Optimal harvesting of a logistic population in an environment with stochastic jumps

J. Math. Biol. (1986) 24:259-277 ,Journal of

Mathematical Biology

�9 Springer-Verlag 1986

Optimal harvesting of a logistic population in an environment with stochastic jumps*

Dennis Ryan ] and Floyd B. Hanson 2

Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA 2 Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago,

P.O. Box 4348, Chicago, IL 60680, USA

Abstract. Dynamic programming is employed to examine the effects of large, sudden changes in population size on the optimal harvest strategy of an exploited resource population. These changes are either adverse or favorable and are assumed to occur at times of events of a Poisson process. The amplitude of these jumps is assumed to be density independent. In between the jumps the population is assumed to grow logistically. The Bellman equation for the optimal discounted present value is solved numerically and the optimal feedback control computed for the random jump model. The results are compared to the corresponding results for the quasi-deterministic approximation. In addition, the sensitivity of the results to the discount rate, the total jump rate and the quadratic cost factor is investigated. The optimal results are most strongly sensitive to the rate of stochastic jumps and to the quadratic cost factor to a lesser extent when the deterministic bioeconomic parameters are taken from aggregate antarctic pelagic whaling data.

Key words: Optimal harvesting - - Exploited renewable resource - - Stochastic jump process - - Dynamic p rogramming- - Feedback control - - Bioeconomics - - Uncertain environments

I. Introduction

Mathematical analysis of random fluctuations in commercially exploited resource populations has received a great deal of recent attention. Beddington and May (1977) and May et al. (1978) have examined problems of harvesting in uncertain environments where the environmental fluctuations have been modeled by Gaussian white noise so that the perturbation response is small, although the long term response of the population may be quite dramatic. Ludwig (1979, 1980)

* Research supported in part by the National Science Foundation under grants MCS 81-01698 and MCS 83-00562.

260 D. Ryan and F. B. Hanson

and Ludwig and Varah (1979) have studied fluctuations in the optimal discounted present value for a Ricker type growth function influenced by lognormal white noise. Gleit (1978) solved a continuous time optimal harvest problem for discounted utility, Malthusian growth, and Gaussian white noise. Discrete time harvesting models have been investigated by Mann (1970), Jacquette (1972), Reed (1974, 1979), Mendelssohn (1979, 1980), and Charles (1983a, b). Reed (1974) in particular, has studied the effects of general discrete time multiplicative noise on harvesting and in his 1979 paper extended the optimality of feedback constant escapement policies to more general cost assumptions. Mendelssohn (1979) has examined the trade-off between risk and return in the general i.i.d, multiplicative noise case and mean-variance trade-off (1980) for the Ricker type stock-recruitment relation with log i.i.d, multiplicative noise. Charles (1983a, b), using a discrete stock-recruitment model, extended the work of Clark, Clarke and Munro (1979) to seasonal fishing, investment delays, investment control, escapement control and high levels of uncertainty, with application to the Australian prawn fishery and the aggregate whale fishery. Andersen and Sutinen (1984) give an excellent survey of stochastic bioeconomics.

In this paper we use dynamic programming to study the effects of large, sudden changes in harvested populations on the economic value of the resource. These large fluctuations are not adequately modeled by diffusion or Gaussian white noise processes. When these large changes are adverse, they are usually called collapses, crashes or disasters. Favorable changes are usually referred to as bonanzas or outstanding year classes. Murphy (1977) has given examples of collapses under exploitation and widely fluctuating landings for seven clupeiod fisheries (such as sardines, herring and anchovies), the most spectacular of which was the 1973 collapse of the Peruvian anchoveta fishery. In 1971, two years before the Peruvian collapse, clupeoids accounted for 27% of the world catch with 15% (54% of the clupeiod total) coming from the Peruvian fishery. Paulik (1971) thoroughly describes this fishery and related ecological systems, giving causes for past collapses and apparently anticipating the 1973 collapse of the "worlds greatest single species fishery". Brongersma-Sanders (1957) presents a long list of geophysical cases of marine mass mortalities ranging from vulcanism to strong vertical currents. In addition, Clark (1981) cites multispecies, nonlinear and stochastic effects in the causes of fishing "catastrophes". Examples of widespread climatic effects on fish stocks, fluctuations and outstanding year classes are found in the comprehensive work of Cushing and Dickson (1976). Other examples of bonanzas are given by Hennemuth et al. (1980), along with eighteen examples of wide recruitment fluctuations for a number of commercial fisheries. They also note that deviations in catch follow deviations in recruitment. Their data also indicate a trend toward periodic crashes and increases for certain fisheries. The bonanza may often be a spectacular multiple of the catch in prior years. Sinder- mann (1958, 1963, 1970) has documented examples of disease resulting in mass mortalities of commercially exploited marine populations, some of which have led to local extinction of the resource. Mass mortalities due to epizootics are generally considered to be density dependent, but Rohde (1982) observes that the effects of disease at low rates of infection tend to be density independent and more prevalent.

Harvesting with stochastic jumps 261

In the present work, we describe a simple single age, single (or aggregate) species model of random disasters and bonanzas against a background of logistic growth. A Poisson jump stochastic differential equation is used to model both the effect and occurrence of these large fluctuations. We note, however, that the actual biological mechanisms of collapse and bonanza are extrordinarily complex, involving many different interacting forces. Thus, our lumped parameter model in which fluctuations are described by a single, density independent, Poisson term is one way of approximating bonanzas and disasters.

Section 2 contains a brief review of a corresponding deterministic harvest model. In Sect. 3 the Poisson jump model and the Bellman equation for the optimal discounted present value are developed. In Sect. 4 the results are presented for the numerical approximation of the Bellman equation for both the Poisson jump model and its analogous quasi-deterministic approximation model. Both the optimal present value and the feedback control are determined. In addition, the sensitivity of the results to the discount factor, the total Poisson jump rate and the quadratic cost factor are discussed.

2. Deterministic model

The harvesting of a renewable resource with logistic growth can be described by a deterministic model of the form

dN ds r N ( 1 - N / K ) - q E ( N , s)N, s > 0 , (2.1)

where the initial population size is

N(0) = x. (2.2)

Here r and K are, respectively, the intrinsic growth rate and the environmental carrying capacity. In addition, q is the catchability coefficient and E, the effort expended in the harvest, satisfies the constraints

Emin ~ E ~ Ema x. (2.3)

Thorough discussion of effort, and harvesting in general, are given by a number of authors, particularly Beverton and Holt (1957), Ricker (1975), Clark (1976), and Rothschild (1977).

We characterize the economic value of the harvest by the discounted present value (Clark (1976))

fO TF v(x; E) = ds e x p ( - & ) N H ( N , E), (2.4)

where TF is the time horizon and ~ the discount rate. In (2.4), the instantaneous net revenue per unit biomass is

H(N, E) = [pqEN - C(E)] /N, (2.5)

where p is the price of a unit of harvested biomass, and C(E) is the total cost of fishing effort when the stock size is N. We assume that C is a quadratic function of effort,

C(E) = ClE + c2E 2, (2.6)


where Ca is the linear coefficient in cost per unit effort and c2 is the quadratic cost coefficient. Holt et al. (1960) examine the quadratic approximation of costs in a great variety of economic situations, while Weitzman (1974) emphasizes that benefit and cost curvatures are critical in comparing price and quantity controls in the presence of uncertainty. Clark (1976) also discusses nonlinear costs in addition to his linear cost models. Sancho and Mitchell (1975, 1977), Smith (1980), Lewis (1981, 1982) and Koenig (1984) use quadratic cost criteria in fisheries applications. Koenig in addition reports estimates for a quadratic cost coefficient for yellow fin tuna, and that stochastic effects alter the tax-quota regulation equivalency of the deterministic problem. Quadratic costs should be more reasonable than linear costs, because significantly increased effort can require the use of less efficient vessels, gear and labor; can cause congestion on the fishing ground; or can result in the use of less desirable fishing grounds (Lewis (1982)). However, the linear cost case can be recovered from (2.6) for c 2--~ 0 +. We examine such an example in Sect. 4.

In this problem, the parameters r, K, q, 8, TF, p, ca and c2 are assumed given positive constants. The effort, E = E (N, s), is our instrument, or feedback control variable.

If we further assume that the goal of the harvest is to maximize profit, then the optimal harvest problem consists of choosing the effort E to maximize the expression in (2.4), i.e. to compute

v*(x) = max[v(x; E)] , (2.7) E

when N and E satisfy (2.1)-(2.3). E depends upon both N and s, so the optimization problem (2.1)-(2.7) constitutes an optimal feedback control problem (Bryson and Ho (1969), Clark (1976)).

To obtain the optimal harvest policy we use methods of dynamic programming (Bellman (1957)) since these are the easiest to employ in the random jump problem described below: Specifically, we perturb the initial condition (2.2) by introduction of a nonzero starting time t, so that now

N( t ) =x, (2.8)

and redefine the present value

V(x, t; E) = ds exp [ -~ ( s - t )]NII(N, E), (2.9) t

where the discounting starts at t. Equation (2.9) is the "current value function" representation for V that is used in economics (see Kamien and Schwartz (1981)). From (2.5), (2.8), (2.9) and the principle of optimality we obtain the well known Bellman equation for the optimal present value V = V*(x, t),

Vt* + rx(1 - x / K ) V*~ - 8V* + S(x, t) = 0, (2.10)

where

S(x, t) = max{E[(p - V*)qx-c~ - c2E]}, (2.11) E


is the concise term from which we determine the optimal control and the points for which it switches to the constraints. In the absence of constraints, we obtain the regular control argument of the maximum in (2.12),

ER(X, t) = [(p -- V*x)qX - c,]/(2c2). (2.12)

Upon imposing the constraints (2.3) for E, we obtain the optimal "bang-regular- bang" control,

f E . . . . Emax ~ ER,

E*(X, t )=JER(X, t), O < E R < E . . . . (2.13) / ( 0 , E R <~0

The substitution of (2.12) and (2.13) into (2.11) yields the following simplified form for the switch term

S(x, t) = c2E*(2ER - E*). (2.14)

It must be emphasized that the quadratic cost property following from (2.6) is essential for the existence of the regular control component of E* in (2.13). In addition, we assume that the quadratic cost coefficient, c2, is positive (additional effort at larger values of E tends to be less efficient), ensuring that E R i s the well defined location of the unconstrained optimum. In the case where the constraints are not in effect, we have that S = c2 �9 E~ and that (2.10) is clearly a nonlinear, hyperbolic partial differential equation for V* because E R is linear in V~* and thus S is in general quadratic in V*.

We also observe that (2.10) is a backward type equation and that instead of an initial condition, we have the final condition

V*(x, TF) = 0, (2.15)

following directly from the integral objective (2.9). From (2.15) it follows that V*(x, T F ) = 0 and hence

ER(X, TF) -- cl(x - x~)/(2x~oc2), (2.16)

where

x~ = Cl/(pq), (2.17)

is the linear cost, zero profit condition of bionomic equilibrium (Clark and Lamberson (1982)). Similarly, (2.16) is the zero marginal profit condition for quadratic costs, given the stock size, x. From (2.16), we can deduce that the "final" switch between 0 and E R o c c u r s at Xmin(X , TF)= x~, while between ER

and Eraax the switch occurs at Xmax(X, T F ) = x~(1 +2c2Emax/Cl). The "extinction" boundary at x = 0 r~quires that N ( s ) = 0 for s I> t from (2.1)

and ER(0, t) = - c~/(2c2) < 0 from (2.12); ultimately implying that E*(0, t) = 0 from (2.13). Consequently

V*(0, t) = 0, (2.18)

is the proper boundary condition for this problem. The numerical computations of the full deterministic problem, (2.10), (2.12)-

(2.15), and (2.18) are presented in Sect. 4.


3. Stochastic model with jumps

In this section we discuss a model that describes the effects of disasters and bonanzas on the optimal feedback strategy for a population with growth-harvest dynamics given by (2.1)-(2.3). By a disaster we mean a collection of randomly occurring effects that lead to a sudden dramatic decline in population size. Similarly, a bonanza represents the effect of beneficial changes and is a sudden increase. The disasters or bonanzas are assumed to occur at times of events of a Poisson process and will be described by a sum of terms accounting for factors that lead to significant changes in biomass. We assume that these changes are in proportion to current population size and so are density independent.

In analogy to the dynamics (2.1)-(2.3) the governing Poisson jump stochastic differential equation for the population size, N, starting from an initial size x at a time t > 0 is given by

d N = [ r N ( 1 - N / K ) - q E N ] ds+N• b~dP~(s;f), N(t) =x, (3.1) i

where the P~(. ; f ) belong to a finite set of independent Poisson processes such that their infinitesimal mean and variance have a common value

MEAN[dPi (s; f ) ] = f ds = VAR[dPi(s; f ) ] , (3.2)

for each i, with f the constant jump rate parameter. The jumps of N occur at the unit jumps of the incremental processes dP~(. ; f ) . For each P~ we assume either that - 1 < b~ < 0 if the process represents disasters or b~ > 0 corresponding to bonanzas. The asymmetry in the b{s follows from the fact that a disaster does not drive the population to zero and below while a bonanza may be extremely large. In either event the actual jump in size is given from (3.1)

[N](t/j) ~ N( t +) -N ( t ~ )= bjN( t~), (3.4)

where t o is the j th jump time of the ith Poisson process. We assume that the parameters b~ and f are known in addition to the deterministic parameters mentioned previously. Hanson and Tuckwell (1981) gave a special case of (3.1) valid for a single density independent disaster. They (1978) also presented a model of constant size disaster with an application to the harvesting of sandhill cranes. The case of optimal harvesting with density dependent jumps is treated in Hanson and Ryan (1985).

Since N is a stochastic process we replace the present value (2.9) with the average,

V(x, t; E) = M E A N ds exp[-~(s- t)]NII(N, E)lN(t)=x , (3.5) t

where MEAN is expectation over the set of processes {P~(.;f~)} on (t, TF] conditioned on N( t ) = x, E is now effort in the random model and the net revenue, H, is given in (2.5). The objective now is to compute the optimal feedback control E*(x, t) and the corresponding optimal value V*(x, t )= l?(x, t; E*).

The principle of optimality takes the form

V*(x, t)=maxMEAN[II(x, E)At+e-~' f '*(x+zlX, t + Zit)], (3.6) E


as A t ~ 0 + where AX ~ [ rx ( 1 - x / K ) - qEx ] A t + x Y. i bia P~ ( t; f-), as an approximation in the mean. The expansion of (3.6) requires a generalized chain rule appropriate for Poisson distributed jumps (Gihman and Skorohod 1972). This results in the addition of a functional argument in V. In particular, the chain rule takes the form

(r*(x + AX, t + A t ) ~ (r*(x, t) + V*(x, t) At + r x ( 1 - x / K ) V*~(x, t) At

+Y~ IV*((1 + b~)x, t ) - l?*(x, t)] APi( t ; f ) , (3.7) i

valid in the mean as At ~ 0 § An intuitive explanation of (3.7) comes from noting that the jumps take place on an infinitely fast scale compared to the continuous grox;vth terms governed by (2.1) so that we may assume that a jump takes place instantaneously. Hence, ignore the jump components when computing the continuous change and then ignore the continuous component when computing the discontinuous change due to a jump. This latter change is computed as the difference in V* at the jump and this difference is triggered by the same Poisson processes as in N. Thus, corresponding to (2.11) we have the Bellman equation.

I?*+ rx(1 - x / K ) V * + E f [ f r * ( ( l + b i ) x , t ) - Q*]- ~ 9 " + S(x, t )=0 , (3.8) i

with E* formally computed as in (2.4) except that V* is replaced by V* and S is given by (2.12) with the same replacement. Equation (3.5) implies the final condition, l?*(x, TF)=0 , as in (2.16) and the boundary condition in (2.19). Hence, the jumps do not affect the explicit form of the optimal policy but manifest themselves implicity in the fact that r~, now satisfies a functional partial differential equation rather than a standard partial differential equation. Equations of the form (3.8) have been derived by a number of authors. See Florentin (1963) and Dreyfus (1965) for derivations using probabiiistic arguments and, for application to a nonrenewable resource model, see Arrow and Chang (1983).

In comparing the deterministic and random models it is more appropriate to use a modified form of (2.1) because the presence of the jump term in (3.1) has the effect of altering the parameters r and K so that the mean of the process in (3.1) differs from the deterministic. Any comparison should reflect this difference. We thus define a "quasi-deterministic" model by the conditioned mean

d N o o ( t) = MEAN[ dN ( t)[N( t) = NeD(t)]

= [FNoo(1 - NoD~K) - qENQo] dt, (3.9)

where ~= r(1 + ~ f b J r ) and /~ = Kf/r . Hence, the process given in (3.9) has the same infinitesimal mean as that given in (3.1). In other words, we set the parameters in the quasi-deterministic model equal to the conditioned infinitesimal mean values of the stochastic model. The numerical solution of (3.8) is compared with that of (2.10) using these modified parameters.

4. Results and discussion

In this section we present our numerical results for the optimal present value, V*(x, t), and the associated optimal feedback control, E*(x, t), of the density


independent, Poisson jump problem. Comparisons are also made to the corresponding quasi-deterministic model whose dynamics are specified by (3.9).

Unless otherwise specified we take the time horizon to be TF = 10 years and the discount factor to be 8 = 10%. For the deterministic biological parameters and some of the other economic parameters, we use the aggregate antarctic pelagic whaling data compiled by Clark and Lamberson (1982): r=0 .05/year , K = 400 x 103 BWU (BWU denotes blue whale units), q = 0.013 • 10-3/catcher days, p = $7 x 103/BWU and Cl = c = $5 x 103/catcher days. Consequently, bionomic equilibrium of a linear cost, common property fishery occurs at Xoo-- 55 • 103 BWU.

However, both Clark and Lamberson (1982) and Allen and Chapman (1977) cite the nonequilibrium ("boom and bust") nature of the history of successive collapses in whaling. Clark and Lamberson (1982) remind us of Larkin's (1977) warning about the weakness in the MSY sustainable yield assumption (biological equilibrium only, with E MSY = r /2q and X M S Y = K / 2 ) when animal populations are subject to large natural fluctuations resulting in fluctuating landings. Lewis (1982) criticizes the use of MSY in both deterministic and stochastic situations. Smith (1980) finds uncertainty significant in whale dynamics, but that uncertainty has a very small influence on the northern U.S. lobster fishery, the primary object of Smith's study. Such criticisms are equally valid for the zero sustainable profit and maximal sustainable profit conditions given in (2.20)-(2.25).

Our numerical procedure takes into account both the intrinsic nonlinear nature of the Bellman equation in (3.8) due to the nonlinear switch term in (2.12), and the functional delay/advance term arising from the Poisson jumps in the dynamic model. A predictor-corrector finite difference method, discussed in general terms in Ames (1977), has been modified for these features integrating backwards from the final time t = TF. We discretize the independent variables with xj -- ( j - 1) Ax, j = 1 , . . . , M and tk = T F - ( k - 1 ) At, k= 1 , . . . , N where /ix = L / ( M - 1 ) , / i t = T F / ( N - 1 ) , and L = max[K, K]. The dependent variable l?*(xj, tk) is replaced by Vjk. Central finite differences are used to approximate all derivatives such that V*(xj, tk) is replaced by DVjk =�89247 Vj-l,k)//iX, with other forms for the boundaries, and ~r tk+o.5) by - (Vj, k+l -- Vj, k)//it. With IV~k denoting the interpolation of 17"((1 + bi)Xj, tk) using the two nearest values of the Vjk'S, the predictor step becomes

V~,g+l,p = Vjkc+ /it{ rx~(1 - x J K )/i Vjmc

+ E f~[IV,~mc - Vjm~] - 8V3mc+ Sjmc[, (4.1) i J

where Vj,,~ denotes the final correction on the kth time step with m-=-k+0.5. This level of interpolation for the functional Poisson term matches the accuracy of the central difference approximation for the derivatives. In the predictor evaluation step, we recalculate D V and IV based on

Vj,, v = ( Vjkc + Vj, k+,,p)/2, (4.2)

resulting in

E o,-p = [(P - DV~,,p) qxj - cl]/(2c2), (4.3)


with E~mp computed from the new (2.14) and Sj,,p computed the new (2.15). The corrector step then becomes

Vj, k+,,c = Vjgc + At ( rxj(1-- X~/k) Vamp-- Vjr, p

q-E fi[ lVijmp - Vj,np] - 8Vitae q- Sjmp l . (4.4) i )

the corrector step is based upon using

Vjmc = ( Vjkc + Wj, k+l,c)/2, (4.5)

to recalculate DV, IV, ER, E* and S on the set { jmc}={j ,k+0.5, c}. The corrections continue with Vj,,~ substituted for V~,,p until

I Vj,,c - Vjmp] < tolerance. (4.6)

The stopping criterion for the corrections is sufficient to give plotting accuracy but corrections are only necessary for the control switching term, S. Usually only 2 or 3 corrections are needed. As far as we can ascertain, our approach for the direct numerical treatment of the partial differential equation of dynamic programming for continuous time stochastic control is relatively novel except for the 4th order collocation method of Ludwig and Varah (1979) applied to optimal harvesting with diffusion type uncertainty and the predictor-corrector-Crank- Nicholson-Galerkin of Hanson and Ryan (1985) applied to the density dependent jump harvesting model.

Our comparison to the quasi-deterministic results using (3.9) will be discussed in detail later, but we find that the results for the stochastic model (3.1) offer a fair to good approximation for the quasi-deterministic model (3.9) in terms of V* and E*. Hence, although our method works for the general case, we will divide our results into only two extreme cases. The first we call the "bonanza- dominated" case in which bl =4.0 and all other b~ are zero, i.e. the bonanza example has the population quadruple at each jump and the expected behaviour should be similar for a set of Poisson processes with the same ~ and bonanzas dominant. This choice is roughly motivated by the data of Cushing and Dickson (1976) and Hennemuth et al. (1980), although the data of the latter also indicate some extraordinarily large multiples for several stocks of haddock. The second case we call "disaster dominant", in which we take b 2 = - 0 . 5 and set all the remaining b, to zero.

The other needed parameters are not readily available for most fisheries. Hence, we take E~ in=0 and Emax = r/q, so that the yield per BWU remains nonnegative and never exceeds the intrinsic growth term, rx. For the "bonanza- dominated" case, the quadratic cost coefficient is taken to be $5/[(catcher day)2/year]. Also it is assumed that the total bonanza rate is ~ifb~ =f ib1 = 2 r (i.e. double the intrinsic growth rate). Consequently, the quasi-deterministic biological parameters are triple the original logistic parameters: ~ = 3 r a n d / ( = 3K. Figure 1 portrays the variation of the optimal 6xpected present value, V*(x, t) in million dollar units relative to the starting population size, x, using optimal feedback effort, E*(x, t). In the presence of bonanzas, the proper x-domain is


x~

>gl

F

%~oo o~2o o~o o~eo ~8o X/L, RELRTIVE POPULRTION SIZE

~'oo

Fig. 1. The optimal expected present value, 17*(x, t), as a function of the relative population size, x~ L, in the "bonanza-dominated" case with L=3K, bl =4.0, f ~ =2.Or~b1, T F = 1 0 years, 8 = 10%, c 2 = 5.0 and other data as given in the text. The units for ~7" are millions of dollars. The five curves correspond to the parameter values, TF- t = 2, 4, 6, 8 and 10 starting from the bottom curve and ending with the top one

unbounded even if the population begins below the carrying capacity, so that we have used the finite, cutoff approximation for the upper bound: x ~< L = / ~ = 3K, i.e. we cut off the state domain at the quasi-deterministic carrying capacity. The near flatness of the curves near x = L demonstrates the reasonableness of this cutoff, as well as the dominance of the maximal control, E = Emax. We have also used L to normalize x. The family of curves in Fig. 1 are parametrized in order of increasing value, 17", by the time-to-go, TF - t = 0, 2, 4, 6, 8 and 10 years, starting at the abscissa which represents ~'*(x, TF)=0. Convergence of the numerical approximation to the solution near T F - t = 10 years and the effect of discounting at long periods is qualitatively demonstrated, because the curves are much closer there than they are near T F - t = 0. The stationarity of the Poisson processes and the starting of discounting at the "current value", s = t, cause the governing PDE for 17" to be autonomous and thus Fig. 1 also gives results for the other t ime horizons, T F - t = 2, 4, 6, and 8, in addition to the original 10 year horizon. This property has an important consequence in that x need not be restricted to use as the starting size, but can also be viewed as a "restarting" size at any time (i.e. especially as the size following a random shock). Thus it can be used with the current time-to-go to compute the expected addition to the present value. The convexity of the curves is most upward near extinction at x = 0, because near x = 0 the net rent, lessened by the expected shadow price term, qx( z*, is likely to be negative making the most likely policy to be zero effort. However, occasional bonanzas in addition to logistic growth allow the use of occasional positive effort, except very near x = 0. The expected width for which harvesting is not economical is estimated to be Xmin exp( -- ~TF) -~ 1% of L because for small x we may approximate the threshold for growth beyond the minimal switch point, Xm~n, by pure exponential growth at the quasi-deterministic intrinsic rate, ~. Hence a populat ion at Xmi n e x p ( - ~TF) would grow to X~in exponentially in TF years.

The optimal feedback effort, scaled as qE*(x, t)/r, is plotted versus the relative starting populat ion level, x/L, in Fig. 2. The family of "control laws" are


kl- tLI

t . ~ O tXtO

tld W t l .

Fig. 2. The scaled optimal feedback effort, " qE*(x, t)/r, as a function of x~ L for the bonanza ~i case with the same bioeconomic data as in Fig. 1. The six curves correspond to T F - t = O, 2, 4, t,. 6, 8 and 10 years, starting from the left and ending on the right

0 0.20 O.tlO 0.60 0.80 1.00 X / L , RELATIVE POPULRTION SIZE

parametrized from left to right by the time-to-go (i.e. time horizons), T F - t = 0, 2, 4, 6, 8 and 10 years. The final switch point at t = TF, where the control switches from the minimal, E* = 0, to regular, E* = ER, occurs at xmi,(TF) -~ 0.046L; while the final switch at t = T F between the regular control and the maximal control, E * = E~ax = r /q , occurs at Xr.a• ~-- 0.40L. Although the control laws are nearly linear for our special choice of bioeconomic parameters here, they begin to show more structure for longer periods. The distance between Xmin(t) and Xm~x(t) strongly depends on the quadratic cost coefficient, c2, but also depends on the "current shadow price" (Clark (1976), p. 101), V*(x, t). As c2~0 +, the control law approaches the singular, "bang-bang" control law with the ideal condition, Xmin = X . . . . The more general quadratic costs permit continuous, if not smooth changes in effort relative to continuous changes in x. In principle, E * ( x , t) may be used to find the optimal, expected populat ion trajectory, _N-*(s), from the forward Kolmogorov equation corresponding to the stochastic differential Eq. (3.1). However, knowledge of N-*(s) would be of limited utility in the fisheries application due to the large random fluctuations in the actual trajectory. Knowl- edge of E * ( x , t) for the current values of x and T F - t would be far more important.

In Fig. 3, the optimal expected present value for the "bonanza-dominated" model (3.1) is compared to the optimal present value for the quasi-deterministic model (3.9), with the same infinitesimal mean, plotted versus x / L at t = 0 years. The quasi-deterministic model does a fair to good job of approximating V* of the stochastic jump model. From x = 0 to x---0.26L, the quasi-deterministic approximat ion underestimates the expected stochastic results by at most 34 million dollars, but the relative error or difference may be as large as 100% or greater, due to the relative error bias at small values of V*. Beyond x ~-0.26L, the quasi-deterministic model overestimates the bonanza results with maximum relative error around 8% at x---0.48L. In the relative error, we take V* as exact, ignoring numerical error. In Fig. 4, the expected switching curves are plotted relative to the time-to-go. The switch curve, X S W = Xmin(t), for the minimal to


g

ii ?

g

~:oo o:ao o:qo o:so o:eo X/L, RELATIVE POPULATION S I Z E

Fig. 3. Comparison of the present values, ~?*(x, 0), for the random bonanza case and V~o(x, 0) for the quasi-deterministic approximation as a function of x~ L with F = 3 r and k = 3 K using the same bioeconomic data as in Fig. 1. V~/9 first underestimates I?* and then overestimates it

regula r con t ro l switch appea r s lowest in the figure, and ha rd ly exhibi ts any difference be tween the b o n a n z a results and its quas i -de te rmin i s t i c a p p r o x i m a t i o n , less than - 4 . 5 % dev ia t ion relat ive to the s tochas t ic result. However , the quasi- de te rmin is t i c m o d e l app rox ima te s the regu la r to ma x ima l cont ro l switch curves, which a p p e a r in the u p p e r par t o f the figure, less be t te r in the sense o f abso lu te differences, bu t be t te r in terms o f relat ive error , less than 1.8%. Thus for the bonanza , we find the quas i -de te rmin is t i c a p p r o x i m a t i o n fair to very good. Our appra i sa l o f our results is somewha t different t han tha t o f Lewis (1982), who found his quas i -de te rmin i s t i c a p p r o x i m a t i o n excellent . However , Lewis ' (1982)

z =, N o . o

- - d "

~o X:="

X

t'O0

Fig. 4. Comparison of the optimal switch curves, Xmln(t ) and Xma• for the random bonanza case to those corresponding to the quasi-deterministic approximation, Xmin, Qo(t) and Xmax.QD(t), analogous to Fig. 3. The minimal switch curves appear near the bottom of the figure, while the maximal ones appear near the top with x~in.Oo overestimating Xmln

,oo z'.oo ,'oo 6"oo 8'.oo ' I0 .00 TF-T, T IHE-TO-GO

Harvesting with stochastic jumps 27t

stochastic model was based upon the locally discretized logistic and had its stochastic noise introduced in a much different fashion.

In the "disaster-dominated" case, we assume that one-half the population is lost in each disaster, i.e. b2 = -0.5. It is further assumed that the total disaster rate - ~ i f b i = - f 2 b 2 = +0.5r, or one-half the intrinsic growth rate. Thus the quasi-deterministic parameters are ?= 0.5r and /( = 0.5K. The quadratic cost coefficient is assumed to be $3/[(catcher day)2/year]. Figure 5 shows the optimal, expected present value, V*, versus the relative population size X / L = x / K . If the population level begins in (0, K), it must remain there, because the carrying capacity is a stable equilibrium for the nonrandom, nonharvested population, while both harvesting and disasters will only lower the population level. Hence, in this case we use L = K as the natural cutoff domain width. The family of curves in the figure are again parametrized by the time-to-go (i.e. time horizons), T F - t --- 0, 2, 4, 6, 8 and 10 years starting at the final condition V*(x, T F ) = 0 and ending up at the initial value '~*(x, 0). We note that the curves appear to be strictly concave up, with much steeper slopes than those for the bonanza case. In the near-extinction segment with x < 0.15 K, .the present value, ~'*, is negligible, because, in contrast with the bonanza case, the disasters subtract from the intrinsic growth rate rather than add to it. The effective intrinsic rate is f = 0.5r = 0.025/year, for the slowly growing megafauna example using whale data here. Our exponential estimate for the width of the zero harvest interval is X~n e x p ( - ~ T F ) ~ 10% of K in this disaster case. In Fig. 6, the "disaster- dominated" control law is presented. In this case, the regular control segment, between Xmin(t ) and Xmax(t) takes up most of the population range, with x ~ , ( T F ) ~- O. 14K and Xma~(TF) ~ - 0.77K being the final locations of the switch points. Most of the apparent differences between the disaster control law in Fig. 6 and the bonanza control law in Fig. 2 are mainly due to the facts that the

Fig. 5. The optimal expected present value, 17*(x, t), as a function of x / L in the "disaster- dominated" case with L= K, b2 = -0.5, f2 = -0.5r/b2, TF=10, 6= 10%, c2=3.0 and other data as given in the text. The units for 17" are millions of dollars. The five curves correspond to T F - t = 2, 4, 6, 8 and 10, starting from the bottom curve and ending with the top one

eJ"

u.lo m~

g

g 00

X/L , RELATIVE POPULATION SIZE


t ,L .O U - UJ

m,o- M.I uJ u -

u J

%.00 0:20 o:qo o'.so o:so 1'.oo X/L, RELBTIVE POPULBTION SIZE

Fig. 6. The scaled optimal feedback effort, qE*(x, t)/r, as a function of x~ K for the disaster case with the same data as in Fig. 5. The six curves correspond to TF - t = 0, 2, 4, 6, 8 and 10 years, starting from the left and ending at the right

domains differ by a factor of 3 and that the quadratic cost coefficients differ by a factor of 5/3. These differences are also reflected in Fig. 5 and Fig. 1, for 9*. However, closer quantitative comparison of the V*(x, 0) reveals that V*(x, 0) is always greater for the bonanza case than for the disaster case. This is consistent with the notion that 9* is the expected, current shadow price which represents the expected value of future harvests. The expected value of future harvests should be greater for bonanzas than for disasters.

Figures for comparing the results of the quasi-deterministic approximation to the stochastic "disaster-dominated" results are omitted, because graphically the differences are much less than in the "bonanza-dominated" case. The relative difference of the quasi-deterministic optimal present value relative to 9*(x, 0) has the values -27 .8%, -8 .2%, -5 .3%, -4 .0% and -3 .0% for x/K = 0.2, 0.4, 0.6, 0.8, and 1.0, respectively. The corresponding relative differences between E~o(x,O) and E*(x, 0) are +1.7%, +0.8%, +0.5%, +0.5% and +0.0% for x~ K = 0.2, 0.4, 0.6, 0.8 and 1.0, respectively, i.e. the quasi-deterministic approximation is much better for the optimal control than for the optimal present value. The relative differences in the switch points at t = 0 are even more negligible.

We have used the data of Clark and Lamberson (1982) for the aggregate antarctic whaling because it represents a somewhat realistic field example of a population that has exhibited evidence of "boom and bust" behavior. However, several parameters, such as the quadratic cost coefficients, r and the total stochastic jump rate, Y~ifb~, are not readily available in this data. Therefore we have explored the sensitivity of our results to these additional parameters, and also to the discount rate, using the same numerical methods as before. In Fig. 7, the variation of the optimal, expected present value, V*(K/2 , 0), with the discount rate, 8, is exhibited. Here, the initial population at t = 0 is assumed to be the MSY level x = K/2 = 200 x 103 BWU, c2 = $3[(catcher day)2/year] and all other parameters are as they were in the "bonanza-dominated" case. The sensitivity of 9* to ~ is greatest at 6 = 0% dropping about 5% for each 15 rise in 8, while dropping only 2.2% at 8 = 5 0 % . Although not shown, the control E*(K/2, O) is not very sensitive to discount rate changes, less than 1% change


Fig. 7. The optimal expected present value, V*(K/2, 0), as a function of the discount rate (%), 6, in the bonanza case with b 1 =4.0, fl = 0.5r/b~, L= 1.5K and c2=3.0

=o

bjP-

.-ho (r

~o uJ~

o .

i..- e--

oi

O0 ,b.oo 2b.oo sb.oo vb.oo '~b.oo DELTA, DISCOUNT RATE

in E* pe r 1% change in 6 for 6 ~ > 1%. The re la t ive sensi t ivi ty o f 17"* and E* to is a p p r o x i m a t e l y the same in the " d i s a s t e r - d o m i n a t e d " env i ronment so we omit

these g raphs also.

In Fig. 8, the sensi t ivi ty o f 9 ' * ( K / 2 , 0) to changes in the quad ra t i c cost factor , c2, is shown for the b o n a n z a case. I t is c lear tha t 9'* is very sensi t ive to c2 when c2 is near zero. Nea r c2 = 0, 9"* decreases by 16.6% for a uni t change in c2, while nea r c2 = 100, 9'* decreases by only 1.04%. In the' l imit that c2 ~ 0 § our p rob l e m

Fig. 8. The optimal expected present value, ~'*(K/2, 0), as a function of the quadratic cost coefficient, c2, in units of year-dollars per square catcher day for the bonanza data in Fig. 7

tlJr .J~ ~>r~

~ d

o_

.= J~

>~.

~o ~o ~b.oo ,*b.oo sb.oo ob.oo C2, QUADRATIC COST FACTOR

tbo.oo


becomes a singular perturbation with the limit being the singular, bang-bang, control problem, c2=0, and the expression for the regular control in (2.12) becomes indeterminate. The limiting process itself is called "cheap control" by Jameson and O'Malley (1975) and the "epsilon-algorithm" by Bell and Jacobson (1975). Lewis (1982) found even greater sensitivity to his zero cost and square root cost (decreasing marginal cost with effort) cases than he did for his quadratic cost (increasing marginal cost with effort) case.

In Fig. 9, the sensitivity of V*(K/2, 0) to the jump frequency, fl, in terms of the scaled total jump rate, fabJr, is exhibited for the "bonanza-dominated" environment. Near flbl/r = 0, V* increases by 75% for a unit change in fib1~ r, while near fabJr = 10, V* increases by 33%. There are comparable changes in the "disaster-dominated" environment of roughly 30%. Hence, for these examples, we have shown that V* is very sensitive to the total jump rate, Y~f~bi/r in general. This also indicates that V* is very sensitive to the variance of the stochastic component of the dynamics, because the ratio of the infinitesimal variance of the stochastic model in (3.1) to the stochastic component of the infinitesimal mean, both conditioned on N(t)=x, is Y~ix2fb~/(Y~jxf.ibj)=

2 ~,ifb~/Y.jfjb~, which reduces to xb~ in the bonanza case. Therefore, for both x and bl fixed, the sensitivity of I3" to the process variance is essentially that in Fig. 9 for the bonanza case. E* decreases from a maximum of 0.46r/q at fl = 0 to zero at fl = 6r/bv For f~ > 6r/b~, E* is zero due to the fact that the expected value of future harvests, V*, increases with flbl/r, so that harvesting ceases in expectation of a bonanza. For disasters f '* (K/2 , 0) is a decreasing function of -f2b2/r, with a 35.8% decrease in I7"* per unit increase in ( -f262/r) near f2 = 0; while E*(K/2, 0) decreases by about 5% for a unit change in (-f2b2/r) . These results are different, especially for bonanzas, from the results of Lewis (1982), who found the stochastic perturbation of r to have a small effect in comparison

oo

=:

Z N . WN

#:-~- ~1~ ~

>,

: '. : - , . O0 2 O0 q O0 S O0 8 0 ~ tO OO FMB/R, $CRLED JUHP RFITE

Fig. 9. The optimal expected present value ~ '*(K/2 , 0), as a function of the scaled bonanza rate, flb~/r, for the bonanza data of Fig. 7, with the exception that the bonanza frequency, f t , has been varied to obtain the curve


to other stochastic perturbations. However, we note that there are many differences between our model and Lewis' models. Lewis (1982) used a locally discretized version of the logistic whereas we use the continuous version, he uses a uniform stochastic distribution for his noise and he applies his results to the very r-selected yellow fin tuna fishery with r----3. The whale fishery is very k-selected with small r. Lewis' (1982) work focuses on price and catchability fluctuations, risk adverse conditions and other cost criterion in addition to the quadratic. Smith (1980) also fotind that blue whales were more sensitive to stochastic effects than were lobsters.

5. Summary and conclusions

The effects of random bonanzas or disasters on the optimal, feedback harvesting of a logistically growing renewable resource has been investigated by computational methods. The performance criterion is the expected present value with nonrandom economic parameters, but features a quadratic cost term. A dynamic programming formulation of the problem has been developed, involving a nonlinear, functional partial differential equation with optimal switching. A predictor- corrector finite difference method with modifications for the functional and switching terms has been found to be suitable for this problem. Rigorous, general results for this type of partial differential equation do not seem to be feasible at this time.

The biological and economic parameters have been partially taken from realistic data, the aggregate antarctic pelagic whaling data of Clark and Lamberson (1982). The sensitivity of our results to several important supplied parameters has been explored computationally. The optimal expected present value, ~ '*(K/2, 0), has been found to be very sensitive to the bonanza-disaster frequency,

f , and consequently to the variance of the process, which is also related to f . Thus we find stochastic effects very important. We have found both I?*(K/2, 0) and E*(K/2, O) also to be very sensitive to the quadratic cost factor, c2. In addition, we have found a moderate sensitivity to the discount rate, 6.

The quasi-deterministic approximation to the stochastic jump model, with matching infinitesimal means, is found to be only a fair approximation with regard to the relative difference, excluding small population levels where the relative difference is not a reliable measure. The quasi-deterministic approximation is much better for disasters than it is for bonanzas; and it is much better in approximating E* then it is approximating V. It is probably better to have an accurate estimate of E* than V*.

We do concur with Lewis (1982) that MSY, or biological equilibrium with optimal yield, is not an efficient policy under deterministic or stochastic dynamics. Similarly, the bionomic equilibrium policy or the maximum sustainable profit policy would not be efficient in the presence of random fluctuations, especially with positive discounting.

Acknowledgements. The authors are grateful to two anonymous referees who suggested several important improvements to this paper. This work has had computational support from the computing facilities of the University of Illinois at Chicago and Wright State University.


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Received January 24, 1984/Revised April 30, 1985 and October 4, 1985

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Optimal harvesting of a logistic population in an environment with stochastic jumps