- Email: [email protected]

Contents lists available at ScienceDirect

Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs

Optimal invasive species management under multiple uncertainties q Koji Kotani a, Makoto Kakinaka a,⇑, Hiroyuki Matsuda b a b

Graduate School of International Relations, International University of Japan, 777 Kokusai-cho, Minami-Uonuma, Niigata 949-7277, Japan Faculty of Environment and Information Sciences, Yokohama National University, 79-7 Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan

a r t i c l e

i n f o

Article history: Received 16 August 2010 Received in revised form 30 May 2011 Accepted 2 June 2011 Available online 17 June 2011 Keywords: Bioeconomic model Invasive species management Growth uncertainty Measurement uncertainty Dynamic programming Value functions

a b s t r a c t The management programs for invasive species have been proposed and implemented in many regions of the world. However, practitioners and scientists have not reached a consensus on how to control them yet. One reason is the presence of various uncertainties associated with the management. To give some guidance on this issue, we characterize the optimal strategy by developing a dynamic model of invasive species management under uncertainties. In particular, focusing on (i) growth uncertainty and (ii) measurement uncertainty, we identify how these uncertainties affect optimal strategies and value functions. Our results suggest that a rise in growth uncertainty causes the optimal strategy to involve more restrained removals and the corresponding value function to shift up. Furthermore, we also ﬁnd that a rise in measurement uncertainty affects optimal policies in a highly complex manner, but their corresponding value functions generally shift down as measurement uncertainty rises. Overall, a rise in growth uncertainty can be beneﬁcial, while a rise in measurement uncertainty brings about an adverse effect, which implies the potential gain of precisely identifying the current stock size of invasive species. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The problems of controlling invasive species have been increasingly important, as every part of the world is intertwined each other in a globalized world and there is no way to perfectly prevent potential entries of invasive species in that environment (see, e.g., [17,18] for general discussions). What we can do best for this problem includes (i) to take countermeasure to prevent invasion and (ii) to manage an established invasive species as a consequence of post-invasion. Once an invasive species succeeds in invasion, serious social damage on indigenous ecosystem and agriculture can occur in many cases. The topic addressed in this paper is concerned with the latter: how to manage the established invasive species, especially focusing on the analysis of optimal strategies in a stochastic dynamic model. Invasive species management in reality consists of several decision processes. The government authorities ﬁrst determine whether to aim at eradication. When eradication is set as a goal, they must determine how to achieve it, i.e., eradication strategies.

q Our gratitude goes to ﬁnancial supports from the Japanese Society for the Promotion of Science through the Grants-in-Aid for Scientiﬁc Research C (Nos. 19530221 and 21530238). We also thank Ken-ichi Akao, Jon Conrad, Toshihiro Oka, and the members of the Global COE Program ‘‘Eco-Risk Asia’’ at Yokohama National University and National Institute for Environmental Studies for valuable comments and encouragement. We are responsible for any remaining errors. ⇑ Corresponding author. E-mail addresses: [email protected] (K. Kotani), [email protected] (M. Kakinaka), [email protected] (H. Matsuda).

0025-5564/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2011.06.002

On the other hand, when the goal of eradication is abandoned or identiﬁed to be infeasible, they need to decide how to manage the invasive species. If controlling costs are not taken into account and eradication appears to be feasible, eradication would be the best option for a society. However, policies aiming at eradication are often judged to be impossible. This problem arises from ‘‘stock-dependent catchability.’’1 The eradication cost is prohibitively expensive when catchability rapidly declines with the existing invasive species stock (see, e.g., [13,1,22]). In particular, there is an anecdote that killing the ﬁrst 99% of a target population can cost less than eliminating the last 1%. To make matters worse, there is another key factor that makes the management decision more complex. The invasive species management is typically subject to various stochasticity, such as ‘‘growth uncertainty’’ and ‘‘measurement uncertainty.’’ In the ﬁeld of resource economics, it is established that growth uncertainty does not generally affect the qualitative feature of optimal control strategies, especially when the current stock can accurately be measured [19]. However, in more realistic settings, the decision of management practices must be made in the informational absence of current states due to measurement uncertainty. Indeed, some papers claim that measurement uncertainty may fundamentally affect optimal strategies (see, e.g., [4,20,11]). Thus, it is important to analyze how measurement uncertainty affects optimal strategies.

1 The term of catchability refers to the proportion of the current stock that can be removed or harvested by one unit of effort [3].

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

Real world cases that exemplify the above issues of invasive species problems include Fili mongoose (Herpestes auropunctatus) management on Amami island, Kagoshima, Japan (see [7] for the details). Thirty individuals of mongooses were initially introduced in the area by means of biological control for habu snake (Trimeresurus ﬂavoviridis) in 1979, since there was a serious problem of the rapid increase in habu snake population, which caused a high risk in human life. However, the original aim could not be achieved at all. Instead, mongoose population has increased up to more than 5,000 individuals and has damaged agricultural productions as well as the unique indigenous ecosystem in that area. Given this circumstance, the Japanese government has organized a program for the eradication of mongooses since 1996. However, it is reported that the catchability depends highly on the existing mongoose population and catch per unit of effort declines due to the decrease in the population. Moreover, the difﬁculty in implementing the eradication program also comes from the fact that the management agency cannot obtain the accurate information about how mongoose population changes. Therefore, some researchers claim that the above two facts related to stock-dependent catchability and uncertainty signiﬁcantly plague the mongoose management aiming at eradication. Several previous research efforts examine the optimal control of invasive species in economic dynamic models in which the objective of a society is to minimize the long-run social cost. Olson and Roy [14] theoretically develop a discrete-time dynamic model under a stochastic invasion growth and study the optimal policy of eradication. Eiseworth and Johnson [5] develop a continuous-time optimal control model, and their focus is mainly on the long-run equilibrium outcomes without analysis on the decision of eradication. Moreover, Eisewerth and van Kooten [6] make the assumption that the current stock is inaccurately known and apply the fuzzy membership function in the invasive species controls. However, all of the above efforts employ the assumption that the cost of removal operations is independent of the current stock size. That is, their analyses neither consider the stock-dependent catchability, nor address when to eradicate in relation to it. Olson and Roy [16] is a pioneering work that considers stockdependent removal costs and derives the conditions under which eradication or non-eradication can be optimal in the deterministic setting. While their innovative model is built under general settings, they do not explicitly examine the implications of stock-dependent catchability. Thus, their analytical results may not directly be applied in real management practices. Kotani et al. [9] focus on analyzing policy implications of stock-dependent catchability by deriving the conditions for various optimal policies in the deterministic setting. More speciﬁcally, our previous work shows that if the sensitivity of catchability is sufﬁciently high, eradication policy is never optimal and in effect the constant escapement policy with some interior target level is optimal. In contrast, if the sensitivity of catchability is sufﬁciently small, eradication policy could be optimal and there may exist a threshold of the initial stock (called a Skiba point) which differentiates optimal actions between immediate eradication and giving-up without controls. If the sensitivity of catchability takes some intermediate values, more complex policies would be optimal. Building upon [9], this paper derives optimal control strategies of the invasive species management in a stochastic environment. Of particular interest is a situation where managers make a decision on controls when the stock of invasive species ﬂuctuate due to growth uncertainty, and also the current stock cannot be precisely identiﬁed due to measurement uncertainty. To the best of our knowledge, this paper is novel in the sense that the model considers both ‘‘stock-dependent catchability’’ and ‘‘multiple stochasticity’’ in the single framework of a bioeconomic model. With this unique model, we seek to clarify the impacts of uncertainties on invasive species

33

management. To achieve this goal, we identify how the degrees of the two uncertainties affect optimal strategies and the corresponding value functions in two distinct scenarios of when (i) eradication and (ii) non-eradication are aimed in the management practices. Our results suggest that an increase in growth uncertainty leads to the optimal strategies that removals should be more restrained. By doing so, the corresponding value functions shift up as the growth uncertainty increases. Furthermore, we also ﬁnd that an increase in measurement uncertainty leads to complex impacts on the optimal strategies in the sense that any systematic pattern of the change in optimal strategies has not been found. However, a rise in measurement uncertainty generally shifts down their corresponding value functions. Overall, these results suggest that an increase in growth uncertainty can be beneﬁcial when the control strategy is optimally adapted. On the other hand, a rise in measurement uncertainty brings about an adverse effect on the management, which implies an importance and potential gain of identifying a precise stock size of invasive species. This paper is organized as follows. In the next section, we elaborate on the basic elements of the model. The section is followed by the analysis of a stochastic model with only growth uncertainty and presents how growth uncertainty affects the optimal strategy. Next, measurement uncertainty is incorporated into the model. We show how the interaction between growth and measurement uncertainties affects the optimal strategy. In the next section, we present how each of growth and measurement uncertainties affects the value functions of the optimal strategy or social welfare. The ﬁnal section offers some discussions and conclusions. 2. The model We consider an inﬁnite-period stochastic model of invasive species management, following a deterministic version of the dynamic model in Kotani et al. [9]. Our model below is developed by posing an invasive species control as the problem of a stochastic dynamic programming with Markovian transitions of multiple uncertainties, especially growth and measurement uncertainties. The speciﬁcation of our dynamic models basically follows the pioneering works of [2,25,24,21,16], all of which employ a stock-recruitment model in renewable resource management. In particular, we follow [21] with respect to the speciﬁcation for various uncertainties. In this paper, we pay attention to growth and measurement uncertainties since an interplay between the two provides an interesting result. We assume that there are two random variables, Z gt and Z m t , capturing growth and measurement uncertainties in each period t, respectively. The random variable Z gt reﬂects uncontrollable stochasticity associated with the stock growth of invasive species, while Z m reﬂects potentially controllable uncertainty.2 t These variables are independent of each other and of period t. We assume that Z gt and Z m t are respectively distributed over some ﬁnite intervals [1 zg, 1 + zg] and [1 zm, 1 + zm] with the mean of unity, where 0 < zk < 1, according to a common distribution function Uk, for k = g, m. The speciﬁcation of uncertainty implies that the distribution is mean-preserving spread with respect to zk. We choose this speciﬁcation since an increase in zk can be interpreted as a rise in the degree of the corresponding uncertainty (see [21]). The responsible ofﬁcials of management agencies are assumed to know the statistical distribution for each of these random variables. The stock (population) of existing invasive species in period t is governed by the following state equation:

xt ¼ Z gt Fðst1 Þ;

ð1Þ

2 For instance, if more efforts on identifying the current stock size of invasive species through extensive ﬁeld survey are devoted, measurement uncertainty is expected to be reduced.

34

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

where F(st1) is the expected or average reproduction function that gives the stock xt as a function of the previous period escapement, st1. We assume that F is differentiable and strictly concave with F(0) = 0 and F0 (0) 2 (1, 1) and that there exists an undisturbed level of the stock of invasive species, s > 0, with s = F(s) such that F(s) > s and F0 (s) P 0 if s 2 (0, s). This speciﬁcation implies that as assumed in [3], the deterministic stock-recruitment relationship holds in terms of conditional expectations, i.e., Eðxt jst1 ¼ sÞ ¼ FðsÞ. We further assume that (1 + zg)F0 (0) > 1 and limx?1(1 zg)F0 (x) < 1. The stock measurement, mt, in period t is given by the following equation:

mt ¼ Z m t xt :

ð2Þ

The society ofﬁcials are assumed to utilize the current measurement when they form expectations since Z m t is Markovian. The current measurement is sufﬁcient information to determine optimal strategies for our dynamic problem.3 Although actual data are not sufﬁcient to distinguish between multiplicative and additive stochastic terms, we believe that the chosen expression with a multiplicative stochastic term is the most convenient since it does not allow for negative values of the stock. Notice that if all the two random variables are constant at unity, our dynamic problem becomes deterministic. We suppose that the social cost in each period consists of the social damage from the escapement of invasive species and the cost associated with the removal operation. The former cost in period t is given by D(st), where D is increasing in s. The latter cost in period t is given by C(wt, xt), where wt is actual removal, and C is increasing in wt with C(wt, xt) P 0 for any wt and xt. The removal cost in each period depends not only on the number of actual removals but also on the stock of existing invasive species in that period. Speciﬁcally, given the stock at the beginning of period t, xt, and the total number of removals during period t, wt, the total cost of removal operations during period t is described by:

Cðwt ; xt Þ ¼

Z

xt

cðzÞdz;

ð3Þ

xt wt

where c(x) represents the unit cost that is a function of the current stock x. This speciﬁcation implies that the feasibility of eradication depends on the functional form of the unit cost function c(x). That is, for given stock x, eradication is feasible if C(x, x) is ﬁnite, and it is infeasible if C(x, x) is inﬁnity. From the above, the payoff for the society in period t is given by:

uðxt ; wt Þ ¼ Dðxt wt Þ Cðwt ; xt Þ:

ð4Þ

For our explanatory purpose to connect our arguments into catchability, we consider the unit cost function:

cðxÞ ¼ kEðxÞ ¼

k k h ¼ ¼ kx ; xqðxÞ pðxÞ

h > 0;

ð5Þ

where k is the constant marginal cost per unit of effort, E(x) is the effort level required to remove one unit of invasive species when the current stock is x, q(x) = xh1 is the stock dependent catchability representing the proportion of the current stock x that can be removed by one unit of effort, and p(x) = xq(x) = xh is the catch per unit of effort (CPUE) representing the stock size that can be removed by one unit of effort. Notice that h = xc0 (x)/c(x) > 0 can be interpreted as the sensitivity of catchability or the dependency of the unit costs in response to a change in the current stock size (see [3,9]). With this

speciﬁcation of catchability and CPUE, the total removal cost can be simpliﬁed as:

( Cðwt ; xt Þ ¼

k 1h

h

x1h ðxt wt Þ1h t

k½ln xt lnðxt wt Þ

[21] justify this assumption based on modeling choice and practical considerations, although they admit that a decision rule could be dependent on past measurements history, which may include some information about the current stock.

for h – 1

ð6Þ

for h ¼ 1:

The expression implies that eradication is infeasible, C(xt, xt) = 1, when h P 1. On the other hand, eradication is feasible, C(xt, xt) < 1, when h < 1. Fig. 1 exhibits the interrelation among removal costs, catchability, CPUE, and current stock size x in two cases of h > 1 and h < 1, each of which corresponds to panels 1(a) and 1(b), respectively. Each panel contains two panels: the upper panel presents the relationship between CPUE and the current stock x, while the lower one exhibits the relationship between unit removal costs and the current stock x. We ﬁrst explain the case of h > 1. In this case, catchability q(x) and CPUE p(x) are increasing in current stock size x (see the upper panel of sub Fig. 1(a)). This implies that a decrease in x leads to more reduction in catchability, and thus a unit removal cost becomes higher as the existing stock decreases. This effect is captured by a rapid increase in c(x) in the lower panel of the subﬁgure as x becomes less. The total removal cost of C(wt, xt), which corresponds to the area under c(x) between xt and xt wt, can be substantially high when eradication is aimed at. Thus, C(xt, xt) becomes inﬁnite when h > 1. On the other hand, catchability q(xt) decreases, but CPUE p(xt) increases in x if h < 1 (see the upper panel of sub Fig. 1(b)). In other words, as the current stock becomes less, CPUE decreases, but catchability increases. Thus, when the current stock declines, the unit cost of c(x) does not increase rapidly compared to the case of h > 1. Due to this property in the unit cost of c(x), the total cost of eradication, C(xt, xt), becomes ﬁnite when h < 1. The implication of parameter h is discussed by Reed [19], Clark [3], and Moxnes [12] for the prototype harvesting problems and by Kotani et al. [9] for invasive species management. For instance, Reed [19] states that the parameter value of h should empirically be between 0 and 1 in the ﬁshery harvesting problems. However, several authors note that the nature of operational costs for controlling invasive species is different and the sensitivity of catchability h is often larger than unity (see, e.g., [3,9]). Given this state of affairs, our analysis proceeds with the possibility that h can be larger than unity. To analyze optimal strategies under multiple uncertainties, we consider a society which extends over the following stages in each period. Given the escapement st1 in the previous period, the true stock xt is randomly determined according to the reproduction process of state Eq. (1). If measurement uncertainty is present, the management ofﬁcials cannot observe the true stock but can obtain the measured stock mt, following the stochastic process of measured stock represented by Eq. (2). Then, given mt, they decide the target removal yt. If yt is equal to or larger than xt, they remove all existing stock, i.e., eradication. If yt is less than xt, they actually achieve the target removal, but they never know the remaining escapement. The actual removal and the true escapement are represented by wt = min{yt, xt} and st = xt wt, respectively. Then, the next period t + 1 proceeds as far as st > 0. The management ofﬁcial is assumed to maximize the expected present value of the payoffs (minimize the present value of the payoff losses) by choosing a sequence of target removals fyt g1 t¼0 :

1 P t max E q uðxt ; wt Þ 06yt

s:t:

t¼0

xt ¼ Z gt Fðst1 Þ mt ¼ Z m t xt

3

i

wt ¼ minfyt ; xt g st ¼ xt wt ;

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

35

Fig. 1. Interrelation among removal costs, catchability and CPUE.

where q 2 (0, 1) is the discount factor and E is the expectation operator. The Bellman equation for this problem is:

g uðxt ; wt Þ þ qv tþ1 Z m ; tþ1 Z tþ1 Fðxt wt Þ

v t ðmt Þ ¼ max E 06y t

ð7Þ

where vt(mt) is the value function given the current stock measurement of existing invasive species, mt. For a given stock measurement, a sequence of the optimal target removals maximizes the expected present value of payoff over time.4 More speciﬁcally, for solving Bellman Eq. (7), the expectation of the right-hand side (RHS), consisting of the current reward and the next period value function, must be derived through calculating the transition probabilities conditional on the current state mt and current action of target removal yt. Representing the transition probabilities by using conditional probability density functions and the corresponding integration, we can derive the RHS of Bellman Eq. (7) as the following conditional expectation:

RHS ¼

Z Z

½uðxt ; wt Þ fxw ðxt ; wt jmt ; yt Þdwt dxt |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Z

Expected current reward

þ q ½v tþ1 ðmtþ1 Þ fm ðmtþ1 jmt ; yt Þdmtþ1 : |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

ð8Þ

Expected value function in the next period

The technique of convolution is applied to identify the transition probabilities with conditional probability density functions of fxw(xt, wtjmt, yt) and fm(mt+1jmt, yt), since xt = wt + st. Utilizing Eq. (8), we solve Bellman Eq. (7), which leads to the identiﬁcation of value functions as well as optimal removal strategies. 4 See [21] for concrete derivations of the conditional probability density function that is required in the Bellman equation.

Before analyzing our stochastic dynamic problem, some theoretical results established in the deterministic models of [9] should be presented as the benchmark for comparison with the stochastic cases. In the deterministic models of the same speciﬁcation, [9] show the following results: R1 The constant escapement policy with an interior target level of s⁄ 2 (0, s) is optimal if the sensitivity of catchability is sufﬁciently high such that h > h⁄ (see proposition 4 in [9] for the detailed derivation of h⁄).5 R2 The eradication policy can be optimal only if the sensitivity of catchability is sufﬁciently low such that h 2 ½0; ^ hÞ with ^ h ¼ 1 þ lnlnF 0qð0Þ < 1. More speciﬁcally, there could exist a threshold of the initial stock, called a Skiba point, which differentiates the optimal action between immediate eradication and giving-up without any control (see Proposition 5 in [9,23] for the detailed explanation on a Skiba point).6 The eradication policy indicates any policy that includes eradication actions at some point in time. Results R1 and R2 show that the optimal policy depends on whether h is large or small. If h is sufﬁciently low, eradication can be justiﬁed. Moreover, there could exist a Skiba point such that immediate eradication is optimal if an initial stock is smaller than the Skiba point, and giving up without any control is optimal otherwise. On the other hand, if h is sufﬁciently large, the constant escapement policy with an interior target level is optimal, which means that eradication is not optimal.

5 We have identiﬁed that h⁄ could be less than one in [9]. This case implies that a constant escapement policy with an interior target can be optimal although eradication is feasible. 6 A Skiba point is deﬁned to be the critical value of the initial state variable which differentiates the long run behavior of optimally controlled stock [23].

36

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

For the purpose of comparison between the deterministic and stochastic cases, this study pays particular attentions to the two cases corresponding to results R1 and R2: (1) a non-eradication case with a relatively large h in which the constant escapement policy is optimal in the deterministic setting; and (2) an eradication case with a relatively small h in which the eradication policy could be optimal in the deterministic setting. We take these two cases as the benchmark scenarios and then incorporate each of growth and measurement uncertainties step by step into the model. 3. Growth uncertainty

( ) ( ) 1 1 X X E qt uðxt ; xt st Þ ¼ Q ðx0 Þ þ E qt Cðst ; Z gt Þ ; t¼0

1þzg

Cðs; zÞdUg ðzÞ

1zg

¼ Q ðsÞ DðsÞ þ q

Z

1þzg

Q ðzFðsÞÞdUg ðzÞ:

ð10Þ

1zg

The maximization of the expected total discounted payoff of g(s) with respect to s is equivalent to ﬁnding a sequence {yt} to maximize the objective function of Eq. (9) subject to the state equation xtþ1 ¼ Z gt f ðxt minfyt ; xt gÞ and the initial stock x0. Thus, our dynamic problem simpliﬁes to a static maximization problem of g(s). Notice that the expected discounted payoff loss in the deterministic case corresponding to the function (10) is given by:

hðsÞ ¼ Q ðsÞ DðsÞ þ qQ ðFðsÞÞ:

This is a special case of the stochastic setting, that is, z = 0 in Eq. (10). The deterministic model of [9] shows that if h is sufﬁciently low, h is strictly convex so that the eradication policy can be optimal for a relatively small s. On the other hand, if h is sufﬁciently high, h is strictly concave so that the interior constant escapement policy can be optimal. In summary, the value of h highly affects the shape of h as well as the optimal policy in the deterministic model. The following subsections present the ﬁndings derived from the stochastic model, which correspond to results R1 and R2 in the deterministic case, as shown before. 3.1. Non-eradication case The examination in this subsection is on the non-eradication case, corresponding to R1, where the sensitivity of catchability is relatively large so that the optimal policy is in a class of interior constant escapement rules in the deterministic setting. Our analysis focuses on how growth uncertainty affects the optimal policy. For this reason, we consider a case where both g and h are strictly concave and unimodal. In a stochastic case, this requires the condition that

0

g

zcðzFðsÞÞF ðsÞdU ðzÞ D0 ðsÞ

xt s when xt > s 0

otherwise

or s ðxt Þ ¼

s

when xt > s

xt

otherwise

where y⁄(xt) and s⁄(xt) are the optimal removal and escapement policy functions, respectively. On the other hand, it is complex to characterize the optimal policy in the stochastic model. Let r denote the escapement level attaining the maximum of g(s), which we call ‘‘short-sighted optimal escapement level.’’ In addition, let S⁄ be the optimal target escapement level where the optimal policy in the stochastic model follows a constant escapement rule expressed as:

xt S 0

when xt > S or s ðxt Þ ¼ otherwise

S xt

when xt > S otherwise:

The unimodality condition requires that r is interior, i.e., g0 (r) = 0 holds with r 2 (0, m). However, the unimodality of g is not enough to guarantee that r is the optimal target escapement level of S⁄ in a stochastic model. Similar to the arguments in [19], we need to make an additional assumption in order for r to be the optimal target escapement S⁄. That is, r has to be self-sustaining, i.e., zF(r) P r for all z such that Ug(z) > 0, or the stock in the next period is required to be always larger than the escapement in the current period.8 If the shortsighted optimal escapement level r is self-sustaining, then it is the optimal target escapement even in a stochastic model, i.e., S⁄ = r. Otherwise, r may not be the optimal target escapement level and becomes the lower bound of the optimal target escapement S⁄, i.e., S⁄ P r (see [19]). Based on the results thus far, we now characterize how the degree of growth uncertainty affects the optimal target level. For this purpose, we take the differentiation of g(s) h(s)

ð11Þ g

#

1þzg

is strictly decreasing in s over [0, m], where m = max{(1 + zg)F(s)j(1 + zg)F(s) P s,s P 0}.7 In the deterministic case where s 2 ð0; sÞ is the escapement level attaining the maximum of h(s) in Eq. (11), s is the optimal target escapement level [9]. In other words, the optimal strategy is the constant escapement rule with the target level s, which is expressed as:

ð9Þ

where Cðst ; Z gt Þ Q ðst Þ Dðst Þ þ qQ ðZ gt Fðst ÞÞ. We denote the expected discounted payoff loss associated with escapement st by:

Z

1zg

y ðxt Þ ¼

t¼0

Z

g ðsÞ ¼ cðsÞ q

y ðxt Þ ¼

This section seeks to characterize how growth uncertainty in the reproduction function affects the optimal strategy in the absence of measurement uncertainty, i.e., Z m t is constant at unity. To characterize the optimal strategy in our stochastic dynamic problem, we transform the payoff in period t by utilizing the form of u(xt, wt) = [Q(xt wt) Q(xt)] D(xt wt), where Q ðxÞ Rm cðwÞdw 2 ½0; 1. The term of Q(x) represents the operational x cost of removing invasive species from the stock level m to some stock level x. Notice that Q0 (x) = c(x) < 0 and Q00 (x) = c0 (x) > 0. Using xtþ1 ¼ Z gt Fðst Þ, we rewrite the objective function as:

gðsÞ ¼

" 0

" 0

g 0 ðsÞ h ðsÞ ¼ qF 0 ðsÞ cðFðsÞÞ

Z

1þzg

# zcðzFðsÞÞdUg ðzÞ :

ð12Þ

1zg

7 Notice that c(s) is the marginal increase in current cost associated with a unit R 1þzg removal at the escapement level s, while q 1zg zcðzFðsÞÞF 0 ðsÞdUg ðzÞ is the expected discounted present value of the marginal increase in sustained future removal cost resulting from a unit increase in the escapement. Thus, the value of R 1þzg BðsÞ cðsÞ q 1zg zcðzFðsÞÞF 0 ðsÞdUg ðzÞ could be regarded as the expected marginal be neﬁ t asso ciated with the unit e scapement in curre nt period. If R 1þzg 0 g q 1z g zcðzFðsÞÞF ðsÞdU ðzÞ is relatively large compared to c(s), then it is more costly to remove the stock in the future so that the policymakers should involve removal action in the current period. 8 Recall that Z gt is distributed over some ﬁnite interval [1 zg, 1 + zg] with the mean of unity, where 0 < zg < 1, according to a common distribution function Ug. It then follows that there exist ﬁnite stock levels, m = max{(1 + zg)F(s)j(1 + zg)F(s) P s, s P 0} and r = max{(1 zg)F(s)j(1 zg)F(s) P s, s P 0}, such that with probability 1, the stock will eventually stay within the interval [r, m] and on which it will attain a stationary probability distribution in the limit. Any escapement level s for which Prob{xt P sjst1 = s} = 1 is called self-sustaining. This implies that any population level in the interval [0, r] is self-sustaining. In the dynamic harvesting models, [19] shows that the constant escapement policy is optimal under some conditions, and the level r at which g(s) is maximized is lower-bound for the optimal level of escapement. He also states that the optimal escapement level is r if r is selfsustaining, and the optimal level is greater than r if r is not self-sustaining.

37

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

Applying Jensen’s inequality, Eq. (12) implies the following three cases with respect to the relation between s and r: Case 1: Case 2: Case 3:

r < s if xc(x) is strictly convex, i.e., h > 1, r ¼ s if xc(x) is linear, i.e., h = 1 r > s if xc(x) is strictly concave, i.e., h < 1.

To see these results, ﬁrst let the random variable Y be such that R 1þzg Y = ZgF(s). Then, it follows that 1zg zcðzFðsÞÞdUg ðzÞ ¼ E½YcðYÞ=FðsÞ. Applying Jensen’s inequality with F0 (s) > 0, it must hold that the value of g0 (s) h0 (s) is negative, zero, or positive for all s if sc(s) is strictly convex, linear, or strictly concave in s. Since our analysis mostly corresponds to case 1 of h > 1 (relatively large h) with strict concavity of h, it must hold that g0 (s) h0 (s) < 0. With these results, we can explain how the degree of growth uncertainty affects the optimal target escapement level. The inspection of Eq. (12) shows that the short-sighted optimal escapement r decreases in zg due to the property of second-order stochastic dominance (see [10] for the details of second-order stochastic dominance). More speciﬁcally, when h > 1, the absolute value of g0 (s) h0 (s) < 0 is increasing in zg, so that the difference between r and s is increasing in zg. Thus, an increase in zg causes the short-sighted optimal escapement r to decline. Since r equals the optimal escapement S⁄ as long as it is self-sustaining, we can derive the following result: Result 1. In the non-eradication case, a rise in zg causes the optimal escapement level S⁄( = r) to decline, as far as r is selfsustaining. It should be noted that as zg increases, the region of the escapement levels that meet the self-sustaining property shrinks. In other words, when zg becomes larger, it is more likely that r is not selfsustaining so that r may not be equal to S⁄, i.e., S⁄ – r. As mentioned earlier, when r is not self-sustaining, it becomes the lower bound of the optimal target escapement level S⁄, i.e., S⁄ P r. In this case, a natural question to be asked is: how does the degree of growth uncertainty affect the optimal escapement level when r is not self-sustaining? When zg is large enough that r is not self-sustaining, what we have as information about S⁄ is only S⁄ P r. In this case, analytical derivation for characterizing S⁄ appears to be difﬁcult as mentioned in [19]. Thus, numerical analysis is employed to further characterize the relationship between growth uncertainty and the optimal target escapement S⁄. To do this, we make the following two speciﬁc assumptions on the functional forms. First, the social damage from invasive species is represented by the linear quadratic form:

DðsÞ ¼ a1 s þ

a2 s2 ; 2

ð13Þ

where s denotes the escapement with a1 P 0 and a2 > 0. Second, the reproduction process of invasive species depends on escapement s, and follows the conventional logistic curve:

s

þ s; FðsÞ ¼ rs 1 K

ð14Þ

where r > 0 is the intrinsic growth rate and K > 0 is the carrying capacity. These two functional forms, which satisfy the assumptions speciﬁed in the previous sections, are also employed in numerical analysis in some studies such as [15,5]. We have also used some alternative reproduction functions, social damage function, and parameter sets in order to check the validity of our results. In all of these different cases, the optimal policy and the corresponding value functions exhibit the same qualitative patterns as the ones we will present in this paper. Thus, our qualitative results presented can be considered robust to a change of functional forms and parameter sets as far as the basic assumptions we posed in the model section are satisﬁed.

Fig. 2. Base case: optimal policy in non-eradication case under growth uncertainty.

For the solution of dynamic programming problems, the value function iteration algorithms introduced in [8] are adopted to approximate the value function v(mt) (or v(xt)) as well as the optimal policy function y⁄(mt) (or y⁄(xt)) that are characterized by the Bellman Eq. (7).9 This algorithm ﬁrst involves the discretization of the state space and then iterates on the Bellman equation with an initial guess for the value function. It is shown that by the contraction theorem, the Bellman equation ﬁxes a unique value function, v(mt), and the iterative process converges to the true value function. Accordingly, a particular optimal policy y⁄(mt) is obtained. For our baseline, we choose k = 250 for the unit cost function, c(x); a1 = 1 and a2 = 2 for the social damage function, D(s); r = 0.3 and K = 10 for the reproduction of invasive species, F(s); and q = 0.95 for a social discount rate. We also set h = 1.1 so that the sensitivity of catchability is relatively large. In this case, the optimal policy in the deterministic case is the constant escapement policy with the interior target level s ¼ 3:9. Given these values and assumptions, we ﬁnd the optimal policy based on each of three different values of the degree of growth uncertainty; zg 2 {0.25, 0.50, 0.75}. Under each of the uncertainty levels, we conﬁrm that the short-sighted optimal escapement, r, is not selfsustaining in the stochastic setting. Fig. 2 illustrates how the degree of growth uncertainty affects the optimal policy. First, it is observed that the constant escapement policy is optimal as in the deterministic case. The numerical result in Fig. 2 suggests that the existence of growth uncertainty still supports the optimal policy to be in a class of the constant escapement rule. Next we realize that a rise in the degree of growth uncertainty monotonically increases the optimal target level of the escapement. The intuition behind these results could be explained as follows. When the degree of growth uncertainty becomes larger, the stock level in the next period is more likely to be in a non-self-sustaining region. This means that the stock tends to decline with a higher probability from the current period to the next period, even without any removal operation. Thus, a larger degree of growth uncertainty induces the current removal to be less necessary. This is in sharp contrast to the result of the case where the optimal short-sighted escapement level, r, is self-sustaining in that an increase in the degree of growth uncertainty decreases the optimal target escapement level S⁄. We summarize the above result as follows. Result 2. In the non-eradication case, the optimal target escapement S⁄ increases in zg when r is not self-sustaining. Combining Results the impact of growth ment in the constant two cases of the 9

1 and 2 provides some implications about uncertainty on the optimal target escapeescapement policy. We have analyzed the self-sustaining and non-self-sustaining

A Matlab code is written for numerical solutions.

38

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

Result 3. In the non-eradication case, the optimal target escapement level S⁄ changes with the degree of growth uncertainty zg in a non-monotonic U-shaped manner. More speciﬁcally, S⁄ declines with zg as far as r is self-sustaining. However, once zg becomes large enough that r is not self-sustaining, then S⁄ increases with zg.

3.2. Eradication case

Fig. 3. Base case: optimal policy in eradication case under growth uncertainty.

short-sighted optimal escapements. When r is self-sustaining, S⁄ declines with zg. However, even if r monotonically declines with zg, it is more likely to be non-self-sustaining with higher zg. This is due to the fact that the region of escapement levels that satisﬁes the self-sustaining property rapidly shrinks as zg rises. In this case, S⁄ increases in zg. Summarizing all of Results 1 and 2 gives us the following result.

This subsection examines how growth uncertainty affects the optimal policy in an eradication case where the eradication policy is optimal. This case corresponds to R2, where the sensitivity of catchability is sufﬁciently small. In this case, it is also difﬁcult to analytically characterize the optimal strategy especially in the presence of uncertainty [3]. Thus, we again approach this case through numerical analysis. As a benchmark, we newly set h = 0.5, keeping other parameters and functional forms unchanged. The optimal policy of the corresponding deterministic model yields immediate eradication for any initial level of stock. Similar to the previous discussion, we examine three different values in the degree of growth uncertainty; zg 2 {0.25, 0.50, 0.75}. The results are shown in Fig. 3. First, the introduction of growth uncertainty yields a Skiba point of the initial stock, which separates

Fig. 4. Base case: optimal policy in non-eradication case under growth uncertainty and measurement error [Fix zm (Left); Fix zg (Right)].

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

39

Fig. 5. Sensitivity analysis: optimal policy in non-eradication case under growth uncertainty and measurement error [Fix zm (Left); Fix zg (Right)].

the optimal actions between immediate eradication and giving-up without any control. More precisely, immediate eradication is optimal if the initial stock is identiﬁed to be lower than the Skiba point. Otherwise, giving up without any control is optimal (see Fig. 3). Moreover, as the degree of growth uncertainty rises, the Skiba point monotonically becomes smaller and approaches to zero. The intuition behind this result is similar to that in the noneradication case. When the degree of growth uncertainty is relatively large, the stock level in the next period is highly likely to fall in a non-self-sustaining region, i.e., the invasive species stock tends to decline with a high probability. A larger degree of growth uncertainty induces the current removal to be less necessary in the optimal strategy. Thus, as zg rises, the government agency with relevant authority should wait more patiently without any control before taking eradication actions. We believe that this simple result is the ﬁrst to show that growth uncertainty brings about the shift of the Skiba point on the invasive species management. Then we summarize the result. Result 4. In the eradication case, the Skiba point approaches zero as zg rises.

4. Measurement uncertainty This section incorporates measurement uncertainty into the previous setup with growth uncertainty. We adopt the same speciﬁcations as in the stochastic model of [21], i.e., measurement uncertainty is Markovian; a signal of true stock is in a multiplicative fashion; and Z m t is uniformly distributed over the interval [1 zm, 1 + zm], where the parameter zm 2 [0, 1) represents the degree of measurement uncertainty. As in the previous section, we use numerical approach to analyze the two cases: (i) the noneradication case and (ii) the eradication case. In each case, we employ two different values of the degree of measurement uncertainty, zm 2 {0.25, 0.50}. The results are compared to the cases in the absence of measurement uncertainty.10

10 In this section, we do not report the case of zg = 0 and zm > 0, i.e., there is no growth uncertainty but measurement uncertainty is present. This is mainly due to the following reasons: First, past literature which deals with measurement uncertainty does not consider this case (see, e.g., [21]). Second, the results are qualitatively similar with those in the case of zg = 0.25 when measurement uncertainty is present. Thus, to avoid redundancy of the same qualitative results, we omit this case.

40

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

4.1. Non-eradication case This subsection analyzes how measurement uncertainty inﬂuences the optimal strategy in a non-eradication case corresponding to R1. Fig. 4 presents the optimal policies in the base case, which consists of six panels. Each panel takes the optimal target removal of y⁄(mt) as a vertical axis and the measured stock mt as a horizontal axis. Three panels at the right side in Fig. 4 describe how optimal strategies change with zm, holding zg ﬁxed at the level of zg = {0.25, 0.5, 0.75}. Similarly, the three panels at the left side of Fig. 4 describe how optimal strategies change with zg, holding zm 2 {0, 0.25, 0.5} constant. Each panel contains three optimal policies, which correspond to the three different levels of the uncertainty. Recall that interior constant escapement rules are optimal in the absence of measurement uncertainty, i.e., zm = 0. The results correspond to the three optimal strategies shown in the panel of zm = 0 in Fig. 4. When we notice that the vertical axis is the optimal target removal y⁄(mt) and the horizontal axis is measured stock mt, the results of panel zm = 0 in Fig. 4 are exactly the same as the ones shown in Fig. 2. Thus we can interpret that the removals are more restrained in the optimal constant escapement strategy as zg rises (i.e., y⁄(mt) shifts down as zg rises in panel zm = 0 of Fig. 4).

Fig. 4 shows that the constant escapement policy is not optimal anymore once measurement uncertainty is present zm > 0. This is illustrated in panels of zg = 0.25, zg = 0.5, and zg = 0.75 in Fig. 4. In these panels, the constant escapement strategy is optimal for zm = 0, but the optimal strategy deviates from the constant escapement rule for zm = {0.25, 0.5}. To check the robustness of this result, we conduct the same analysis with another set of parameters, where a2 = 1 with other parameters set as before. Fig. 5 consisting of six panels in a same manner illustrates the optimal strategy under the set of the new parameters and shows the qualitatively similar result that the optimal strategy deviates from the constant escapement rule for zm = {0.25, 0.5}. Then we summarize this ﬁnding as follows: Result 5. In the non-eradication case, the constant escapement strategy is not optimal when measurement uncertainty is present, i.e., zm > 0. Our focus now turns to the three panels of zm = {0, 0.25, 0.5} at the left in Fig. 4. The left panels illustrate how an increase in zg affects the optimal strategy, holding zm constant at some level. Recall that the optimal constant escapement level increases in zg without measurement uncertainty. The panel of zm = 0 shows that the optimal target removal is decreasing in zg for a given measured stock, i.e., removals should be more restrained as zg rises.

Fig. 6. Base case: optimal policy in eradication case [Fix zm (Left); Fix zg (Right)].

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

41

Fig. 7. Sensitivity analysis: optimal policy in eradication case under growth uncertainty and measurement error [Fix zm (Left); Fix zg (Right)].

The two other panels of zm = 0.25 and zm = 0.5 exhibit the different transition of the optimal strategies. The panel of zm = 0.25 shows that the graphs of the optimal target removals cross each other. Moreover, the panel of zm = 0.5 shows that the three graphs never cross anymore, and more speciﬁcally, the optimal target removal is increasing in zg for a given measured stock when zm = 0.5. This is in sharp contrast with the case of zm = 0 where the optimal target removal is decreasing in zg for a given measured stock. The same qualitative results of this reversal are conﬁrmed in the three panels at the left side of Fig. 5, which show the graphs of the optimal strategies from the analysis of the set of the different parameters. The above arguments suggest that a rise in zg causes the optimal target removal to decrease for zm sufﬁciently small, but once zm becomes sufﬁciently large, this relation becomes the opposite so that a rise in zg causes the optimal target removal to increase. One possible explanation may be that measurement uncertainty gives rise to a new important role of growth uncertainty in our dynamic problem. When measurement uncertainty is present, growth uncertainty affects the expectation of ‘‘current rewards’’ in

the Bellman Eq. (7). However, once there is no measurement uncertainty, growth uncertainty does not have any impact on the current rewards, but only affects the expectation of the continuation value in the Bellman equation (value in the next period). This is the critical difference between the two cases with and without measurement uncertainty. In the presence of measurement uncertainty, a rise in growth uncertainty intensiﬁes the uncertainty on current rewards through an interplay between two uncertainties, and thus reduces the expected current reward associated with the stock of invasive species.11 Therefore, the target removals must be implemented more aggressively to control the risk of current rewards when growth uncertainty rises in the presence of large measurement uncertainty. Overall, these results imply that the way of adaptation to growth uncertainty in the optimal policy can get reversed depending on

11 This holds when the strict convexity of social damage is large enough. Jensen’s inequality can be applied for the proof.

42

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

the degree of measurement uncertainty. Then we summarize this result as follows: Result 6. Consider the non-eradication case. Given a small degree of measurement uncertainty, fewer removals are required in the optimal policy as the degree of growth uncertainty rises. In contrast, given a large degree of measurement uncertainty, more removals are required in the optimal policy as the degree of growth uncertainty rises. Finally, panels at the right in Fig. 4 also show that intensiﬁed measurement uncertainty fundamentally changes the shape of the graphs of the optimal target removal, or the sensitivity of the optimal target removal in response to the measured stock. Generally, a rise in zm causes the graphs of the optimal target removal to become ﬂatter. That is, a rise in zm causes the optimal target removal to become less sensitive to a change in the measured stock. We also conﬁrm the same results from panels in Fig. 5. To understand this, we need to notice the two points. The ﬁrst is that informational quality of the measured stock in hand gets worse as the measured stock increases due to the multiplicative stochastic term m of Z m t . The second is that z can be considered as a measure of how fast informational quality of the measured stock deteriorates.

Given the arguments, the presence of measurement uncertainty enforces the management authority to make the current removal decision based on more unreliable information as the measured stock increases. As a result, the optimal target removal becomes less sensitive to the measured stock as measurement uncertainty becomes larger. Then we summarize the discussion as follows: Result 7. In the non-eradication case, a rise in measurement uncertainty causes the optimal target removal to become less sensitive to a change in the measured stock.

4.2. Eradication case This subsection discusses how measurement uncertainty together with growth uncertainty affects the optimal policy in the case where eradication is optimal at least for some levels of the measured stock. This case corresponds to R2 in the deterministic setting. Figs. 6 and 7 respectively present the base case and sensitivity analysis case with h = 0.5. The difference between them is only the value of a2, which represents the degree of convexity in the social damage function, as in the previous subsection. In the

Fig. 8. Base case: value function in non-eradication case [Fix zm (Left); Fix zg (Right)].

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

43

Fig. 9. Sensitivity analysis: value function in non-eradication case [Fix zm (Left); Fix zg (Right)].

absence of growth and measurement uncertainties, the optimal policy in the base case is immediate eradication for any stock level, while in the sensitivity analysis case it is immediate eradication or giving-up without any control depending on whether the stock level is smaller or larger than the Skiba point which is around 4.0. All graphs in each panel of Figs. 6 and 7 show that there exists a Skiba point such that immediate eradication is optimal if the measured stock is smaller than the Skiba point, and otherwise givingup without any control is optimal.12 These ﬁgures show no clear pattern of the impacts of the uncertainties, i.e., the optimal policy becomes more elusive. For instance, the panels in Figs. 6 and 7 show that the Skiba points are affected by the degrees of measurement uncertainty as well as growth uncertainty, but they do not display 12 Some panels in Fig. 6 describe a situation where the optimal target removal is constant at some positive level over some region of the measured stock. This constant target removal arises from the fact that given some degree of measurement error, there exists a possible upper level of true stock. The society ofﬁcials just set the target removal at the upper level when the current measured stock is relatively large so that the next period’s stock can be reached to the upper level with some positive probability, and when their optimal policy is immediate eradication. Thus, even though the constant target removal is observed over some region of the measured stock in panels, it is aimed for eradication.

any systematic shift of the Skiba point at all. We also ﬁnd that the same type of complexity arises from various other eradication cases in which parameters and functional forms are changed as alternative speciﬁcations. Thus, the issue remains as an open question and would be an interesting topic to be addressed in the future. 5. Values of optimal removal policies The previous sections have focused on a qualitative change of the optimal policy in response to a change in the degrees of growth uncertainty and/or measurement uncertainty. This section examines the impact of multiple uncertainties on social welfare or the value of optimal policies by employing numerical analysis. We show the results in both non-eradication and eradication cases in order. 5.1. Non-eradication case This subsection focuses on the non-eradication case which corresponds to R1. Figs. 8 and 9 respectively show the values of optimal policies in the base case and sensitivity analysis case, as in the

44

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

previous section. They present some systematic features about the impacts of growth and measurement uncertainties on the value function. First, given a ﬁxed degree of measurement uncertainty, a rise in the degree of growth uncertainty causes the graph of the value functions to shift up (see panels at the left in Figs. 8 and 9). This result might be surprising since an increased uncertainty is generally considered as a negative event for a society. However, this case clearly illustrates that a rise in growth uncertainty can improve social welfare or lead to cost saving of removal operations if management ofﬁcials optimally adapt the policy to a change in the uncertainty. As mentioned earlier, a rise in the degree of growth uncertainty enlarges the non-self-sustaining region of the stock, which implies that the future stock of invasive species is more likely to decline even without any control. Thus, removal actions become less necessary, so that the cost reduction is attained in the optimal strategy. Second, holding the degree of growth uncertainty ﬁxed, a measurement uncertainty causes the graph of the value functions to shift down (see panels at the right in Figs. 8 and 9). In other words, an increased measurement uncertainty would deteriorate social

welfare, which is in sharp contrast with the impact of a change in the degree of growth uncertainty. This result is consistent with the one in the stochastic ﬁshery models of [4]. More accurate information about the current status of the stock generally beneﬁts a society. What kind of policy implications can we suggest from these results concerning the value of optimal policies in the noneradication case? Concerning growth uncertainty, our results suggest that when an invasive species stock appears to ﬂuctuate at a higher rate due to high growth uncertainty, the authority can attain the cost effective management and improve social welfare by adopting more restrained removal actions. Regarding measurement uncertainty, the analysis suggests that more precise information on the current stock size of invasive species helps improve social welfare. The informational quality on the measured stock can be improved by devoting more efforts on stock survey. Thus, our result can be of some guidance for how much efforts on stock survey should be devoted to improve the practice of invasive species management. Finally, we summarize the results associated with the values of optimal policies.

Fig. 10. Base case: value function in eradication case [Fix zm (Left); Fix zg (Right)].

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

Result 8. In the non-eradication case, a rise in the degree of growth uncertainty generally increases the values of optimal policies, but a rise in the degree of measurement uncertainty generally decreases the values.

5.2. Eradication case This subsection turns to the eradication case which corresponds to R2 in the deterministic setting. Fig. 10 shows the graph of the value functions in the base case and exhibits complex patterns of shifts in the value functions. One thing we notice is that in the absence of measurement uncertainty, the graph of the value functions shifts up as the degree of growth uncertainty rises (see the panel of zm = 0 in Fig. 10). This implies that a rise in growth uncertainty improves social welfare even in an eradication case. However, once measurement uncertainty is present in the model, this feature disappears, and the graphs of the value functions cross each other in a complex manner. In contrast to the non-eradication case, we cannot draw any clear conclusion on the impact of growth and measurement uncertainties on social welfare in the eradication case. An increase in growth uncertainty and a decrease in measurement uncertainty, which would improve social welfare in the noneradication case, are not necessarily beneﬁcial to the society in the eradication case. We summarize the results with respect to the values of optimal policies. Result 9. In the eradication case, a rise in the degree of growth uncertainty increases the value of optimal policies in the absence of measurement uncertainty. On the other hand, when measurement uncertainty is present, a change in the uncertainties gives complex impacts on the values of optimal policies.

6. Conclusion This paper has examined how growth and measurement uncertainties affect optimal policies and the corresponding value functions on invasive species management. Although there might be other factors that complicate the management decisions, such multiple uncertainties are among the main factors that government ofﬁcials should take into account in advance. We have shown that such uncertainties could signiﬁcantly alter not only qualitative features of optimal policies, but also their value functions. While some of our results are common to the ﬁndings of stochastic ﬁshery models in the past literature, we have found a series of novel results, focusing on the two cases: non-eradication and eradication cases. We believe that our results would be valuable for real management practices. The analysis has demonstrated that in the non-eradication case, growth uncertainty does not impact the qualitative features of the optimal policy which is in a class of constant escapement rules, but it alters the optimal target escapement level, as shown in [19]. One of the novel ﬁndings in this paper is that the target escapement level does not monotonically vary with the degree of growth uncertainty. In fact, the impact could be U-shaped in the sense that the optimal target escapement level is decreasing and then is increasing in the degree of growth uncertainty. On the other hand, in the eradication case, growth uncertainty could generate the existence of a Skiba point in the optimal policy. Furthermore, the Skiba point approaches zero as the degree of growth uncertainty increases. This suggests that as the degree of growth uncertainty rises, the management authorities should wait more patiently without any control before taking eradication actions. Concerning measurement uncertainty, we have identiﬁed that the introduction of measurement uncertainty signiﬁcantly affects

45

the qualitative features of optimal policies in both the non-eradication and eradication cases. In the non-eradication case, measurement uncertainty induces the optimal policy to deviate from the constant escapement rules. In particular, given a small degree of measurement uncertainty, fewer removals are required as the degree of growth uncertainty rises. In contrast, given a large degree of measurement uncertainty, more removals are required as the degree of growth uncertainty rises. These imply that the way of adaptation to the growth uncertainty in the optimal policy can get reversed in the presence of measurement uncertainty. On the other hand, in the eradication case, we have found that the presence of measurement uncertainty signiﬁcantly affects the optimal policy in a highly complex manner. Unfortunately, no systematic pattern of changes has been observed. This study has also examined the values (or social welfare) of optimal policies in the presence of growth and measurement uncertainties. In the non-eradication case, the value of optimal programs is generally monotone increasing in the degree of growth uncertainty, while it is monotone decreasing in the degree of measurement uncertainty. In the eradication case, an increase in growth uncertainty shifts up the value function in the absence of measurement uncertainty. However, the presence of both growth and measurement uncertainties causes the value function to change in a highly complex manner. These ﬁndings suggest that management agencies have to adapt their removal actions to the uncertainties in order to improve social welfare. In particular, it is important to note that less removal actions are more likely to yield cost saving when an invasive species stock appears to ﬂuctuate more due to higher growth uncertainty. Concerning measurement uncertainty, obtaining more precise information on the current stock size of invasive species pays off, which is illustrated by the shift-up of the value functions in the non-eradication case. As discussed earlier, informational quality on the measured stock is controllable and can be improved by devoting more effort on stock survey. Therefore, our result can be of some guidance for how much survey effort should be put to improve the practice of invasive species management. Although we have conﬁrmed that the same qualitative results in this paper could be obtained by employing other different functional forms and parameter sets that satisfy the basic assumptions of the model, some attention must be paid to the other issues we did not consider. First, the social damage could be considered cumulative as in the case of stock pollutant, and also the regeneration function may be subject to a minimum viable population as in the critical depensation growth in reality. We believe that such a change in the functional forms deserves some further exploration and may yield a different result. Second, we assume that growth and measurement uncertainties are Markovian, and the probability distributions are fully known with parameters. In reality, these suppositions can be questioned. What we can do instead in future research may be to apply the Bayesian learning model for unknown parameters or the Kalman ﬁlter for alternative assumptions of state space modeling. However, it is our belief that this paper could be considered some guidance on real practices and a starting point of researches on invasive species management under multiple uncertainties. We hope that some potential extensions as listed above would be made in the near future.

References [1] M. Bomford, P. O’Brien, Eradication or control for vertebrate pests, Wildlife society bulletin 23 (1995) 249. [2] C.W. Clark, A delayed-recruitment model of population dynamics, with an application to baleen whale populations, Journal of Mathematical Biology 3 (1976) 381.

46

K. Kotani et al. / Mathematical Biosciences 233 (2011) 32–46

[3] C.W. Clark, Mathematical Bioeconomics, 2nd ed, John Wiley and Sons, Inc.,, 1990. [4] C.W. Clark, G.P. Kirkwood, On uncertain renewable resource stocks: optimal harvest policies and the value of stock surveys, Journal of Environmental Economics and Management 13 (1986) 235. [5] M.E. Eisewerth, W.S. Johnson, Managing nonindigenous invasive species: insights from dynamic analysis, Environmental and Resource Economics 23 (3) (2002) 319. [6] M.E. Eisewerth, G. van Kooten, Uncertainty, economics, and the spread of an invasive plant species, American Journal of Agricultural Economics 84 (5) (2002) 1317. [7] N. Ishii, Controlling mongooses introduced to Amami-Oshima island: a population estimate and program evaluation, Japanese Journal of Conservation Ecology 8 (2003) 73 (in Japanese). [8] K.L. Judd, Numerical Methods in Economics, MIT Press, 1998. [9] K. Kotani, M. Kakinaka, H. Matsuda, Dynamic economic analysis on invasive species management: some policy implications of catchability, Mathematical Biosciences 220 (1) (2009) 1. [10] J.-J. Laffont, The Economics of Uncertainty and Information, The MIT Press, 1989. Translated by John P. Bonin and Helene Bonin. [11] C. Loehle, Control theory and the management of ecosystems, Journal of Applied Ecology 43 (2006) 957. [12] E. Moxnes, Uncertain measurements of renewable resources: approximations, harvesting policies and value of accuracy, Journal of Environmental Economics and Management 45 (2003) 85. [13] J.H. Myers, A. Savoie, E. van Randen, Eradication and pest management, Annual Review of Entomology 43 (1998) 471. [14] L.J. Olson, S. Roy, The economics of controlling a stochastic biological invasion, American Journal of Agricultural Economics 84 (5) (2002) 1311.

[15] L.J. Olson, S. Roy, Controlling a biological invasion: a non-classical dynamic economic model, 2004, Working Paper. [16] L.J. Olson, S. Roy, Controlling a biological invasion: a non-classical dynamic economic model, Economic Theory 36 (2008) 453. [17] C. Perrings, M. Williamson, S. Dalmazzone, The Economics of Biological Invasions, Edward Elgar, 2000. [18] D. Pimentel, R. Zuniga, D. Morrison, Update on the environmental and economic costs associated with alien-invasive species in the United States, Ecological Economics 52 (2005) 273. [19] W.J. Reed, Optimal escapement levels in stochastic and deterministic harvesting models, Journal of Environmental Economics and Management 6 (1979) 350. [20] J. Roughgarden, F. Smith, Why ﬁsheries collapse and what to do about it?, Proceedings of National Academy of Sciences of the United States of America 93 (10) (1996) 5078 [21] G. Sethi, C. Costello, A. Fisher, M. Hanemann, L. Karp, Fishery management under multiple uncertainty, Journal of Environmental Economics and Management 50 (2005) 300. [22] D. Simberloff, Today tiritiri matangi, tomorrow the world! Are we aiming too low in invasion control?, in: C.R. Veitch, M.N. Clout (Eds.), Turning the tide: the eradication of invasive species (2002) 4. [23] A.K. Skiba, Optimal growth with a convex-concave production function, Econometrica 46 (1978) 527. [24] A.K. Supriatna, H.P. Possingham, Optimal harvesting for a predator–prey metapopulation, Bulletin of Mathematical Biology 60 (1998) 49. [25] G.N. Tuck, H. Possingham, Optimal harvesting strategies for a metapopulation, Bulletin of Mathematical Biology 56 (1994) 107.