ ∈ [ P¯m , P¯M ] there exists a T˜ ∈ [Tmax , Tmin ] such ˜ ¯ T˜ ). We now approach (x, ˜ E) ˜ most that p( ¯ T ) = p. ˜ Let x˜ = x( ¯ T˜ ), E˜ = E( ˜ P, rapidly and can stay there forever if we choose the tax T = T˜ after reaching ˜ E). ˜ Thus, we can maintain the predator at any level (between P¯m and P¯M ) (x, ˜ P, by suitable tax policy. However, this procedure does not guarantee maximum benefit to the society (as explained earlier).
A CKNOWLEDGEMENTS Part of this paper was written while S. V. Krishna was visiting Middle East Technical University, Ankara, Turkey. The visit was supported by TUBITAK, Turkey. S. V. Krishna acknowledges the support of METU and TUBITAK.
R EFERENCES
Cesari, L. (1983). Optimization—Theory and Applications, Applications of Mathematics, Vol. 17. New York: Springer-Verlag. Clark, C. W. (1976). Mathematical Bioeconomics: The Optimal Management of Renewable Resources, New York: Wiley. Clark, C. W. (1979). Mathematical models in the economics of renewable resources. SIAM Rev. 21, 81–99. Choudary, K. and T. Johnson (1990). Bioeconomic dynamics of a fishery modeled as an S-system. Math. Bio. Sci. 99, 231–249 Glendinning, P. (1994). Stability Instability and Chaos: an Introduction to the Theory of Nonlinear Differential Equations, Cambridge: Cambridge University Press. Pontriagin L.S., V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko (1964). The Mathematical Theory of Optimal Processes, London: Pergamon Press. Received 30 January 1997 and accepted 12 October 1997
Conservation of an Ecosystem through Optimal Taxation S. V. KRISHNA, P. D. N. SRINIVASU Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India B. KAYMAKCALAN Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey In this paper we study a harvesting problem in the presence of a predator and a tax. The objective is to maximize the monetary social benefit as well as prevent the predator from extinction, keeping the ecological balance. c 1998 Society for Mathematical Biology
1.
I NTRODUCTION
Economic progress and ecological balance always have conflicting interests. Catering to the necessities and comforts of human beings invariably robs the ecological structure of the nature. This, more often than not, leads to the extinction of a species of life. Often it is possible to prevent such extinction by proper planning. Such a planning has to be either by force or dissentive. For example, if a particular activity by individuals of a region is causing severe damage of the ecosystem of that region and if the activity is inevitable then the governing authority of the region should plan a regulating policy which would keep the damage to the ecosystem minimal. If a species is becoming extinct the regulating policy should prevent it. One such activity is harvesting. Every bit of the catch is not edible and harvesting harms some of the marine species which live on the other species from the sea and which are edible to human beings. Thus, the predator species are likely to become extinct with an indiscrete increase in harvesting activity. To avoid this the regulating authority levies a tax on the catch of the harvesting agency. This acts as a deterrent to the fisher and helps the predator to grow. It sometimes acts as an incentive to the fisher (when the tax takes the form of a subsidy). The aim of this paper is to find the proper taxation policy which would give the best possible benefit through harvesting to the community while preventing the extinction of the predator. This is different from the usual optimal harvesting problem whose objective is purely monetary. It differs from the problem of 0092-8240/98/030569 + 16
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maintaining the predator at a high level without caring for the harvesting activity. It also differs from the other problems in that due to the imposition tax the effort is a dynamic evolutionary variable. Hence, the problem is three-dimensional and some useful theorems such as Du Lac theorem (Glendinning, 1994) cannot be used here. Harvesting problems with tax have been studied by Clark (1976, 1979) and Choudary and Johnson (1990). In the absence of a predator these problems were just two-dimensional. The main feature of this paper is using a natural expression for the catch which allows meaningful conditions for the existence and stability of interior equilibrium. These conditions are later used as constraints on the tax policy. We have also obtained a characteristic equation for the optimal equilibrium. This paper is organized as follows. The problem is clearly stated in the next section. In Section 3, we analyse the existence and stability of equilibria. In Section 4, we describe the path of the system; particularly how a path approaches the interior equilibrium for various tax levels. Section 5 is devoted to the main results of the paper, namely the optimal tax policy and the corresponding paths. Section 6 contains numerical verification of some key results. Conclusions and discussion are presented in Section 7.
2.
T HE P ROBLEM
The ecological set-up is as follows. There is a prey which is harvested continuously. There is a predator, living on the prey. It is assumed that the harvesting does not affect the growth of the predator population directly, and the harvesting agency does not adjust the effort due to the presence of the predator. Thus, the interaction between the harvesting agency and the predator is through the third party, namely, prey. However, there is a conflict of interests between the fisherman and the predator, i.e., a competition for a common resource. Although this situation appears to be game theoretic, the predator is unable to evolve a strategy for its survival. Hence the regulating agency comes to the rescue of the predator through a suitable tax policy. The harvesting agency’s aim is to obtain as much revenue as possible through its activity, whereas the community needs the food through harvesting and is also keen on protecting the predator from extinction. Thus the benefit to the community consists of the revenue through the harvest and the retained predator population. Thus the problem of optimization of the community’s benefit is a conditional optimal control problem in the sense that the revenue is to be maximized subject to the condition that the predator population is larger than a positive quantity as t → ∞. In order to achieve this goal a regulating agency has to curb arbitrary growth of harvesting. This is done by levying a tax on the catch (which can also be a subsidy). The problem, now, is to find the optimal taxation policy which gives maximum benefit to the society, i.e., a taxation policy which gives the maximum benefit
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through the harvesting and at the same time takes the predator population to a level from which it will not become extinct. The harvesting is affected by the predation. Tax (or subsidy) makes the harvesting effort a dynamic variable. The net benefit, from the harvesting, to the society is the revenue before the deduction of tax, obtained by the harvesting agency. One would wish for a variable taxation through an initial ‘plan’ period and settle down to a ‘constant’ taxation thereafter. The initial period is utilized to iron out the initial hiccups one might encounter at the beginning. Thus if [0, A] is the initial plan period our objective can be to find a taxation policy T (t) defined on [0, A] such that the social benefit through harvesting in [0, A] is maximized subject to the constraint that the predator population p(t) reaches a level p ∗ at t = A, such that under a suitable taxation T (t) = T ∗ , p(t) cannot approach 0 as t → ∞. We now formulate this problem mathematically. We denote by x(t), p(t) and E(t), the population of prey, the predator population and the effort at time t. The prey–predator dynamics with harvesting is given by: x x 0 (t) = γ x 1 − − αx p − h(E, x) K p 0 (t) = −βp + ρpx where γ , K , α, β and ρ have the usual interpretation and h(E, x) is a harvesting term referring to the catch when the prey population is x and harvesting effort is E. It is assumed that harvesting h is not affected by the predation directly, nor is any part of the effort E devoted to control predation. Traditionally, h(E, x) is taken as q E x, where q is the catchability coefficient. We do not take h in this form, our objection being that h(E, x) = q E x leads to the fallacial conclusion that h(E, x) tends to infinity as the effort goes to infinity with the prey population finite and fixed, or as the prey population goes to infinity with finite and fixed effort. It may be argued that as the effort and the prey population remain bounded, there is no danger of letting E or x go to infinity. Mathematically this may be granted. When we combine economics, a more realistic harvesting function would lead to very important conclusions, as we see in Section 3. We therefore take the harvesting term as (Clark, 1979): h(E, x) = q
Ex a E + 1x
where q is still the catchability coefficient and a and l are constants. We will discuss the importance of a and l in Section 3. Here we point out that lim E→∞ h(E, x) = qa x, and lim X →∞ h(E, x) = ql E. Using this harvesting term, we rewrite the equation for x as Ex x x0 = γ x 1 − − αx p − q . K a E + lx
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The controlling agency, like the government, levies a tax T on the harvesting agency. The purpose of the tax (which may be a subsidy) is to regulate the harvesting effort. We assume that the tax T is bounded: Tmin ≤ T ≤ Tmax . Introduction of tax makes harvesting effort E a dynamic variable governed by the equation Ex 0 E =λ q (m − T ) − cE a E + lx where m is the price of unit catch, c is the cost of the unit effort and λ is a factor which converts savings into capital. Thus in our problem, the state variables x, p, E are governed by the equations Ex x − αx p − q x0 = γ x 1 − (1) K a E + lx p 0 = −βp + ρpx Ex 0 E =λ q (m − T ) − cE a E + lx
(2) (3)
these are supplemented by initial conditions x(0) = x0 , p(0) = p0 , E(0) = E 0
(4)
and the positivity constraints (x, p, E) ∈ R3+ , the positive octant in R3 . The control variable is T , and the constraint on T at this moment is T ∈ [Tmin , Tmax ]. We note that the behavioral response of the fisher to the introduction of tax is to retard the growth of his effort with the increasing tax rates. This is an important and deciding factor in determining the optimal tax policy. As stated earlier, the goal is to maximize the social benefit. The monetary benefit to the society due to the harvesting activity is Z ∞ f (T ) = H (t, x, p, E, T ) dt 0
Ex where H (t, x, p, E, T ) = e−δt q a E+lx m − cE where δ is the discounting factor. As the social benefit includes preservation of predator population we should maximize f (T ) such that p(t) approaches a nonzero value as t → ∞. Let X ∗ = (x ∗ , p ∗ , E ∗ ) be such that there exists an admissible tax policy T (t) such that (1)–(3), with X (A) = X ∗ has a strictly positive solution under the policy ∗ T (t) for t ≥ A. Let T be the set of all such R ∞ policies. Let T ∈ T (subject to existence) be a tax policy which maximizes A H dt with constraints (1)–(3) and X (A) = X ∗ . We now consider the problem of approaching X ∗ most rapidly (i.e., most rapid approach—see Clark, 1976). Let T¯ be such a policy and A be the shortest time obtained thereof. Then, the optimal tax policy T0 we seek consists of T¯ (t), t ∈ [0, A] T0 = ∗ T (t), t ≥ A.
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An interesting and important path is the equilibrium or steady-state solution. It is often convenient to find an optimal equilibrium, i.e., the equilibrium corresponding to the optimal tax, so that the optimal path would consist of a path leading to the optimal equilibrium level as quickly as possible and then stay at that optimal equilibrium level with that optimal tax subsequently. We remark that for such an optimal policy to be practicable, it is necessary that the optimal equilibrium point be at least locally asymptotically stable.
3.
E QUILIBRIUM A NALYSIS
We now study the existence and nature of equilibria of the system. Particularly we are interested in the interior or positive equilibrium of the system. Since tax is a system parameter, positive equilibria exists only for some restricted tax levels. Stability of the positive equilibrium adds to the constraint on the tax, assuming the other system parameters are known. Furthermore, we wish to know the bioeconomic ramification of the bounds on the tax levels. Interior equilibrium is important for the main problem of this paper because, to control optimally we obtain the optimal tax levels and drive the system to the optimal equilibrium level most efficiently and in the shortest time possible. To begin with we list all possible equilibria. (i) The trivial equilibrium (0, 0, 0). (ii) Equilibrium in the absence of effort (E = 0) x E = β/ρ,
γ pE = α
β 1− ρK
.
(iii) Equilibrium in the absence of the predator ( p = 0) xp =
K [cl + τ (aγ − q)] , aγ τ
Ep =
K [cl + τ (aγ − q)](gτ − cl) . a 2 γ cτ
(iv) The interior (positive) equilibrium x(T ¯ ) = β/ρ, β qτ − cl γ 1− − p(T ¯ )= , α ρK αaτ ¯ ) = β(qτ − cl) E(T ρaτ ¯ )) by e(T where τ = m − T . We denote (x(T ¯ ), p(T ¯ ), E(T ¯ ). To facilitate the discussion of equilibria, we need to introduce some notions.
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(1) MEC at x (maximum effort catch at x) is the limit of the catch as the effort E → ∞. Thus MEC at x = (q/a)x, MEC at x =
q x, a
q MEC = . a unit pop
(2) MPC at E (maximum population catch at E) is similarly defined. (3) PGR at x = γ x (pure growth rate at x) is the growth rate of the prey when its population is x, in the absence of predator, harvesting, and intraspecific competition (Malthusian growth). (4) PGRRC at x = γ (K − x) (pure growth rate of the remaining capacity at x) is the PGR at K − x. (5) PGRRC at Eq (PGRRC at equilibrium) is γ (K − βρ ) 4(PGR at Eq) (6) Recouping quotient (RQ) = θ = . MEC at K Equilibrium (ii) is the usual prey–predator equilibrium, whereas (iii) is the equilibrium with harvesting and without predator. Neither these nor the trivial equilibrium is of interest to us. However, the equilibrium (x p , 0, E p ) exists, that is x p and E p are positive, only if 0 < qτ − cl < aγ τ . The existence of the positive equilibrium puts restrictions on the tax level T . These and the additional bounds we obtain later, will be constraints on T in the optimal control problem discussed in Section 5. For the equilibrium (iv) to be positive we first need β < ρ K . This is independent of the tax. Next we must have qτ > cl. This leads to the upper bound on T , T < m − cl/q.
(5)
¯ ) positive. For p(T Such a restriction on T makes E(T ¯ ) to be positive we have to discuss two distinct situations. (A) MEC at K > PGRRC at Eq. In this case we have the lower bound for T , namely ρ K cl T >m− . (6) βγ a − ρ K (aγ − q) Thus the constraints on T , for the existence of the positive equilibrium are m−
ρ K cl < T < m − cl/q. βγ a − ρ K (aγ − q)
(7)
(B) MEC at K < PGRRC at Eq. Then no lower bound on T can nor needs to be imposed. Thus, the only constraint on the tax will be the upper bound given by (5). The inequality (5) can be interpreted as follows. ‘Tax’ plus the ‘cost-to-catch’ ratio at arbitrary large populations should be smaller than the price quoted by the fishing agency.
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Suppose this condition is not satisfied, then the fishing agent loses steadily and the disinvestment in the harvesting activity occurs. This leads to a decrease in the effort, and as a consequence the prey population increases. Thus a ‘positive’ equilibrium cannot be reached. This is an almost obvious interpretation. The inequality (6) gives a lower bound for T . But for this it is necessary that MEC at K > PGRRC at Eq. Let us examine this situation closely. Assume that MEC at K > PGRRC at Eq.
(8)
Suppose that T < m − βγ a−ρρ KK cl(aγ −q) , i.e., the government or the regulating agency has given a subsidy T to the fishing agency. Then the net price m − T to the fisherman exceeds βγ a−ρρ KK cl(aγ −q) > 0. This encourages the fishing agency to put in larger effort (E → ∞). By (8), the catch (loss to the population) is more than the pure growth of the remaining capacity. Hence the prey tends to extinction so losing positive equilibrium. Thus no equilibrium is possible (E ↑ ∞, x ↓ 0). If we have case (B) any amount of subsidy given by the regulating agency does not cause constant increase or decrease in the effort or population and hence we can get a positive equilibrium. Thus inequalities (7) in case (A) and (5) in case (B) are the constraints on the tax for the existence of the positive equilibrium. The effect of harvesting on the prey–predator system is clear. It lowers the predator equilibrium level. The level of tax affects the equilibrium positions of the predator and harvesting effort, but the equilibrium positions of the prey is not affected. Since the reduction due to harvesting in the equilibrium level of −cl the predator population, qταaτ , increases with τ = m − T , it is obvious that the equilibrium p(T ¯ ) increases with reduced tax and decreases with increased tax. This is sensible because reduced tax levels encourage harvesting effort, leading to depletion of prey which in turn causes starvation of predator, thus lowering their equilibrium level. The equilibrium level of the harvesting effort is clearly higher for lower tax levels. The equilibrium (iii) is not positive if aγ τ −lc. However, this and (5) or (7) are not compatible. Thus equilibrium (ii) exists (as a positive equilibrium) whenever the system has a positive (interior) equilibrium (iv). Combining all these results we have the following. T HEOREM 1. The prey–predator system with harvesting described by equations (1)–(3) has a unique interior equilibrium for any tax level T with m − βγ a−ρρ KK cl(aγ −q) < T < m − cl/q if MEC at K > PGRRC at Eq; and for any tax T with T < m − cl/q if MEC at K < PGRRC at Eq. The equilibrium is ¯ )) given in (iv). The equilibrium level of the predator, namely (x(T ¯ ), p(T ¯ ), E(T ¯ ) p(T ¯ ) increases with a raise of tax level. The equilibrium level of the effort E(T increases with a reduction of tax. Proof. Simple computations show that MEC at K > PGRRC at Eq implies that βγ a − ρ K (aγ − q) is positive. p(T ¯ ) in (iv) is only positive if (βaγ − ρaγ K + K ρq )τ < kρcl.
(9)
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Kρcl Thus m − T < βaγ −ρaγ , which gives the upper bound in (7) for T . If K +K ρq MEC at K ≤ PGRRC at Eq, then (9) is true for any non-negative value of τ and hence for any T < m. Thus, the only constraint in this case is given by (5). The second part of the theorem is almost trivial. We now discuss the stability of the equilibria (ii)–(iv).
T HEOREM 2. The boundary equilibrium points (iii) and (ii) are unstable under the hypotheses (7). The interior equilibrium (iv) is locally stable if recouping quotient > 1 or
ρ K cl ρ K cl ρ K cl ρ K cl T ∈ R\ m − − θ, m − + θ . 2βγ a 2βγ a 2βγ a 2βγ a
(10)
Proof. The Jacobian of the system (1)–(3) at e(T ) is
cl(qτ − cl) βγ + ρK aqτ 2 (qτ − cl)2 J (e(T ¯ )) = λ aqτ γ aτ (ρ K − β) − ρ K (qτ − cl) αaτ K −
c2l qτ 2 (qτ − cl) −cλ qτ −
0
−
αβ ρ 0 . 0
We employ the Routh–Hurwitz criterion to determine local stability. The Hurwitz matrix is −(A + E) 0 0 CFE −(B D − AE + C F) −(A + E) 0 0 CFE where A=−
βγ cl(qτ − cl) + , ρK aqτ 2
C =−
αβ , ρ
E = −cλ
D=λ
(qτ − cl) , qτ
B=−
c2l , qτ 2
(qτ − cl)2 , aqτ F=
γ aτ (ρ K − β) − ρ K (qτ − cl) . αaτ K
For the positivity of the principal minors of this matrix, we need: A + E < 0, (A + E)(B D − AE) + AC F > 0, and C F E{(A + E)(B D − AE) + AC F} > 0.
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All of these conditions are satisfied if A < 0. A < 0 is equivalent to: aβγ qτ 2 − ρ K clqτ + ρ K c2l 2 > 0. At τ = 0 the value of the expression is ρ K c2l 2 which is positive. Thus for the quadratic expression to remain positive it is sufficient if it has a negative discriminant. This means that the recouping quotient is larger than > 1. If this is not true, the quadratic expression aβγ qτ 2 −ρ K clqτ +ρ K c2l 2 = 0 has two positive roots τ1 , τ2 and the expression is negative in (τ1 , τ2 ) and positive outside. This leads to (9). This establishes the asymptotic stability of the interior equilibrium (iv). The instability of the equilibria (ii) and (iii) easily follows by considering the derivatives of the causal functions at the respective equilibrium points and observing that the determinant in each case is positive; which rules out the possibility of all the eigenvalues having negative real parts.
4.
T HE PATHS AND THE G EOMETRICAL C ONFIGURATION OF THE S YSTEM
We give a description of the geometry of the paths of the system (1)–(3). This brings out the difficulties to be encountered in a three-dimensional system. The three isoclines are: x—isocline is the surface
γl K
x2 +
γ a K
x E + (αa) pE + (αl) px − (γ l)x + (q − aγ )E = 0.
(11)
p—isocline is the plane: x = β/ρ. E—isocline is the plane:
x=
a qτ − cl
(12) E
(13)
where as the prey and the predator isoclines are independent of the tax levels, the E isocline is a plane through the p-axis and moves from nearer to the x pplane at the lowest tax levels to the E p-plane at higher tax levels. Geometrically, condition (7) assures that the curve of the intersection of plane (12) and surface (11) does meet plane (13) in the positive octant. The observation made about the E isocline now tells us that the interior equilibrium e(T ¯ ) moves from the px plane towards the p E plane as the tax level increases. From (iv) the interior equilibrium traces a smooth curve as τ varies continuously on τ > 0. The isoclines given by (11)–(13) divide R3+ into eight regions R1 , . . . , R8 . Each region is characterized by a combination of growth patterns of the state variables x, p and E. R1 − {(x, p, E) : x is increasing, p and E are decreasing}. We use the notation R1 = {+, −, −}. Similarly R2 = {+, −, +}, R3 = {+, +, +}, R4 = {−, +, +}, R5 = {−, −, +}, R6 = {−, −, −}R7 = {+, +, −}, R8 = {−, +, −}. Let P(x0 , p0 , E 0 ) be an initial state of the system. By Theorem 2, the path
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through P with tax T , will approach e(T ¯ ) asymptotically if P is sufficiently close to e(T ¯ ). The journey of the material point is described by: (6) (5)
(8)
(1) e (T)
(4)
(7)
(2)
(3)
If P is in R4 , it can either go directly to e(T ¯ ) as if it is a node or go through the cycle indicated in the above diagram. If the starting point is in R7 , then the path goes in to the region R3 and then joins the diagram. Thus, the path either goes through the cycle to reach e(T ¯ ) as if e(T ¯ ) is a spiral point or goes to R4 and then to e(T ¯ ) as if it is a node. A path starting in the region R8 goes to R4 or R6 and either goes to e(T ¯ ) as if it is a node or goes through the cycle to reach e(T ¯ ) asymptotically as if it were a spiral point. We now fix a T ∗ satisfying (10) and consider the paths, which reach e(T ¯ ∗ ), corresponding to all possible T satisfying (10). Through a lengthy but routine procedure, it is observed that the paths corresponding to T < T ∗ reach e(T ¯ ∗) ∗ from ‘below’ the surface (11) and the paths corresponding to T > T reach e(T ¯ ∗) from ‘above’ the surface (11).
5.
T HE O PTIMAL TAX P OLICY
In this section we prove the following. T HEOREM 3. Suppose the cubic equation [−{γρ 2 K (aq K − aβ) + δβ(δa Kρ + γβa − ρ K q)}q − δβρ K q 2 −ρ 2 K 2 q]τ 3 + [ρ K clq(βδ − ρ 2 K )]τ 2 +[{aρ 2 K γ (K q − β) + δβ(δaρ K + γβa − ρ K q)}mcl −δβρ K c2l 2 − ρ 2 K 2 q]τ − (ρ 3 m K 2 c2l 2 + βδρ K c2l 2 ) = 0 has a positive solution τ ∗ = m − T ∗ , T ∗ satisfying the conditions of Theorems 1 and 2. Then there exists a time A > 0 and optimal tax policies such that the system approaches the optimal equilibrium e(T ¯ ∗ ) in the period [0, A] most rapidly. The optimal taxation problem on the infinite horizon, namely, max f (T ), T subject to (1)–(4) and T satisfying conditions of Theorems 1 and 2, has a solution. Proof. We obtain the optimal tax policy to maximize the social benefit, in two stages. First we consider the problem: Z ∞ (P1 ) max f (T ) = H (t, x, p, E, T ) dt T
0
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(1), (2), (3), (4) and x, E, P > 0, T ∈ [Tmin , Tmax ]
and satisfies the hypotheses of Theorems 1 and 2 with Tmax and Tmin obtained in the last section. We use the maximum principle to obtain the optimal solution of this problem. The Hamiltonian H is H(t, x, p, E, T ) = H + µ1 x˙ + µ2 p˙ + µ3 E˙
(14)
with x, ˙ p˙ and E˙ are given by (1)–(3) respectively and µi , i = 1, 2, 3 are adjoint variables corresponding to the variables x, p, E, respectively. Using maximum Ex principle, ∂∂H = 0 gives λµ3 q a E+lx = 0. We use a singular control and find T the singular path. For this we take µ3 = 0. Continuing the use of maximum principle, 0 = µ˙ 3 = − ∂∂H , we obtain E (q E + lx)2 µ1 (t) = e−δt m − c . (15) qlx 2 Comparing this with the µ˙ 1 derived from maximum principles, we obtain qma E 2 1 µ2 (t) = −e−δt ρp (a E + lx)2 2γ x qa E 2 − µ1 (t) γ − δ − . (16) − αp − K (a E + lx)2 Finally, we obtain µ˙ 2 from the maximum principles and compare it with the µ˙ 2 obtained from (16) and (15), we are left with qma E 2 1 −δt αx(t)µ1 (t) − µ2 (t)(ρx(t) − β) = δe ρp (a E + lx)2 2γ x qa E 2 +µ1 (t) γ − δ − − αp − K (a E + lx)2 (17) x, p, E in (17) are positive solutions of (1)–(3) with given initial conditions. Suppose T ∗ is the (singular) optimal control. Then the corresponding equilibrium (x ∗ , p ∗ , E ∗ ) is a solution of (1)–(3) (with initial values (x ∗ , p ∗ , E ∗ )) (x ∗ , p ∗ , E ∗ ) is given by (iv) in Section 3. Using the solution in (17) leads to the following equation for T ∗ : [−{γρ 2 K (aq K − aβ) + δβ(δa Kρ + γβa − ρ K q)}q − δβρ K q 2 − ρ 2 K 2 q]τ 3 +[ρ K clq(βδ − ρ 2 K )]τ 2 + [{aρ 2 K γ (K q − β) + δβ(δaρ K +γβa − ρ K q)}mcl − δβρ K c2l 2 − ρ 2 K 2 q]τ − (ρ 3 m K 2 c2l 2 + βδρ K c2l 2 ) = 0, (18)
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with τ = m − T . Let T ∗ (= m − τ ∗ ) be a solution of (18) which is in the admissible set of controls obtained in Section 2. Then the optimal path for all t subsequent to an instant A would be x(t) = x ∗ (T ∗ ) for all t subsequent to an initial time A. Thus, we would have an optimal policy and an optimal path, if we could arrive at X ∗ = (x ∗ , p ∗ , E ∗ ) in a finite time A with maximum social benefit in [0, A] starting from any initial state. The period [0, A] might be dictated by several necessities; we may envisage this period as a plan period or as the shortest period of approach to X ∗ . Solution of this problem constitutes the second stage. We take A to be the shortest time to reach X ∗ . Let (x0 , p0 , E 0 ) ∈ R3+ \{0}. (x ∗ , p ∗ , E ∗ ) ∈ R3+ \{0} be as obtained in the last paragraph. We now seek the solution of the problem: (P2 ) min A(T ) T
Subject to: (1)–(3) X (0) = x0 ,
p(0) = p0 ,
E(0) = E 0 ,
x(A) = x ∗ ,
p(A) = p ∗ ,
E(A) = E ∗ , (x, p, E) ∈ R3+ \{0}
t ∈ [0, A].
This time we would be looking for nonsingular (bang–bang) control. Using maximum principle (Pontriagin et al., 1964; Cesari, 1983) we obtain the equations for the adjoint variables µ1 , µ2 , µ3 as µ˙ 1 = −
∂H ; ∂x
µ˙ 2 = −
∂H ; ∂p
µ˙ 3 = −
∂H . ∂E
(19)
The side conditions on µ1 , µ2 , µ3 are given by the equations m(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = 0;
t ∈ [0, A]
(20)
where M(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = [ Sup ]H(t, x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t), T ). Tmin ,Tmax
˙ Equation (20) prescribes a set H being the Hamiltonian 1 + µ1 x˙ + µ2 p˙ + µ3 E. of initial conditions µ10 , µ20 , µ30 for µ1 , µ2 , µ3 . µ1 (0) = µ10 etc.
(21)
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Since (20) is an identity on [0, A] it determines a set of terminal conditions for µ1 , µ2 , µ3 . It is easy to observe that the unique solution determined by (19) and (21) has to satisfy the terminal condition determined by (20). It is now clear that the optimal tax policy is Tmax for all t ∈ [0, A] such that µ3 (t) > 0 ¯ T (t) = (22) Tmin for all t ∈ [0, A] such that µ3 (t) < 0. The optimal path is [0, A] are solutions 0¯ of (1)–(3) with the given initial states. We now combine the two stages to obtain the optimal tax policy and the optimal paths in the infinite horizon as: T (t) = T¯ (t), T (t) = T ∗ ,
t ∈ [0, A], ¯ t > A0(t) = 0(t),
0(t) = (x ∗ , p ∗ , E ∗ ),
6.
t ∈ [0, A],
t > A.
N UMERICAL V ERIFICATION OF S OME K EY R ESULTS
From Theorem 3, it is clear that once the state reaches the optimal equilibrium point X ∗ and tax is then maintained at T ∗ , the social benefit will be maximized for subsequent times. Hence, the numerical analysis will concentrate on supporting our theory about reaching X ∗ optimally, i.e., most rapidly. We have used the Runge–Kutta method of order four to compute solutions of the differential equations. From Theorem 3 of the last section, the optimal control policy is a bang–bang policy with the switches affected at those times when µ3 changes its sign. Through the numerical simulation we wish to exhibit the control policy which will take a given initial state to X ∗ in the shortest time. Due to several factors such as truncation and propagation of errors etc, the numerical result will not show the path reaching X ∗ . Thus, we proceed along the numerical path (of Theorem 3) until we are closest to the optimal equilibrium level and then switch to T ∗ . This proximity is very much dependent on the step length chosen for the nu∗ ∗| ∗| merical computation. We define the distance d(X, X ∗ ) = |x−x + | p−p∗p | + |E−E . x∗ E∗ ∗ With step length 0.01, the closest we could come to X was 0.01. We take the approximating neighborhood as {X : d(X, X ∗ ) < 0.01} which is the sphere with the center as X ∗ and radius 0.01. The values of the parameters involved are α = 0.02, λ = 0.002, δ = 0.01, ρ = 0.01, K = 5000, c = 0.1, γ = 0.1, m = 16, l = 2, a = 40, q = 1, β = 2. The sufficiency conditions for the existence of the equilibrium are verified by this set. We obtain T ∗ from the cubic equation (18). There is only one root T ∗ (= 15.4) satisfying conditions of Theorems 1 and 2. The corresponding optimal equilibrium X ∗ is (200, 66.7, 3.71). The next task is to take different sets of initial conditions. Whereas x(0), p(0)
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and E(0) can be arbitrarily chosen, µ1 (0), µ2 (0) and µ3 (0) are constrained by the relation M(x(t), p(t), E(t), µ1 (t), µ2 (t), µ3 (t)) = 0 for all t. We discuss five sets of initial conditions and tabulate the initial values, number of switches, time taken to reach the optimal equilibrium (optimal time): Table 1.
Initial values x(0) p(0) E(0) 314 3.50 51.8 117 3.34 44.9 203 22.5 47.1 196 0.06 48.2 258 1.09 124 248 13.6 11.5
No. of switches 7 7 7 7 7 27
Starting tax policy Tmin Tmin Tmin Tmin Tmin Tmax
Optimal time 4250 4307 4289 4278 3655 4719
6.1. Comparison with the suboptimal policies. Let T0 be the optimal policy obtained through Theorem 3. Let X¯ be any state vector X¯ 6= X ∗ . t1 is smaller than the optimal time, be any time such that X (t1 ) = X¯ under the control function T (t) = T0 (t), 0 ≤ t ≤ t1 . Then the tax policy Ts given by: Ts (t) =
T0 (t), T ∗,
t ∈ [0, t1 ] t > t1
is a suboptimal policy, i.e., we switch to T ∗ a bit earlier than in the case of optimal policy. A test that the optimal policy given by Theorem 3 really is optimal is that the time taken for the path under the suboptimal policy Ts , to reach the same proximity as with T0 is larger than with T0 . We remark that when the tax policy is switched to T ∗ far away from the optimal equilibrium level, as in the suboptimal policy, the ensuing path may never reach the optimal equilibrium level X ∗ . This is because X ∗ may not be globally stable. In this case the suboptimal policy is obviously not viable. Thus, the interest lies in finding out: even if we switch to T ∗ when the state is quite close to X ∗ (within the region of attraction of X ∗ , as verified by numerical procedure) whether the suboptimal policy takes longer to come as close to optimal equilibrium level as in the case of optimal policy. For this we consider the initial-state level (x(0), p(0), E(0)) as (314, 3.5, 51.8). The switch to T ∗ is made at 4243 time units when the state level is (202, 3.63, 66.4) which is separated from the optimal equilibrium level by 0.04. We find that this suboptimal policy takes 73.9 units of time since the switch to T ∗ to reach the same degree of proximity as under the optimal policy. Thus, the total time under this suboptimal policy is 4317 units of time, whereas under the optimal policy the time is 4250 units of time. This validates our claim of optimality of the policy proposed in Theorem 3.
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S UMMARY AND C ONCLUSIONS
The objective of this work is mixed, namely economic and ecological. The economic goal is to maximize the monetary benefit to the community and ecologically we wish to keep the predator and prey from extinction; or better keep the ecological balance. The instrument used to drive the system towards such a state is tax. We do not wish to forgo the food provided by the fishing (harvesting) or allow the predator to become extinct. High predator levels may be obtained at the cost of food and the other way too. If the objective was only to maintain the predator at a high level we would face an entirely different problem. The basic theme of this problem takes into cognizance the inherent competition between the predator and fisher for the common prey. As an impartial referee, the regulating agency tries to do the best for both the fisher and predator by implementing the optimal tax policy. For, under any other policy the risk exists of the predator becoming extinct or the harvest not being the best. Thus, we observe that the basic element of this problem is the inherent competition between the predator and the fisher. As we bunch all the fishing activity, ignoring the internal competition between them, into one unit, we can view this competition between two agencies, which cannot be divided into smaller units. Bio-economically we have looked for an optimal tax policy and an interior equilibrium corresponding to this tax policy. Next, we drove this system to this interior equilibrium in the shortest possible time and most beneficially as in Section 5. The optimal tax policy is a combination of bang–bang and singular control policies. Because we reach the interior equilibrium in the shortest time possible, the number of switches in the bang–bang control are decreased and thus we have a smooth tax policy. This and the optimality combine to make the policy obtained in Theorem 3, the most efficient tax policy. Mathematically, a multigoal optimization problem is reduced to a sequence of optimization problems. One on the infinite horizon, one on a finite horizon and finally a time optimal control problem. Theorem 3, and the ensuing optimal policy are obtainable only if an interior equilibrium exists and is sufficiently rough. Constraints on tax policies have to be imposed to ensure the existence and stability of the interior equilibrium. These mathematical constraints allow sensible bio-economic conclusions and thus are validated (cf. Section 3). One interesting conclusion (Theorem 2) is that the existence of interior equilibrium strongly depends on the tax level, but its stability may not depend on tax level if the prey has sufficiently large recouping quotient. Use of a nonconventional harvest function helps in obtaining such conclusions. From the numerical results, we find that the switches occur with some regularity, to start with, and then there is a long period of time during which no switch (sign change of µ3 ) takes place. Thus, the tax policy remains unchanged for a very long time. This, as we pointed out earlier, is a sign of a good and a practicable policy. We find that this situation repeats for each set of initial values considered.
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Thus, the optimal tax policy given by Theorem 3 is not only optimal theoretically, but seems to be sound from a administrative angle also. A referee suggested that we comment on the predator population to be maintained at ‘adequately large values’. In this sequel we make the following remarks. Let us assume that Tmax is smaller than m. Then the equilibrium as a function ¯ ) of T is continuous on [Tmax , Tmin ]. p(T ¯ ) is an increasing function of T and E(T ¯ ¯ is a decreasing function of T . Let PM and Pm be the maximum and minimum values of p(T ¯ ). Then for any p˜ ∈ [ P¯m , P¯M ] there exists a T˜ ∈ [Tmax , Tmin ] such ˜ ¯ T˜ ). We now approach (x, ˜ E) ˜ most that p( ¯ T ) = p. ˜ Let x˜ = x( ¯ T˜ ), E˜ = E( ˜ P, rapidly and can stay there forever if we choose the tax T = T˜ after reaching ˜ E). ˜ Thus, we can maintain the predator at any level (between P¯m and P¯M ) (x, ˜ P, by suitable tax policy. However, this procedure does not guarantee maximum benefit to the society (as explained earlier).
A CKNOWLEDGEMENTS Part of this paper was written while S. V. Krishna was visiting Middle East Technical University, Ankara, Turkey. The visit was supported by TUBITAK, Turkey. S. V. Krishna acknowledges the support of METU and TUBITAK.
R EFERENCES
Cesari, L. (1983). Optimization—Theory and Applications, Applications of Mathematics, Vol. 17. New York: Springer-Verlag. Clark, C. W. (1976). Mathematical Bioeconomics: The Optimal Management of Renewable Resources, New York: Wiley. Clark, C. W. (1979). Mathematical models in the economics of renewable resources. SIAM Rev. 21, 81–99. Choudary, K. and T. Johnson (1990). Bioeconomic dynamics of a fishery modeled as an S-system. Math. Bio. Sci. 99, 231–249 Glendinning, P. (1994). Stability Instability and Chaos: an Introduction to the Theory of Nonlinear Differential Equations, Cambridge: Cambridge University Press. Pontriagin L.S., V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko (1964). The Mathematical Theory of Optimal Processes, London: Pergamon Press. Received 30 January 1997 and accepted 12 October 1997