Optimal harvesting of stochastically fluctuating ... - Semantic Scholar

J. Math. Biol. (1998) 37: 155—177

Optimal harvesting of stochastically fluctuating populations Luis H. R. Alvarez1, Larry A. Shepp2 1 Institute of Applied Mathematics, University of Turku, FIN-20014 Turku, Finland 2 Columbia University and AT&T Labs, Room 2c-374, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Received 24 June 1996; received in revised form 7 April 1997

Abstract. We obtain the optimal harvesting plan to maximize the expected discounted number of individuals harvested over an infinite future horizon, under the most common (»erhulst-Pearl) logistic model for a stochastically fluctuating population. We also solve the problem for the standard variants of the model where there are constraints on the admissible harvesting rates. We use stochastic calculus to derive the optimal population threshold at which individuals are harvested as well as the overall value of the population in the sense of the model. We show that except under extreme conditions, the population is never depleted in finite time, but remains in a stationary distribution which we find explicitly. Needless to say, our results prove that any strategy which totally depletes the population is sub-optimal. These results are much more precise than those previously obtained for this problem. Key words: Optimal harvesting — Stochastic logistic model — Threshold population density — Value of the opportunity to harvest — Itoˆ’s theorem. Subject Classification: AMS 60J70, 60J60, 92D25

1 Introduction Establishing both biologically and economically reasonable harvesting policies is indisputably one of the most difficult problems in modern governmental decision making. Here we neglect competition between harvesters and assume throughout that the population is under our total and exclusive control. There are well-reported occurrences where short-sighted unconstrained harvesting has led to both local and global extinction of even entire species (see [C, LES]). On the other hand, we also have the discouraging examples,

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L. H. R. Alvarez, L. A. Shepp

such as the Peruvian anchovy crisis in 1973 (see [C, pp. 32—33]), where scientific predictions based on economical arguments clearly overestimated the actual population density and the warnings of ecologists about the potential collapse of the fishing industry were proved to be correct. This, of course, weakens the political argument for the creation of scientifically based harvesting planning. However, the different objectives among members of a pluralistic society are thus seemingly in constant conflict and the resolution of this conflict surely lies in scientific decision making based on a solid mathematical footing. The decision to harvest is irreversible, so that if the harvester decides to harvest today, he is aware that it will take time for the population to recuperate its current level. The population of harvested individuals fluctuates randomly and continuously over time, depending on climate and other biological factors, thus creating a gap between the actual population density and the predicted (or expected) one. Therefore, the harvester is continuously faced with the possibility of extinction [LES], if the harvester over-estimates the current population. Moreover, the harvester who depletes the population simultaneously loses the option to profitably harvest sometime in the future when the population increases. Motivated by the arguments presented above it is our purpose in this study to analyze the optimal harvesting plan, one maximizing the present value of the expected cumulative yield of individuals. We discount in time an individual harvested later by including a factor e~kt where k"k #k , with 1 2 k the rate of incidence of a major catastrophe affecting the entire population 1 which can be assumed to occur at an exponentially distributed time, and k is 2 the usual rate of decreasing time-value of dollars. We apply this criterion for optimality to each of the usual variants in the standard harvesting literature [BM, MBHS, LES, L", C, R]. We consider three separate constraining regimes: unconstrained harvesting, bounded catchper unit of effort, and bounded quotas. Using techniques of stochastic calculus and the principle of smooth-fit to divine the answer, we derive the optimal population threshold at which to harvest the intrinsic value1 of the population. We show that if the harvesting rate is constrained by an upper bound, then a finite population threshold density exists and it is such that harvesting at lower levels is never optimal (see [L"] for similar results). These type of strategies are called singular and they are difficult to be implemented in practice since they require a perfect knowledge of the density of the harvested population at all times. Moreover, they lead to erratic changes from periods of intense harvesting activity to periods of complete inactivity. This result has an important role in the case where the population is expected to go extinct even in the absence of harvesting. In that case our model actually predicts that harvesting will never be initiated in finite expected time as long as the initial density is below the harvesting threshold. However, we prove that for a ————— 1By intrinsic we mean the value of an individual measured in terms of a numeraire. Thus, we neglect other values (for example, preservational) which a population may have

Optimal harvesting of stochastically fluctuating populations

157

growing population the stationary distribution of the population density is such that the population is expected to remain close to the harvesting threshold most of the time. Therefore, in that case nonactivity will never be permanent. We show that in a regime of bounded harvesting rates the population never becomes extinct in finite time under the optimal harvesting rule. Surprisingly, the same conclusion holds also in the case of unconstrained harvesting, except in an extreme case when discounting is so severe that the population evolves almost surely towards extinction independently of its initial state (which occurs when k is larger than the intrinsic population growth rate at low densities). In that case, which certainly doesn’t hold except in strange circumstances, it is optimal to harvest the entire population instantaneously. However, if this condition actually does hold, then of course other ecological and preservational issues may enter to preclude this tactic. The contents of the paper are as follows: In the second section we present the model is and study its consequences under the assumption of an unconstrained harvesting plan. In the third section, we study variants of the problem subject to constrained harvesting strategies. Finally, section four contains our main conclusions.

2 Optimal unconstrained harvesting Consider a large population having a density X"MX(t); t70N which in the absence of harvesting evolves according to the Itoˆ stochastic differential equation, or Verhulst-Pearl diffusion (see [May]), dX(t)"rX(t)(1!cX(t))dt#pX(t)d¼(t), X(0)"x ,

(1)

where both the per capita population growth rate at low densities, r, and the carrying capacity, c~1, are assumed to be nonnegative known parameters, and p2 is the variance parameter measuring the fluctuations (or volatility) which are modelled by the Brownian motion process, ¼. It is well-known and easily shown that X does not reach either zero or infinity in finite time and that, provided that r'1p2, then the process, X, has a stationary distribution 2 s2 (4rc2 x) (a chi-square distribution with parameter g) with continuous density, g p see [BM] (4rc)g@22~g@2x(g~2)@2e~(4rc@pÈ)x@2 , p(x)" p2 C( g ) 2 where g/2"(2r/p2)!1. In the opposite case, g(0, and X(t)P0 almost surely as tPR. If the population is subject to harvesting, and if Z(t) is the total number of individuals harvested up to time t, then the density of the harvested population, XK "MXK (t); t70N satisfies the stochastic differential equation dXK (t)"rX] (t)(1!cX] (t))dt#pX] (t)d¼(t)!dZ(t)

(2)

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where the harvesting strategy Z(t) is assumed to be non-negative, nondecreasing, right-continuous, and MF N-adapted (nonclairvoyant), in the sense t that the future increments of ¼ are independent of the present value of Z. X] and Z are not assumed to be Markovian, although it will be proved that the optimal harvesting plan will be Markovian and Z will be a function of X] . The behavior, including the boundary behavior, of the process X] is dependent on the nature of the harvesting rate dZ(t) and, therefore, nothing can be said about it yet. In this section we are going to solve the optimal harvesting problem which is to maximize the expected total time-discounted value of the harvested individuals starting with a population of size, x, i.e. we will find q* e~ksdZ(s) , »(x)" sup E x Z(t)| A 0

P

(3)

where A denotes the set of admissible harvesting strategies (that is, the set of harvesting strategies satisfying the set of assumptions stated above), k is a nonnegative parameter, q*"infMt70: X] (t)"0N is the time at which extinction occurs, and E denotes the expected value evaluated with respect to x the law of the harvested population X] . The role of k in the model is two-fold: part of k represents the rate at which the entire population is destroyed by a major catastrophe such as increased predation and part of k simply represents the decreasing value of money due to inflation. We assume that k is known (we allow k"0 under the further assumption that r(p2/2) and we note that »"»(x, k). Thus we want to maximize the expected cumulative yield in present dollars of harvesting from the present up to extinction. The harvesting process of course has nonnegative increments and in the first model may be unbounded so that the population can be arbitrarily harvested. On devising a strategy, a reasonable person has to consider that, at each moment, there are two possible courses of action, namely, whether to harvest now or to wait and harvest sometime in the future. As in [RS] and [SS] we begin the analysis by constructing an upper bound for the value, », of harvesting. This nonnegative majorant is presented in the following lemma and it will be proven later to also be a lower bound for ». Lemma 1. ¸et º : R >R be a twice continuously differentiable function ` ` satisfying the conditions (i) º(0)"0; (ii) º@(x)71 for all x 3 R ; ` (iii) 1p2x2ºA(x)#rx(1!cx)º@(x)!kº(x)60 for all x3 R ; 2 ` ¹hen, »(x)6º(x) for all x3 R . ` Proof. Assume that the conditions of the lemma are valid and let Z(t)3 A be an admissible harvesting strategy. Then, according to the Dole´ ans-DadeMeyer formula for discontinuous semimartingales (also known as the


159

generalized ItoL -formula; see [H] Sects. 4.7 and 6.4, [K, L", P], Sect. II.7) E [e~kT*º(X] (¹*))]"º(x)#E x x !E x

P

T*

P 0 e~ks((A!k)º)(X] (s))ds

T* e~ksº@(X] (s))dZc(s) 0

(4) + e~ks[º(X] (s))!º(X] (s!))] , 06s6T* where ¹*"R'infMt70: X(t)7RN'q*, Zc(t) denotes the continuous part of the harvesting strategy Z(t), and A"1p2x2(d2/dx2)#rx(1!cx)(d/dx) is the 2 differential operator representing the infinitesimal generator of X. Therefore, #E x

T* (***) E [e~kT*º(X] (¹*))]6 º(x)!E e~ksº@(X] (s))dZc(s) x x 0

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+ e~ksDº(X] (s)) 06s6T* T* (**) 6 º(x)!E e~ksdZc(s)#E + e~ksDº(X] (s)) , x 66 * x 0 s T 0 #E x

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where Dº(X] (s))"º(X] (s))!º(X] (s!)). Since º is continuously differentiable on R and satisfies the inequality º@(x)71 for all x 3 R we obtain that ` ` Xª (s~) º(X] (s!))!º(X] (s))" º@(y)dy7!DX] (s)"DZ(s) Xª (s)

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Multiplying the inequality above by !1 finally gives us that Dº(X] (s))6 !DZ(s). Therefore, by collecting terms we obtain that for any admissible harvesting strategy Z(t) 3A and for all x 3 R ` T* (5) E e~ksdZ(s)6º(x)!E [e~kT* º(X] (¹*))]6º(x) , x x 0

P

since º was assumed to be nonnegative. By letting now R tend to infinity we obtain that for any admissible harvesting strategy Z(t)3A and for all x 3 R ` * q E e~ksdZ(s)6º(x) . (6) x 0

P

Since (6) is valid for any admissible harvesting strategy it has to be also valid for the optimal one and, therefore, for all x3R ` q* »(x)" sup E e~ksdZ(s)6º(x) , (7) x Z(t)| A 0

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which completes the proof.

K

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At this point it is worth mentioning that the argument of Lemma 1 can be easily extended to the case where an artificial lower extinction boundary l3 R is imposed (see [LES]). In that case the function º is constructed as in ` Lemma 1 with the exception that then the state-space is (l, R) and assumption (i) becomes º(l)"0. Our main results are summarized in Theorem 1. (i) If k7r, then the population is immediately driven to extinction2 (that is Z(0)"x and q*"0 almost surely for all x3 R ) and the value of ` harvesting is »(x)"x for all x 3 R . ` (ii) If 0(k(r, then there exists an unique threshold density x*3 (0, R) satisfying the smooth-fit condition tA(x*)"0 and defined as x*"infMx3 R : ` tA(x)"0N, where t(x)"xh`M(h`, 2h`#2r/p2, (2rc/p2)x), h`"1!(r/p2)# 2 J(1!(r/p2))2#(2k/p2), and M is the confluent hypergeometric function (see 2 [AS, DP]). For a, b3R M is defined as ` C(b) 1 = (a) (cx)n n " ecxtta~1(1!t)b~a~1dt , M(a, b, cx)" + C(a)C(b!a) (b) n! n n/0 0

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where (a) "a(a#1) . . . (a#n!1) and (a) "1. In this case, the optimal n 0 harvesting strategy is (x!x*)`, if t"0 Z(t)" L(t, x), if t'0

G

where L(t, x*) is the local time of the process X at x*. ¹herefore, Z(t) corresponds to the ’’local time push’’ (or ’’one-sided regulation’’) exerted at the threshold x* so as to keep the population within the boundaries 0 and x* (see [H, and ¸"]). In this case, the value of harvesting is

G

t(x) , if x(x* »(x)" t{(x*) * x!x*#t(x *) , if x7x*. t{(x ) Moreover, the population does not get depleted in a finite time under the optimal harvesting strategy (i.e. q*"R). Proof. Assume first that k7r. Define then the semimartingale D"MD(t); t70N as t D(t)"e~ktX] (t)# e~ksdZ(s) . 0

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Note, that D(t) can be interpreted as the net population value. Then, according to the Dole& ans-Dade-Meyer formula for discontinuous semimartingales dD(t)"e~kt[!(k!r)X] (t)dt!rcX] (t)2dt#pX] (t)d¼(t)]. ————— 2It is worth pointing out that instantaneous extinction is caused by harvesting, not because of the underlying population dynamics, since in the absence of harvesting extinction could never be attained in finite expected time


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Therefore, for any admissible harvesting strategy Z(t), E [D(q*)]" x E :q*e~ksdZ(s)6x (Note: the process e~ktX] (t) is a nonnegative supermartinx 0 gale and, therefore, convergent). It is now clear that »(x)"x satisfies the properties of Lemma 1 (»(0)"0, »@(x)"1 and rx(1!cx)!kx60 for all x3 (0, R), »(x) is bounded on compacts and X] 6X, thus lim C E [e~kT*X] (¹*)]"0 by bounded convergence). Thus, if the discount R= x rate is greater than or equal to the intrinsic population growth rate, then the ’’take the money and run’’ strategy is optimal. As mentioned earlier, other ecological considerations may preclude the use of the full-harvest strategy. Assume now instead that 0(k(r. For the sake of distinctness, we divide the proof in three separate parts. ¹he value function and existence of x*: As in standard optimal stopping theory, we now guess that the (continuation) region where harvesting is unoptimal (i.e. dZ(t)"0) must be of the form (0, x*) and, furthermore, that on that region the value must be k-harmonic and satisfies the conditions »(0)"0 and »@(x)71. In other words, we guess that on (0, x*) the ordinary differential equation, 1 p2x2»A(x)#(rx!rcx2)»@(x)!k»(x)"0 , (8) 2 subject to the boundary conditions »(0)"0 and »@(x*)"1 has to be satisfied. Try a solution of the form »(x)"xhh(x). Then, the following conditions: and

1 p2h(h!1)#hr!k"0 , 2

(9)

1 p2xhA(x)#(r#p2h!crx)h@(x)!chrh(x)"0 , (10) 2 have to be met. By making the transformation y"2crx we obtain the ordinary p2 differential equation

A

yhA(y)#

B

2(r#p2h) !y h@(y)!hh(y)"0 , p2

(11)

which is the well-known Kummer’s equation (see [DP, Sect. 5A, pp. 161—167] or [AS, Chap. 13, pp. 503—536]). Its solution is given by the confluent hypergeometric function. Denote, the positive and negative root of the quadratic equation (9) by h` and h~, respectively. For simplicity, denote the general solution of the ordinary differential equation by »(x)" ct(x)#d/(x), where c and d are constants, t(x)"xh`h(h`, 2crx/p2), and /(x)"xh~h(h~, 2crx/p2). Then, since lim B /(x)"R the boundary conx0 ditions imply that »(x)"(t(x)/t@(x*)) on (0, x*). Now, we guess that on the set [x*, R) the value is of the form »(x)"x#f, where f is a constant. Since we guessed that » 3C2(R ), » should be twice continuously differentiable at ` x*, we should have »(x)"x!x*#(t(x*)/t@(x*)) and tA(x*)"0. We have to now prove that such x* exists. Notice now that since h`(1, lim B tA(x)"!R and lim C tA(x)"R (this is a straightforward x0 x=

162


consequence of the definition of t(x)). Therefore, the continuity of tA(x) implies the existence of at least one root for the equation tA(x)"0. »erification of the conditions of ¸emma 1: Since »(x) is k-harmonic on (0, x*) and »@(x)"1 for x3[x*, R), it is sufficient to now verify that »@(x)71 on (0, x*) and that »(x) satisfies the inequality ((A!k)»)(x)60 on [x*, R). The condition lim B tA(x)"!R and the monotonicity of t(x) ([IM, x0 Sect. 4.6]) imply that if we choose x*"infMx 3R : tA(x)"0N, then t@(x) is ` nonnegative, continuous and monotonically decreasing on (0, x*). Therefore, t@(x)7t@(x*)70, proving the inequality »@(x)71. For x7x*, ((A!k)»)(x)"rx(1!cx)!k(x!x*#t(x**) ), and t{(x ) ((A!k)»)(x*)"0. Now, (d/dx)((A!k)»)(x)"(r!k)!2rcx(0 for x'(r!k)/2rc (see [L"]). Thus, if x*7(r!k)/2rc, then ((A!k)»)(x)60 for all x7x*. To prove that this is the case, notice that tA(x)(0 for all x(x*. Therefore, the k-harmonicity of the value on (0, x*) implies that for all x(x* 0"0.5p2x2»A(x)#rx(1!cx)»@(x)!k»(x) (rx(1!cx)»@(x)!k»(x) . However, since rx*(1!cx*)»@(x*)!k»(x*)"0 (proving that x*(c~1, since »(x*)'0, and »@(x*)"1), we obtain that d D C * [rx(1!cx)»@(x) !k»(x)]60. dx x x That is, (r!k)62rcx* proving the inequality ((A!k)»)(x)60 on [x*, R). This proves that » satisfies the conditions of Lemma 1 and, therefore, is an upper bound for the value of the optimal harvesting strategy. »erification of optimality: Let Z(t)3A be defined as in the theorem. Then, the Skorokhod’s stochastic differential equation (2) with reflection at x* has a solution on (0, x*) (see [F, Sect. 1.6]). The optimality of the proposed harvesting strategy then follows from the fact that if the proposed policy is applied, we obtain that (see [H, L"]) E [e~kT*»(X] (¹*))]"»(x)#E x x

!E x

T*

P 0 e~ks((A!k)»)(X] (s))ds

T*

P 0 e~ks»@(X] (s))dZ(s)

+ e~ks[D»(X] (s))#»@(X] (s!))DZ(s)] , 06s6T* where ¹*"R'infMt70: X(t)7RN'q*. Now since harvesting occurs only (except for the initial state) at the boundary x*, the value is linear on [x*, R) and k-harmonic (except for a set of ¸ebesgue-measure zero), we obtain that #E x

E [e~kT*»(X] (¹*))]"»(x)!E x x

T*

P 0 e~ksdZ(s) .


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Therefore, the required result is obtained by letting R tend to infinity. Moreover, the uniqueness of Z(t) is clear from [F] (Theorem 6.1, page 89). Finally, our results imply that the state-space of the optimally harvested population is now (0, x*], where the lower boundary 0 remains natural and the upper boundary x* is reflecting. It is then a standard result from the theory of diffusions that under these boundary conditions the population density X] of the harvested population can never become extinct in finite time (see [IM, M]). K Theorem 1 has important practical implications. First, it proves the intuitively clear result that independently of the size of the carrying capacity c~1, a population possessing a extinction rate which is higher than or equal to the intrinsic population growth rate at low densities (i.e. if the population is becoming almost surely extinct) should be immediately depleted. The reason for this result is intuitively clear. If the population density is decreasing faster (due to catastrophes, intraspecific competition and natural mortality) than new individuals are born, then waiting can never be optimal since the value of harvesting today will always be greater than the expected cumulative yield of harvesting at any future date. Informally, why should the harvester wait and lose his opportunity (i.e. save the option) to harvest today if sooner or later nature is going to carry out the job anyway. The conservationist ecologists may object to this conclusion, especially if the harvesting threatens some endangered species, and we are assuming here that eliminating the entire population will not incur the exogeneous, or extra-model loss incurred by alienating the conservationist ecologists or worried citizens; i.e., if this turns out to be the optimal policy some re-thinking ought to be done here. This argument is of essential importance since we have implicitly assumed that the harvester does not have an alternative source of income. Therefore, immediate depletion of the population implies a simultaneous loss of a source of livelihood. Second, if r'k, then there always exists an optimal threshold x* at which harvesting is initiated (see [BM, LES]). This threshold is below3 (r!k)/rc but above (r!k)/2rc. It is of interest to notice that if one considers a population density XI (t) evolving according to the deterministic law XI @(t)"rXI (t)(1!cXI (t))!kXI (t) (subject to an extra mortality term with ————— 3 By differentiating twice the integral representation of the function t and collecting terms one obtains that xh`~2 tA(x)" B(h`, h`#2r2 ) p

1

P0e

4rch` 2rc2 ` ` 2r p xtth ~1(1!t)h `p2~1[h`(h`!1)# p2

D

4r2c2 xt# x2t2 dt , p4

where B denotes the Beta-function. By invoking the mean-value theorem for integrals and integrating we obtain that for any fixed x 3 (0, R) there is a fixed t' 3[0, 1] such that

C

D

4rc(h`)2 4r2c2h`(h`#1) 2rc tA(x)"xh`~2 e p2 xtˆ h`(h`!1)# x# x2 . p2(2h`#2r2 ) p4(2h`#2r2)(2h`#2r2#1) p p p ~ ) The required results are then obtained by setting x"(r~k)"(1~h`)(1~h rc ch`h~

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a constant rate k) these points form the long-run steady state and the state where the maximum sustainable yield is attained, respectively. Thus, our results suggest that neither of these extreme points is (generally) optimal but instead the true optimum will be found somewhere between them. It is particulary pleasant that the optimal solution is compatible with ecological thinking — the population is never eliminated under the optimal strategy. A limiting property following from Theorem 1 is summarized in the following proposition: Proposition 1. ¸et r'k'0. ¹hen, lim »(x)"x . pC= Proof. It is clear from Theorem 1 that

C

SA

B

D

1 r 2 2k # ! "1 . 2 p2 p2

1 r lim h`"lim ! # pC= pC= 2 p2

(12)

Therefore, by letting p tend to infinity, we obtain that for all x3 (0, R) lim »(x)"xM(1, 2, 0)"x , pC= which is the required result.

K

Proposition 1 shows how the value of harvesting reacts as environmental stochasticity approaches the extreme limit where stochastic fluctuations are so severe that the predictability of the population density is completely lost. In that case, the value of harvesting approaches the value which is attained by instantaneously depleting the population. However, by numerical experimentation it is possible to conjecture that (r!k) . lim x*" rc C p= Unfortunately, we have not been able to either prove or disprove this claim analytically. It is at this point of interest to compare our results with the ones obtained under complete certainty. Consider now the optimal harvesting problem q* »] (x)" sup e~ksdZ(s) , Z(t) | A 0

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subject to deterministic population growth modelled by the standard Verhulst-Pearl model (2) with p"0. The results of the deterministic model are summarized in Proposition 2. (a) If r6k, then for all x 3(0, R), Z(0)"x, q*"0, and »(x)"x.


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(b) ¸et r'k'0. ¹hen, 1 ckr ~1(k#r)kr `1(r!k)1~kr ( x )kr , 1~cx »] " 4rk x#(r~k)2, 4crk (r!k) x*" , 2rc

G

and

A

if x((r~k) 2rc if x7(r~k) 2rc

B

(r!k) ` Z(0)" x! 2rc

G

dZ(t)"

0, if x((r~k) 2rc 2 2 (r ~k ) dt, if x"(r~k) 4rc 2rc

Proof. By following the spirit of this paper, it is our purpose to now derive a nonnegative majorant for the value of the deterministic harvesting problem. By invoking the deterministic counterparts of Lemma 1 we obtain that in the non-action region the majorant has to satisfy the ordinary differential equation rx(1!cx)»] @(x)!k»] (x)"0 subject to the boundary conditions »] @(x*)"1, and »] (0)"0. Combining these rersults with the smooth-fit principle then gives the value and the optimal threshold of Proposition 2. Part (iv) of Lemma 1 is now obvious. Thus, the proposed value function is a majorant for the harvesting problem. The optimality of the proposed harvesting strategy then follows by noticing that if x'(r~k), then 2rc »] (x)"Z(0)#:= e~ksrx*(1!cx*)ds"x!x*#(r2~k2)"x#(r~k)2 as was 0 4rkc 4crk claimed. The case x6x* is proved similarly. K Since the harvesting decision is irreversible, it makes waiting valuable and creates a value of preservation (or option value, see [DP]). This value can be defined as the difference between the value of exercising the harvesting opportunity immediately and the value of (postponement) harvesting sometimes later in the future. It is clear from Theorem 1 that on (0, x*) the value can be written as »(x)"x#(t(x)/t@(x*))!x. Therefore, the value of preservation can now be explicitly defined as (t(x)/t@(x*))!x":x ((t@(y)/t@(x*))!1)dy, 0 where t@(x)/t@(x*)!1 can be interpreted as the marginal value of preservation. What Theorem 1 then essentially states is that harvesting should be initiated only at that point where the marginal value of preservation vanishes. At population levels which are below it, harvesting is clearly sub-optimal. It is now of interest to slightly qualify and discuss our theoretical results. At this point, we rely on the example of the Antarctic fin whale (Balaenoptera physalus) presented in ([C, p. 50]) and set r"8% per annum, c~1"400000 whales and x"70000 whales. This example is illustrated in Table 1. As can be directly seen, our results suggest that the relationship between enviromental uncertainty (i.e. volatility) and harvesting is negative in the sense that increased p increases the population threshold x* and, therefore, prolongs waiting. Similarly, increased uncertainty decreases the value of harvesting

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Table 1. Antarctic fin whale: optimal harvesting threshold, expected present cumulative yield (EPCY) and option value Discount rate k

Volatility coefficient p

Optimal threshold x*

EPCY »(x)

Option value »(x)!x

0 0 0 0.01 0.01 0.01 0.01 0.01 0.04 0.04 0.04 0.04 0.04

0.5 1 R 0.01 0.1 0.4 1 R 0.01 0.1 0.4 1 R

400000 400000 400000 175194 191926 309906 346131 350000 100083 107587 160379 192148 200000

322408 95063 70000 669411 660482 286031 89156 70000 119656 119161 99666 76605 70000

252408 25063 0 599411 590482 216031 19156 0 49656 49161 29666 6605 0

»(x). As p tends to zero, x* decreases towards r!k/2rc corresponding to the state where the maximum sustainable yield is attained in the presence of catastrophes appearing at a constant rate k (see Propostion 2). Similarly, as p tends to infinity (complete unpredictability), the threshold density x* increases towards the state r!k/rc, as was conjectured above. It is of interest to notice that these results are in accordance with the results of the modern theory of irreversible investment (see [DP]). In Theorem 1 we proved that if k7r, then q*"0 independently of the size of 1p2. The expected long-run evolution of the harvested population in the 2 opposite case is summarized in the following proposition: X] (t)"0 a.s. Proposition 3. (a) If 0(k(r(1p2, then q*"R a.s. but lim t?= 2 (b) If r'1p2 and k'0, then the density of the harvested population 2 evolves towards a long-run stationary distribution with continuous density p*(x) which is defined as (h!1)xh~2e~hcx p*(x)" x*h~1M(h!1, h, !hcx*) where 1 M(h!1, h, !hcx)"(h!1) e~hcxtth~2dt 0

P

is the confluent hypergeometric function, and h"2r/p2. In this case,

A B A BC

1 M(h, h#1, !hcx*) E[XK ]" 1! x* = h M(h!1, h, !hcx*)

D

1 1 e~hcx* " 1! 1! . h c M(h!1, h, !hcx*)


167

Moreover, the expected long-run population density E[X] ] satisfies the = inequalities

A B

A B

1 1 1 1 [1!e~hcx*]6 1! "E[X ] . 06E[X] ]6 1! = = h c h c Proof. If k(r(1p2, then x*'0, and as was proved in Theorem 1, q*"R. 2 However, the population density is bounded above by the geometric Brownian motion process defined by the sde dX(t)"rX(t)dt#pX(t)d¼(t) which decreases almost surely to 0 as tPR, if r(1p2. 2 If r'1p2, then in the absence of harvesting the population density 2 approaches the long-run stationary s-square distribution presented in the beginning of this section. However, the state-space is now of the form (0, x*], where 0 is natural and x* reflecting so that the stationary distribution is altered. By following the techniques presented in [H, Sect. 3.1]) we obtain that the stationary distribution has to satisfy the ordinary differential equation (the formal adjoint equation) (1 p2x2p*(x))A!(rx(1!cx)p*(x))@"0 2 subject to the conditions :x* p*(x)dx"1, and rx*(1!cx*)p*(x*)" 0 (1p2x*2p*(x*))@. The required density is then obtained by integrating this 2 equation twice wrt x, and invoking the constraints. The expected long-run population density E[X] ] can then obtained by = standard integration and by using the recurrence properties of confluent hypergeometric functions (see [AS, pp. 506—507]). Specially, integrating by parts the function M(h, h#1, !hcx*) gives 1 M(h, h#1, !hcx*)" [M(h!1, h, !hcx*)!e~hcx*] cx* Therefore, E[X] ] can be written in the form =

A BC

D

1 1 e~hcx* E[X] ]" 1! 1! = h c M(h!1, h, !hcx*)

By invoking the mean value theorem for integrals we obtain that there is a tJ 3 [0, 1] such that M(h!1, h, !hcx*)"e~hct3 x* implying that

A B

1 1 [1!e~hcx* (1~t3 )] . E[X] ]" 1! = h c The required inequality is then obtained by letting tJ approach 0 and 1, respectively. K Proposition 1 shows how the population density of the harvested individuals is expected to behave in the long-run. It proves that if the percentage

168


population growth rate4 without intraspecific competition is positive, then the harvested population will never become extinct and the population density approaches a stationary distribution which is strongly right-skewed and, therefore, assigns greater probabilities to those values of X] which are close to x*. Thus, as time passes the process is expected to remain most of the time close to the harvesting threshold x* and away from the extinction level 0. Moreover, as intuitively is clear, the expected long-run population density is higher in the absence of harvesting than when it is present. However, it is also clear from Proposition 1, that the difference between the expected long-run * %~hcx is a destationary densities D(x*)"E[X ]!E[X] ]"(1!1)1 = = h c M(h~1,h,hcx*) creasing function of the threshold density x*. Thus, the greater the optimal threshold is, the closer is the expected density of the harvested population to the one attained in the absence of harvesting. There are strong ecologically sensible arguments supporting that one should not use discounting in problems of optimal management of renewable resources (see [C, LES]). In the absence of discounting all harvested generations are equally valuable and, therefore, the harvester expects that any current losses can be completely covered by future expected gains and vice versa. In the following corollary of Theorem 1 we show how our results are altered as the extinction rate k approaches zero. Corollary 1. If the population is free of catastrophes and time-discounting of money can be ignored, that is, if kB0, then a finite and unique harvesting threshold exists if the expected percentage growth rate at which the population increases without intraspecific competition is negative (that is, if r(1/2p2). In that case the optimal threshold is equal to the stable (in the deterministic sense) carrying capacity c~1. ºnder the optimal plan, the value of harvesting is

where

G A

S(x) if x(c~1 »(x)" S{(c~1) ) x!c~1#S(c~1 , if x7c~1 S{(c~1)

B P

x 2r 2r 2rc 2r S(x)"x1~2rp2 M 1! , 2! , x " e2rc p2 uu~p2 du. p2 p2 p2 0 If r71/2p2, then the value of harvesting is infinite and an optimal harvesting strategy does not exist. Proof. If r(1 p2, then the lower boundary 0 is attracting and S(0)"0. The 2 results follow now from Theorem 1 by letting kB0 and noticing that

A

B

2r 2rc x 2rc 2r kB0 "x1~2rp2 1 e2rc xh`M h`, 2h`# , x &" e p2 uu~2rp2 du. p2 xyy~p2 dy" p2 p2 0 0

P

P

K

————— 4 The percentage growth rate is defined as the expected increment of the process X] in logarithmic scale. That is, the percentage growth rate is defined as dt~1E [d ln(X] (t))]. In the x absence of intraspecific competition the population fluctuates according to a geometric Brownian motion and, therefore, dt~1E [d ln(X] (t))]"r!p2/2 x


169

Corollary 1 shows that an optimal threshold exists and is equal to the carrying capacity c~1 provided that r(1 p2 (This result was also obtained in 2 [LES].) Thus, in the absence of a natural extinction rate, a finite threshold exists only if the population is doomed to die out in the long run. This follows from the fact that the population density is bounded from above by a geometric Brownian motion process which tends to 0 (if r(1 p2) as the calendar time 2 tends to infinity. Moreover, if the initial population density is below the carrying capacity, then the date at which harvesting is initiated has infinite expected value and, therefore, it is not reached in finite expected time. A result which, at least partly, supports conservationist arguments and shows that ecology and economy do not necessarily have to be in conflict.

3 Optimal constrained harvesting In the previous section the harvesting rate could be chosen to be infinite. Therefore, we implicitly assumed that the population could be instantaneously harvested below the optimal threshold density, would the initial population density exceed it. In reality that assumption can be valid only if either the harvested population is perfectly observable or if the harvesting capacity is unlimited. Thus, that assumption is clearly not always valid in reality. Especially, for imperfectly observed populations the harvesting success usually decreases as the population density decreases. Motivated by this idea, it is our purpose in this section to study our model subject to bounded harvesting rates. In the first model we assume that the bounded catch-per-unit-efforthypothesis of the Schaefer — model is valid (see [BM, MBHS, LES, C, P]) and assume that 06dZ(t)6dX] (t)dt, where d is a positive constant. Therefore, we assume that the maximum rate at which the population can be harvested is proportional and positively related to the current population density. As intuitively is clear, this case does not significantly change the results of the previous section, since proportional harvesting affects the population density more or less just by adding an extra ‘‘extinction rate’’ to the already existing one. Thus, it can be interpreted as a pure discount rate effect. In the second model we assume that the maximum harvesting rate is density-independent and constant over time. This harvesting strategy is generally called one of bounded quotas (see [BM, MBHS, LES, C]). Put formally, we assume that 06dZ(t)6kdt, where k is a positive constant. A well-known problem related to this assumption is that if harvesting is continuously present then it creates an Allee-effect (see [BM]) and makes the model one of critical depensation. Moreover, as shown in [BM], in a regime of constant quotas the population is always ’’doomed to eventual extinction’’. This result is essentially based on the fact that in models maximizing sustainable discounted yield, harvesting is continuously present. However, if the objective is to maximize the present expected value of the cumulative yield, then harvesting is not always optimal as was shown previously in this study. Therefore, the

170


’’hyperopia’’ (or long-sightedness) of the harvesting decision may exclude the Allee-effect from population dynamics. In fact, as we will later see, this indeed is the case. We begin the analysis of the harvesting problems subjected to bounded harvesting rates by stating the following auxiliary lemma: Lemma 2. Assume that dZ(t) 3 [0, k(X] (t))dt], k(X] (t))70 and that there is a (a.e. in R ) twice continuously differentiable and bounded function ` u: R >R satisfying the conditions ` ` (i) u(0)"0; (ii) 1p2x2uA(x)#rx(1!cx)u@(x)!ku(x)6!k(x)(1!u@(x))` for all x3R . ` 2 ¹hen, »(x)6u(x) for all x 3 R . ` Proof. Define the process y"My(t); t70N by t

P 0 e~ksdZ(s)

y(t)"e~ktu(X] (t))#

The nonnegativity of u imply that y(t)7:t e~ksdZ(s) a.s. for all t70. By 0 taking expectations q* e~ksdZ(s) . E [y(q*)]"E [ekq*u(X] (q*))]#E x x x 0 Now, since u was assumed to be bounded and u(0)"0, we obtain that

P

E [y(q*)]"E x x According to Itoˆ’s lemma

q*

P 0 e~ksdZ(s) .

dy(t)"e~kt[((A!k)u)(X] (t))dt#u@(X] (t))pX(t)d¼(t)#(1!u@(X] (t)))dZ(t)] , where A"1 p2x2 d22#rx(1!cx) d . By integrating from 0 up to extinction 2 dx expectations, dx time q* and taking we finally obtain that E [y(q*)]"u(x)#E x x #E

x

q*

P 0 e~ks((A!k)u)(X] (s))ds

q*

P 0 e~ks(1!u@(X] (s)))dZ(s) .

It is clear that (1!u@(x))6(1!u@(x))` for all x 3R . Therefore, since any ` admissible control dZ(t)3[0, k(X(t))dt]-R ` q* q* E e~ks(1!u@(X] (s)))dZ(s)6E e~ks(1!u@(X] (s)))`dZ(s) x x 0 0 q* 6E e~ks(1!u@(X] (s)))`k(X(s))ds . x 0 The required result follows now from assumption (ii). K

P

P P


171

Lemma 2 shows us that if there is a harvesting strategy for which the value satisfies the conditions of the lemma, then it is an upper bound for the value. Specially, it shows that if there is an admissible harvesting policy for which the condition (ii) is satisfied as an equality, then that strategy is optimal since the process y(t) becomes an uniformly integrable martingale. This lemma has the following trivial corollary: Corollary 2. Assume that u: R >R is a twice continuously differentiable ` ` function satisfying the following properties: (i) u(0)"0; (ii) 1 p2x2uA(x)#(rx(1!cx)!k(x))u@(x)!ku(x)"!k(x) for all x3 R . 2 ` (iii) u@(x)61 for all x3 R . ` ¹hen, u(x)"»(x) and the optimal strategy is to harvest always. The result of Corollary 2 is intuitively clear. It states that if the value of harvesting always is both k-superharmonic and satisfies the growth condition u@(x)61, then the maximum harvesting strategy dZ(t)"k(X] (t))dt is optimal for all t70. This result is, however, dependent on the form of the maximum harvesting strategy k(X] (t))dt and, therefore, to prove the existence of such function more information on the function k is required. We can now summarize the central results of this section in the form of the following theorem: Theorem 2. (i) ¸et dZ(t)3 [0, dX] (t)dt] and k'0. ¹hen there is a unique threshold density y 3R where harvesting is initiated. ¹his threshold is given by ` the condition = /K (s)dsm ' (ds)"B , t@(y) y

P

where B'0 is the constant ¼ronskian determinant of t and /K , m' (s) is the speed measure of the controlled diffusion X] , t is defined as in ¹heorem 1, /K (x)"xaM(a, 2a#2(r~d) , 2rcx ), and a"1!r~d!J(1!r~d )2#2k . ¹he 2 2 2 2 p2 by its density 2 p2 speed measure mˆ (s) is defined 2rc

x2rp2~2e~ p2 x if x(y 2rc 2d2 2(r~d) 2 2 y p x p ~2e~ p x, if x7y

G

mˆ @(s)"

In this case, the value can be written in the form

G

t(x) , if x(y »(x)" t{(y) ª t(y)(/ª (x) #:= G (x, s)dsmL (ds), if x7y t{(y) (y) y k where G (x, s) is the Green-kernel of the controlled diffusion X] . ¹he kernel is k defined as B~1/K (x)tK (s) if x's G (x, s)" k B~1/K (s)tK (x) if x(s `

G

where tK (x)"xa M(a`, 2a`#(2(r!d)/p2), (2rc/p2)), and a`"1!r~d # 2 p2 J(1!r~d )2#2k2 . 2 2 p

172


(ii) ¸et dZ(t)3 [0, kdt] and k'0. ¹hen there is a unique threshold density xJ 3 R where harvesting is initiated. ¹his threshold is given by the condition ` k BI S@(xJ ) "! . k /I @(xJ )t@(xJ ) where /I is the decreasing solution of the ordinary differential equation 1 p2x2uA(x)#(rx(!cx)!k)u@(x)!ku(x)"0, t(x) is defined as in ¹heorem 1 2 and BI is their constant ¼ronskian. ¹he value of the optimal harvesting strategy is t(x)8 , if x(xJ »(x)" t{(x8 ) 3 3 t(x)((x) #k (1!(3 (x)8 ) , if x7xJ . t{(x8 ) /3 (x8 ) k ((x)

G

Proof. Assume that dZ(t)3[0, kdt]. As in Theorem 1, we guess that the continuation region where the population is left unharvested is of the form (0, xJ ). It is then straightforward to show that on (0, xJ )»(x)"t(x)"t(x) . On t{(x) the region [xJ , R) where harvesting is optimal, the value is k-superharmonic and satisfies the equation »(x)"E x

q(x8 )

P0

e~kskds#»(xJ )E [e~kq(x8 )] , x

where q(xJ ) is the first passage time through xJ . Since » is continuous over xJ we obtain that »(xJ )"t(x8 8 ) . On the other hand, E :q(x8 ) e~kskds" x 0 t{(x) k (1!E [e~kq(x8 )]). Thus, (see [IM, Sect. 4.6], [M, Chap. V]) x k

A

B

t(xJ )/J (x) k /J (x) »(x)" # 1! , J t@(xJ )/(xJ ) k /J (xJ ) where /J (x) is the decreasing solution of the ordinary differential equation 1 p2x2uA(x)#(rx(1!cx)!k)u@(x)!ku(x)"0. We guessed that » is differen2 8 ) J )"!k . By tiable over xJ . Therefore, lim B 8 »@(x)"1, implying that I B3 S{(x J xx k ( {(x )t(x combining these results we notice that » is continuous, »(0)"0 and 1 p2x2»A(x)#rx(1!cx)»@(x)!k»(x)"!k(1!»@(x))`. Therefore, the 2 conditions of Lemma 2 are satisfied and the proposed value function is an upper bound for the value of harvesting. On the other hand, if the proposed harvesting strategy 0, if x(xJ dZ(t)" kdt, if x7xJ

G

is implemented, then the process y(t) becomes a nonnegative (uniformly integrable) martingale, implying the optimality of the proposed policy. Assume that dZ(t)3 [0, dX] (t)dt]. Then, on (0, y) the value is evaluated as in the first case. However, on [y, R), the strong Markov property of diffusions


173

imply that (see [IM, D1, M]) »(x)"E x

P0

=

Py

"

q(y)

e~ksdX] (s)ds#»(y)E [e~kq(y)] x

t(y)/K (x) G (x, s)dsm' (ds)# . k t@(y)/K (y)

The alledged continuous differentiability of » then implies that the necessary condition t@(y):=/K (s)dsm' (ds)"B has to be satisfied. It is clear that our results y satisfy the conditions of Lemma 2. The rest of the proof is then carried out as in the previous case. K Theorem 2 describes how the optimal harvesting thresholds can be determined and presents their corresponding values. Unfortunately, in the case of constant quotas, we were not able to determine the explicit form of the fundamental solutions of the ordinary differential equation 1 p2x2uA(x)#(rx(1!cx)!k)u@(x)!ku(x)"0. However, since the upper 2 boundary is natural, we know that lim C /J (x)"lim C /J @(x)"0 (see [IM, x= x= Sect. 4.6). By invoking the necessary condition for harvesting we can rewrite the value on (0, x8 ) as /J (xJ ) t(xJ ) # »(x)" J /@(xJ ) t@(xJ )

A

B

t(xJ ) /J (x) !1 7 "»(xJ ) . J/(xJ ) t@(xJ )

It is clear from this expression that as the population density increases towards infinity the value of harvesting increases towards a finite constant value which is higher than the value at the harvesting threshold. Formally, t(xJ ) /J (xJ ) »@(x)70, lim »(x)" ! , and lim C »@(x)"0 . x= /@(x J) t@(x J ) C x= In the case of proportional harvesting the problem becomes more tractable in the sense that both /K (x) and tK (x) are explicitly known. Unfortunately, it is impossible to evaluate the functional in Theorem 2 explicitly. However, it can be evaluated numerically for approximate results. It is obviously clear that since the harvesting rate is now bounded from above the population cannot be immediately depleted. However, there still is a remarkable difference between the long-run implications of our results and the ones obtained in the standard theory of exploited biological resources (see [BM, C, MBHS]). These results are stated in the following proposition: Proposition 4. If the expected percentage growth rate at which the population increases without intraspecific competition is nonpositive (r61 p2), then 2 q*"R and lim X] (t)"0 almost surely. If the opposite holds, then the t?= harvested population X] is never depleted and X] (t) evolves towards a long-run stationary distribution.

174


(i) If dZ(t)3 [0, dX] (t)dt] then the long-run stationary distribution p' (x) is given by mˆ @(x) pˆ (x)" 2r , 2rc2 2rc 2 :y Sp ~2e~ p sds#y2dp2 := S2(r~d) p2 ~2e~ p2 sds 0 y where y is the threshold density defined in ¹heorem 2, and m' @(x) is the speed density of the population density X] . m' @(x) is defined as in ¹heorem 2. (ii) If dZ(t)3[0, kdt] then the long-run stationary distribution pJ (x) is given by mˆ @(x) , pL (x)" J 2r 2k 2(r~d) 2rc 2k 2 sds#e 8 2 := S :x Sp2~2e~2rc p xp p2 ~2e~ p2 s`s8 p2 ds 0 y where xJ is the threshold density defined in ¹heorem 2, and the speed-density, denoted by m8 @(x), is defined as

G

m8 @(x)"

x2rp2~2e~2rc if x(xJ p2 x, 2r2 2rc2 1 1 2 ( ~ ), xp ~2e~ p x`2k if x7xJ p x x

Proof. If r61 p2, then the population density X] (t) is attracted towards the 2 lower boundary 0 and bounded above by a almost surely asymptotically stable geometric Brownian motion. Therefore, lim X] (t)"0 almost surely. t?= Since 0 is natural, q*"R almost surely. If r'1 p2, then the long-run 2 stationary distributions are derived as in Proposition 3. K It is clear from Proposition 4 that if the population of the harvested individuals is subjected to a bounded harvesting rate then the population will never become extinct in finite time. Specially, if the population growth rate is sufficiently great, then the process converges towards a stationary random variable having a well-defined continuously differentiable stationary distribution with a finite mean. This result is of special interest in the case of constant quotas. As proved in [BM], a population subjected to everlasting constant harvesting is doomed to eventually die out. However, our proposition proves that the contrary argument is valid if the population has ’’time to recuperate’’, that is, if there is a non-empty interval (0, xJ ) where harvesting does not occur. This result shows how extremely sensitive the population process is with respect to the implemented harvesting strategy. The next corollary summarizes our results as the natural extinction rate k decreases to zero (see [LES]). Corollary 3. Assume that k"0 and that the expected percentage growth rate at which the population increases without intraspecific competition is negative (r61p2). 2


175

(i) If dZ(t) 3 [0, dX] (t)dt], then the optimal threshold y is given by the equation = rs(1!cs) 2(r~d) 2rcs s p2 e~ p2 ds"0 . p2

Py

¹he corresponding value function is

G

S(x) , if x(y »(x)" S{(y) S(y) #:= G(x, s)dsmˆ (ds), if x7y S{(y) y where m' (s) is the speed measure of the controlled diffusion,

G

G(x, y)"

S(x), if x(y , S(y), if x'y

and S(x)":x S@(y)dy is the scale measure of the controlled diffusion. In this case 0 the scale density of the controlled diffusion is

G

x~2rp2 e~2rcx p2 ,

if x(y . 2r 2d 2rcx x~p2(x)p2 e p2 if x7y y (ii) If dZ(t)3 [0, kdt], then the optimal threshold xJ is given by the condition m8 [xJ ,R)"1/kS@(xJ ), where m8 denotes the speed-measure of X] . Explicitly, the optimal thershold is defined by the condition S@(x)"

=

Py

p2 2k y2rp2~2e~2rcy p2 `p2y dy"

2k

2rcx8 2k 2r2 e~ p2 `p2x8 . xJ p

In this case the value is

G

S(x)8 , if x(xJ »(x)" S{(x8 ) S(x8 ) #k:=8 G(x, y)m8 (dy), if x7xJ , S{(x) x where G(x, y) is defined as above. ¹he scale density is now of the form,

G

S(x)"

x~2rp2 e~2rcx p2 , 2k ~1 8 ~1 x~2rp2 e2rcx p2 ~ p2 (y ~x ),

Proof. Follows directly from Theorem 2.

if x(xJ . if x'xJ K

4 Summary and conclusions In this paper we studied the optimal harvesting of a population which fluctuates randomly over time and follows a linear (Verhulst-Pearl) diffusion process. We studied the optimal harvesting problem under three separate constraints familiar from standard optimal harvesting problems. Namely, under an unbounded harvesting rate, under a bounded catch-per unit of effort and under the regime of bounded harvesting quotas.

176


In the first one, where the harvesting rate was assumed to be unbounded, we proved that a finite harvesting threshold exists if the net per capita population growth rate at low densities is positive. Thus, our result suggest that a ’’growing’’ population should never be completely harvested. This result was shown to have an interesting option value interpretation. It states that the harvester should never exercise the option to harvest as long as its value remains below the opportunity cost of irreversibly making that decision. Therefore, the higher the opportunity cost is, the higher is the optimal harvesting threshold and the longer will the harvester wait. This shows clearly that while the addition of an extinction (i.e. discounting) rate may lead to biological overexploitation of the population, it does not necessarily lead to extinction. In fact our results show that if the net population growth rate at low densities is positive and the expected percentage growth rate of the population in the absence of harvesting is negative, then harvesting will never be initiated in finite expected time. In the cases where the harvesting rate is bounded above by a constant catch-per unit of effort or by a constant quota, a finite well-defined threshold exists. However, due to the boundedness of the harvesting strategies there are no immediate severe jumps in the population and it does not become instantaneously depleted. In fact, in the case of our study the population does never become extinct in finite time. Therefore, while bounded harvesting affects the dynamics of the population density it does not lead to immediate extinction. In this study it was assumed that the harvested population can be homogenized and, therefore, that its evolution could be described by only one simple diffusion equation. However, in reality this is seldom the case since harvesting generally has a strong effect on the entire population structure. A question of special interest is whether harvesting can lead to rapid (local) extinctions of populations if harvesting affects the subpopulations (say, youngsters, for example) in an asymmetric fashion. A study of that generality would require the use of a general structured population model involving density dependent extinction and recruitment rates, etc. which at the present is out of the scope of our study. Acknowledgements. The authors are grateful to Prof. Mats Gyllenberg, to Prof. Bernt "ksendal, and to Prof. Carlos A. Braumann for their helpful comments and for their suggested improvements on the contents of the paper. The research of L. H. R. Alvarez was funded by Academy of Finland grant 31041 awarded to Prof. Mats Gyllenberg.

References [AS] [BM] [BSW]

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