A stochastic differential Fishery game for a two species fish population with ecological interaction

ARTICLE IN PRESS Journal of Economic Dynamics & Control 34 (2010) 844–857

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A stochastic differential Fishery game for a two species fish population with ecological interaction Wen-Kai Wang a, Christian-Oliver Ewald b,c, a b c

Department of Economics, University of St. Andrews, UK School of Mathematics and Statistics, University of Sydney, Australia Center for Dynamic Macro Economic Analysis, University of St.Andrews, UK

a r t i c l e i n f o

abstract

Article history: Received 7 May 2008 Accepted 17 November 2009 Available online 6 January 2010

We combine and extend two existing lines of research in game theoretic studies of fisheries, building up on Quirk and Smith (1977), Anderson (1975), Fisher and Mirman (1996), Sumaila (1997) and most recently Datta and Mirman (1999) who developed either static or discrete time models, not including ecological uncertainty and Jorgensen and Yeung (1996) who do include uncertainty but do not capture any features of ecological interaction. In this article we develop a continuous time framework, where ecological interaction is described by a stochastic dynamics, including the cases of predator–prey and competition. We obtain a stochastic differential game and derive Markov feedback Nash-equilibrium strategies in semi-analytic form. Furthermore we compare the results with the case where fisheries regulations restrict each fishery to harvest only one species and study the inefficiencies which arise from this. In addition to that, we also consider the case where fisheries cooperate. Here we observe quite different effects on the ecosystem, depending on whether the system is competitive or predator–prey. & 2009 Elsevier B.V. All rights reserved.

JEL classification: C73 Q22 Q57 Keywords: Differential games Fisheries Environmental and resource economics Stochastic optimal control

1. Introduction Various recent works in game theoretic modeling of fisheries focus on continuous time and are based on differential game models. Among them are Dockner et al. (1989), Haurie et al. (1994), Jorgensen and Yeung (1996) and Kaitala (1989). All consider the case of a single species. Kaitala has presented a deterministic game model in which he assumes that the price of fish is given by a constant p while costs for each agent i are defined by a proportionality constant ci . He studied the non-cooperative case and derived a feedback Nash-equilibrium. Hamalainen, Haurie and Kaitala have studied a slightly different model and derived an open-loop Nash-equilibrium. These authors also considered the cooperative case and provided two numerical examples. Dockner et al. (1989) considered a continuous time framework involving uncertainty. He studied the non-cooperative case and followed the idea in Clark (1980) and Simaan and Takayama (1978) for the specification of the price function. Dockner et al. assume that the price depends on the quantity harvested by all agents. Jorgensen and Yeung (1996) studied a model where the concept of price is similar and costs specified by a function which is decreasing in the stock of biomass. These authors derived a feedback Nash-equilibrium and also considered the cooperative case. Moreover, they analyzed surplus maximization and optimal market size.

 Corresponding author at: Center for Dynamic Macro Economic Analysis, University of St. Andrews, UK. Tel.: + 44 1334 462435.

E-mail address: [email protected] (C.-O. Ewald). 0165-1889/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jedc.2009.12.001

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In our article, we study a model where a number of fisheries are harvesting two different species of fish. Ecologically these two species are assumed to interact with each other. The assumed interactions include the cases of predator–prey and competition. The case of a two species predatory game theoretic fishery model has been studied before by Quirk and Smith (1977), Anderson (1975), Fisher and Mirman (1996), Datta and Mirman (1999) and furthermore by Sumaila (1997). Sumaila focuses on the case of the Barents sea and the species Cod and Capelin. In this particular case Cod preys on Capelin and Sumaila provides various results which document the importance of studying fisheries within a general multi-species ecological context. Other examples include the case of sharks in Tasmania, which when decreasing in numbers cause their main prey, octopus, to increase in numbers, which then results in a decrease of spiny-lobster. In fact there are claims that increased shark finning driven by demand in South East Asia has brought the Tasmanian spiny lobster fishery to near collapse, see http://www.sharkattacks.com/sharkvictims.htm. The University of Tasmania Marine Research Laboratories however could not confirm this claim, and it is likely to be political motivated, rather than based on scientific facts. Nevertheless, the example demonstrates very nicely what Datta and Mirman (1999) refer to as a biological externality. Such externalities can be negative or positive, depending on the underlying ecological system, and are present next to externalities which arise from the fact that the fisheries are common property as well as dynamic externalities. Obviously biological externalities do not occur in single-species models, which is why the case of multi-species is so important. Without including a component of ecological interaction, game theoretic models for fisheries will produce inefficient fishery strategies and ecological disasters become more likely. The general idea that one should adopt an ecosystem approach to the sustainable use of natural resources in fact underpins many of the resolutions passed by the 2002 World Summit on Sustainable Development (WSSD) in Johannesburg. In particular, the WSSD plan of implementation requires signatory nations to develop and implement an ecosystem approach to fisheries (EAF) by 2012. However, the models considered by Sumaila, Smith, Anderson, Fisher and Mirman as well as Datta and Mirman are neither continuous time nor do they include ecological uncertainty. Both of the latter two aspects are however crucial for the setup of a realistic dynamic model. The aspect of continuous time can in principle be mimicked by using a discrete time model with sufficiently small time increments. In our case however the continuous time setup allows us to compute semi-analytic solutions which are numerically tractable. The inclusion of ecological uncertainty is fundamental and new in this context. In a world of climate change and an increase of sea temperatures with unpredictable effects we consider it asabsolutely necessary. Within our model we derive explicit forms for the strategies and value function, depending on certain constants, which we need to compute numerically by an iteration process. Following this, we discuss how different parameters effect the solution and discuss their economical interpretation. In addition to the setup where each fishery can harvest both species, we also discuss the case where fisheries are restricted to harvest only one of the two species as well as the case where the fisheries cooperate, e.g. are jointly managed. In both cases we discuss the economic consequences as well as the ecological impact. A significant observation is that a competitive ecological system thrives better under cooperative management, while a predatory ecological system thrives better under competition between the fisheries. These we think are very striking policy implications. The remainder of the article is organized as follows. In Section 2, we introduce a model of two ecologically interacting species with uncertainty. We analyze this model and derive a feedback Nash-equilibrium using the Hamilton–Jacobi– Bellman approach. In Section 3, we provide a sensitivity analysis and in Section 4 we consider the cases where (a) regulation restricts each fishery to concentrate on one particular species and (b) fisheries cooperate. We study the economic and biological inefficiencies and consequences resulting from this. Numerical results and examples are presented in Section 5. The main conclusions are summarized in Section 6. 2. A stochastic differential fishery game with ecological interaction The following dynamic is inspired by Hofbauer and Sigmund (1998) (p. 11 for a predator–prey dynamic and p. 26 for a competitive dynamic) adapted to account for ecological uncertainty and harvesting through a number of fisheries N, each harvesting both species1: sffiffiffiffiffiffiffiffiffiffi# " ( ) N X a1 x2 ðtÞ  u1i ðtÞ dt þ s1 x1 ðtÞ dW 1 ðtÞ ð1aÞ dx1 ðtÞ ¼ x1 ðtÞ pffiffiffiffiffiffiffiffiffiffib1 g1 x1 ðtÞ x1 ðtÞ i¼1 sffiffiffiffiffiffiffiffiffiffi# ) N X x1 ðtÞ 2  x2 ðtÞ pffiffiffiffiffiffiffiffiffiffib2 g2 ui ðtÞ dt þ s2 x2 ðtÞ dW 2 ðtÞ x2 ðtÞ x2 ðtÞ i¼1

( dx2 ðtÞ ¼

"

a2

ð1bÞ

Here xi ðtÞ represents the biomass of species i and uji ðtÞ Z0 the rate at which the i-th fishery harvests species j. All pffiffiffiffi coefficients are assumed to be constant. The birth rate of species j is given by aj = xj and the death rate by bj . There are effects on the size of biomass of both species which are due to predation of one species on the other as well as competition. In the absence of fisheries, i.e. N ¼ 0, the case of a pure predator–prey system with species 1 preying on species 2 is 1 The particular functional form making use of the square root function is inspired by Jorgensen and Yeung (1996) who present a single species dynamic (Eq. (1a)) which is related.

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represented by a1 o 0, a2 4 0, b1 ¼ b2 ¼ 0, g1 o 0 and g2 4 0. A purely competitive system is represented by the case where all constants are positive. W1 ðtÞ and W2 ðtÞ are two Wiener processes which for simplicity we assume to be uncorrelated. The parameters si control for the amount of uncertainty in our model. Each fishery tries to maximize its objective functional, given by 9 8 =
where r is the discount rate. Pj ðtÞ and Cij ðtÞ are price and cost functions of species j for agent i, respectively. We assume that the market price of fish species j is determined by the inverse demand function 1 ffi; P j ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN pj i ¼ 1 uji ðtÞ

j ¼ 1; 2

ð3Þ

and the costs for fishery i incurred from fishing for species j is given by cij ; Cij ðtÞ ¼ pffiffiffiffiffiffiffiffiffi xj ðtÞ

j ¼ 1; 2

ð4Þ

where pj and cij are constants. Essentially we choose the same cost functions as in Jorgensen and Yeung (1996) but multiply pffiffiffiffi the price functions by 1= pj , since we study a two species model. Eq. (4) implies that a larger value for the biomass of species j results in lower harvesting costs for this species. We assume that cij ¼ cj and hence our model is time homogeneous and symmetric. Within this framework our objective will be to identify a stationary symmetric feedback Nash-equilibrium. In order to find one, we apply a variation of the finite horizon approximation method, that has been indicated within a specific deterministic example in Dockner et al. (2000, p. 70), and extended to a general stochastic framework in Ewald and Wang (2009) to which we refer for more details on this approach. Following this approach we assume that T is a finite time horizon, and fisheries maximize (2), where the integral runs until T rather than 1. We denote the corresponding value function with Vi ðt; x; TÞ. The Hamilton–Jacobi–Bellman equation for fishery i and a feedback Nash2 1 2 equilibrium ððu1 1 ; u1 Þ; . . . ; ðuN ; uN ÞÞ in this framework is characterized by 8 2 > > < 2 6 X uji cj uj s2j x2j @2 @ 6rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffii þ V ðt; x; TÞ rV i ðt; x; TÞ Vi ðt; x; TÞ ¼ max   4 1 2 P 2 @x2j i @t xj ui ;ui > > :j ¼ 1 pj uji þ kai uj k !#) X pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi @ þ ð5Þ Vi ðt; x; TÞ aj xj bj xj gj x1 x2 uji  uj k @xj kai of the right hand side of (5) is where x ¼ ðx1 ; x2 Þ. A necessary condition for maximizers uj i uj i ¼ 4pj

N3



ð2N1Þ2 cj @ pffiffiffiffi þ Vi ðt; x; TÞ xj @xj

2 ;

j ¼ 1; 2

ð6Þ

Substituting Eq. (6) into Eq. (5), we obtain a partial differential equation 2  2 2 X6 c ð2N1Þ2 cj 2N1 @ 6   j rV i ðt; x; TÞ ¼ 6 pffiffiffiffi þ Vi ðt; x; TÞ p ffiffiffiffi 3 cj @ 4 xj @xj 4pj N xj j ¼ 1 2p N 2 pffiffiffiffi þ V ðt; x; TÞ j xj @xj i 1 0 þ

3

C s B pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi @ ð2N1Þ2 C B þ V ðt; x; TÞBaj xj bj xj gj x1 x2   2 C A @ 2 @xj i cj @ 2 4pj N pffiffiffiffi þ Vi ðt; x; TÞ xj @xj

2 2 j xj

7 @ @2 7 Vi ðt; x; TÞ7 þ Vi ðt; x; TÞ 5 @t @x2j ð7Þ

with boundary condition given by Vi ðT; x; TÞ ¼ 0

ð8Þ

We make a sophisticated guess2 and assume that the solution of (7) with boundary condition (8) is of the following type: pffiffiffiffiffi pffiffiffiffiffi ð9Þ Vi ðt; x; TÞ ¼ A1 ðt; TÞ x1 þ A2 ðt; TÞ x2 þ A3 ðt; TÞ 2 Our ‘‘sophisticated guess’’ is motivated from the functional form of the (infinite horizon) value function in the single species model of Jorgensen and Yeung (1996, Eq. (12)).

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Substituting Eq. (9) into Eq. (7), we obtain a system of ordinary differential equations A10 ðt; TÞ ¼ k1 A1 ðt; TÞ

A20 ðt; TÞ ¼ k2 A2 ðt; TÞ

A30 ðt; TÞ ¼ rA3 ðt; TÞ

2N1 p1 N 2 ð2c1 þ A1 ðt; TÞÞ 2N1 p2 N 2 ð2c2 þ A2 ðt; TÞÞ

þ

þ

ð2c1 þNA1 ðt; TÞÞð2N1Þ2 2p1 N3 ð2c1 þ A1 ðt; TÞÞ2 ð2c2 þNA2 ðt; TÞÞð2N1Þ2 2p2 N3 ð2c2 þ A2 ðt; TÞÞ2

þ

g2 A2 ðt; TÞ

þ

g1 A1 ðt; TÞ

2

2

a1 A1 ðt; TÞ þ a2 A2 ðt; TÞ

ð10Þ

2

where kj ¼ ðr þ bj =2 þ s2j =8Þ. We first consider the case of a two species competitive ecological system, i.e. all coefficients are positive. Following the finite time horizon approximation approach, we have to find out the fixed points of system (10), as these under the appropriate circumstances (see Ewald and Wang, 2009 and the discussion below) correspond to the coefficients of the value function of the infinite time horizon case (2). To derive the fixed points, we have to solve the rational system k1 A1 

k2 A2 

rA3 

2N1 p1 N 2 ð2c1 þ A1 Þ 2N1 p2 N 2 ð2c2 þ A2 Þ

a1 A1 þ a2 A2 2

þ

þ

ð2c1 þNA1 Þð2N1Þ2 2p1 N3 ð2c1 þ A1 Þ2 ð2c2 þNA2 Þð2N1Þ2 2p2 N3 ð2c2 þ A2 Þ2

þ

g2 A2

þ

g1 A1

2

2

¼0

ð11Þ

¼0

ð12Þ

¼0

ð13Þ

In Eq. (13), A3 can be easily derived from A1 and A2 . Therefore, we only concentrate on Eqs. (11) and (12). Note that if we fix A2 in ½0; X2 Þ, with X2 ¼

2N1 2c1 g2 p1 N3

Eq. (11) after multiplication with ð2c1 þ A1 Þ2 becomes a cubic polynomial and it can be shown that Eq. (11) has a unique positive solution. Existence can be proved by the intermediate value theorem and uniqueness can be shown by contradiction. A similar argument works for Eq. (12) if A1 is given and lies in ½0; X1 Þ, where X1 is defined by X1 ¼

2N1 2c2 g1 p2 N3

The following proposition says that the system of Eqs. (11) and (12) has a unique positive pair of solutions. Lemma 2.1 (Competitive case). Eqs. (11) and (12) have a unique positive pair of solutions ðA1 ; A2 Þ, if for ðj; j0 Þ ¼ ð1; 2Þ and (2,1) one of the following conditions holds: ð2N1Þð2N3Þ N2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 c2 pj0 gj ð2N1Þð2N3Þ 2cj kj þ  pj 2pj N 2

Condition ð1Þ

4c1 c2 pj0 gj r

Condition ð2Þ

Xj Z

kj

with Xj defined above. Proof. For j ¼ 1; 2 we define the polynomials   cj ð12NÞ ð2N1Þð2N3Þ fj ðxÞ ¼ kj x3 þ 4cj kj x2 þ 4cj2 kj þ xþ 2pj N2 pj N 3 gj ðxÞ ¼

gj0 2

ð2cj þ xÞ2

ð14Þ

and note that Eqs. (11) and (12) are equivalent to f1 ðA1 Þ þ g1 ðA1 ÞA2 ¼ 0 f2 ðA2 Þ þ g2 ðA2 ÞA1 ¼ 0 Substituting Xj into fj ðxÞ and using either condition (1) or (2), we obtain    ð2N1Þð2N3Þ 2c1 c2 pj0 gj fj ðXj Þ ¼ Xj kj Xj2 þ 4cj kj Xj þ 4cj2 kj þ 40  pj 2pj N2

ð15Þ

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In addition to this we have that fj ð0Þ ¼

cj ð12NÞ o0 pj N3

ð16Þ

We can therefore conclude that there exist zeroes Aj rXj s.t. fj ðAj Þ ¼ 0 and fj ðxÞ o 0 for all x 2 ½0; Aj Þ. We now define the function F11 ðA1 Þ ¼ 

f1 ðA1 Þ g1 ðA1 Þ

ð17Þ

and conclude that for all A1 the pair ðA1 ; F11 ðA1 ÞÞ solves Eq. (11). In the following we show that the function F11 ðA1 Þ is strictly decreasing and strictly concave on ½0; A1 . Computing the derivative leads to 0 F11 ðA1 Þ ¼

f10 ðA1 Þg1 ðA1 Þ þ f1 ðA1 Þg10 ðA1 Þ g12 ðA1 Þ

ð18Þ

We have f1 ðA1 Þ o0 for A1 2 ½0; A1 Þ, while f10 ðA1 Þ; g1 ðA1 Þ; g10 ðA1 Þ and g12 ðA1 Þ are all positive for obvious reasons. Hence we 0 0 ðA1 Þ o0 on ½0; A1 Þ. In order to show concavity, we show that F11 ðA1 Þ is strictly decreasing on ½0; A1 Þ. As the conclude that F11 denominator in (18) is obviously strictly increasing, it is sufficient to show that the nominator is strictly decreasing. Its derivative is given by d ½f10 ðA1 Þg1 ðA1 Þ þf1 ðA1 Þg10 ðA1 Þ ¼ f100 ðA1 Þg1 ðA1 Þ þ f1 ðA1 Þg100 ðA1 Þ dA1

ð19Þ

As before, we conclude from f1 ðA1 Þ o0 and f100 ðA1 Þ; g1 ðA1 Þ and g100 ðA1 Þ all positive by inspection, that expression (19) is indeed negative. We have therefore shown that F11 ðA1 Þ is strictly decreasing and strictly concave on ½0; A1 . Furthermore we have that F11 ðA1 Þ ¼ 

f1 ðA1 Þ ¼0 g1 ðA1 Þ

ð20Þ

In complete analogy to the above, it can be shown that the function G12 ðA2 Þ ¼ 

f2 ðA2 Þ g2 ðA2 Þ

is strictly decreasing and strictly concave on ½0; A2  with G12 ð0Þ ¼ X1

and

G12 ðA2 Þ ¼ 0

ð21Þ

F12 ðA1 Þ ¼ G1 12 ðA1 Þ

We define as the inverse function of G12 . As such F12 ðA1 Þ is strictly decreasing and strictly convex on ½0; X1 . Using Eqs. (21) and (20) we obtain that F11 ð0Þ ¼ X2 Z A2 ¼ F12 ð0Þ

ð22Þ

F11 ðA1 Þ ¼ 0 r F12 ðA1 Þ

ð23Þ

where in the second inequality we used that G12 ð0Þ ¼ X1 implies that F12 ðX1 Þ ¼ 0 and since F12 is strictly decreasing and A1 rX1 that 0 rF12 ðA1 Þ. It now follows from (22) and (23) in combination with the monotonicity and concavity/convexity properties of F11 and F12 , that there exists exactly one point A1 2 ½0; A1  such that F11 ðA1 Þ ¼ F12 ðA1 Þ ¼: A2 By construction this pair ðA1 ; A2 Þ is the unique positive solution of Eqs. (11) and (12).

&

Based on the concavity/convexity combination of F11 and F12 established in the proof of Lemma 2.1 it is easy to derive an iterative scheme, which computes the solution of Eqs. (11) and (12) numerically. We applied this, in order to obtain the numerical results in Section 5, but omit the details here. The following lemma will help us to identify the value function of the infinite horizon problem. Lemma 2.2. Under the assumptions of Lemma 2.1 we have limT-1 Ai ðt; TÞ ¼ Ai for i ¼ 1; 2 independent of t. Proof. The system of differential equation (10) does not explicitly involve calendar time nor the horizon. This allows us to use time to go t ¼ Tt as an argument and to consider A1 ðtÞ :¼ A1 ð0; TtÞ and A2 ðtÞ :¼ A2 ð0; TtÞ instead. Under this setup, system (10) becomes A10 ðtÞ ¼ 

A20 ðtÞ ¼ 

f1 ðA1 ðtÞÞ þg1 ðA1 ðtÞÞA2 ðtÞ ð2c1 þ A1 ðtÞÞ2 f2 ðA2 ðtÞÞ þg2 ðA2 ðtÞÞA1 ðtÞ ð2c2 þ A2 ðtÞÞ2

;

A1 ð0Þ ¼ 0

;

A2 ð0Þ ¼ 0

ð24Þ

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where fj ðxÞ; gj ðxÞ are defined in (14). In the following we will show that limt-1 Aj ðtÞ ¼ Aj from which the statement of Lemma 2.2 follows directly. To do that, we will derive the phase diagram of system (24). Since both denominators in (24) are positive, we have that Aj0 ðtÞ Z 0 3 fj ðAj ðtÞÞþ gj ðAj ðtÞÞAj0 ðtÞ r0

ð25Þ

with equality on both sides, if equality holds on one of the two sides. Using the results obtained in the proof of Lemma 2.1 we obtain that for 0 r A1 ðtÞ rA1 the equivalence (25) implies that A10 ðtÞ Z 0 3 A2 ðtÞ r F11 ðA1 ðtÞÞ

ð26Þ

with equality on both sides, if equality holds on one of the two sides. Similarly we obtain that for 0 r A2 ðtÞ r X1 A20 ðtÞ Z 0 3 A2 ðtÞ r F12 ðA1 ðtÞÞ

ð27Þ

In the proof of Lemma 2.1 we have also shown that the functions F11 and F12 are both strictly decreasing, with F11 being strictly concave and F12 being strictly convex and further that there exists exactly one point of intersection ðA1 ; A2 Þ with A1 2 ½0; A1  and A2 2 ½0; A2 . The phase diagram of the system (24) therefore looks like Fig. 1. Note that on the blue curve F11 we have vanishing A10 ðtÞ and on the red curve F12 we have vanishing A20 ðtÞ. In the area indexed with I, we have A10 ðtÞ Z 0 and A20 ðtÞ r 0 moving the state to the south-east, moving only south if the state hits the blue line. In the area indexed with II, we have A10 ðtÞ Z 0 and A20 ðtÞ Z 0 moving the state north-east, and only east if the state hits the red line. In the area indexed with III, we have A10 ðtÞ r 0 and A20 ðtÞ Z 0 moving the state north-west, moving only west if the state hits the red line. Starting from ðA1 ð0Þ; A2 ð0ÞÞ ¼ ð0; 0Þ it is now already clear that the state will never enter the area indexed with IV, neither will it move outside the picture. It is then obvious, that in fact starting at (0,0) the state will eventually end up in ðA1 ; A2 Þ, the intersection of F11 and F12 which by Lemma 2.1 is the unique positive pair of solutions of Eqs. (11) and (12) and the fixed point of the system (10). This concludes the proof. & It now follows from Lemma 2.2 and Ewald and Wang (2009) that stationary symmetric feedback Nash-equilibrium strategies for our infinite horizon model are given by uj i ðxj Þ ¼

ð2N1Þ2 xj pj N3 ð2cj þ Aj Þ2

;

j ¼ 1; 2

ð28Þ

for agent i ¼ 1; 2. The corresponding equilibrium utility functions are given by pffiffiffiffiffi pffiffiffiffiffi Vi ðx1 ; x2 Þ ¼ A1 x1 þ A2 x2 þ A3

ð29Þ

Note that in Eq. (28), ui ð0Þ ¼ 0. This guarantees that the strategy pair derived is admissible in the sense of Jorgensen and Yeung (1996, p. 384). It can be examined that Eq. (29) is indeed a solution of the Hamilton–Jacobi–Bellman equation for our infinite horizon model. A minor adaptation of a standard verification theorem such as Oksendal (2007, Theorem 11.2.1) implies that the function Vi ðx1 ; x2 Þ is in fact the value function for the stochastic differential game. We omit the details. Note that the finite horizon approximation technique has been applied here to rigorously derive the value function, and not because we need to restrict ourselves to catching-up optimality as criterion, see Dockner et al., p. 62. Jorgensen and pffiffiffi Yeung (1996) simply derive the coefficient A in their value function VðXÞ ¼ A x þ B arguing that A needs to be positive, in order to guarantee that the value function is concave, by which A is uniquely determined from an equation that

5

F11 (A 1)

4.5

F12 (A 1)

4 3.5

A2

3 I

2.5

IV

2 1.5 II

1 0.5 0

III 0

1

2

3 A1

Fig. 1. Phase diagram of the system (24).

4

5

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corresponds to Eqs. (11)–(13) in our model. In our case, we have a far more complex dynamic with interaction between the two states, and it is a priori not clear (neither mathematically nor economically) why the value function should be concave. Now we move to the case of a predator–prey system. Without loss of generality, we assume that a1 o 0, a2 40, b1 ¼ b2 ¼ 0, g1 o0 and g2 4 0, i.e., species 1 is the predator and species 2 is the prey. The following lemma guarantees that Eqs. (11) and (12) have a unique positive pair of solutions in ½0; A1   ½A2 ; X2 , using the same notation as before. Lemma 2.3 (Predator–prey case). The system of Eqs. (11) and (12) has a unique pair of positive solutions in ½0; A1   ½A2 ; X2  if one of the following conditions: Condition ð1Þ

4c1 c2 p1 g2 r

ð2N1Þð2N3Þ N2

2c2 k2 þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 c2 p1 g2 ð2N1Þð2N3Þ  p2 2p2 N2

Condition ð2Þ

X2 Z

Condition ð3Þ

48c22 k2 p2 N 3 Zð2N1Þð2N3Þ

k2

and

hold. Proof. The proof of this lemma is very similar to the proof of Lemma 2.1 and we keep using the same notation. Note that either condition (1) or (2) implies that f2 ðA2 Þ has a root at A2 o X2 and that it still holds that F11 ðA1 Þ ¼ f1 ðA1 Þ=g1 ðA1 Þ is decreasing in ½0; A1 . Condition (3) implies that G12 ¼ f2 ðA2 Þ=g2 ðA2 Þ is a strictly increasing function in ½A2 ; 1Þ, which in return implies that the inverse function F12 exists and is increasing in ½0; 1Þ. Moreover, F11 ð0Þ ¼ X2 ZA2 ¼ F12 ð0Þ and F11 ðA1 Þ ¼ 0 rF12 ðA1 Þ. Therefore, there exists exactly one point A1 2 ½0; A1  such that F11 ðA1 Þ ¼ F12 ðA2 Þ ¼: A2 By construction the pair ðA1 ; A2 Þ is the unique positive solution of Eqs. (11) and (12).

&

Lemma 2.4. Under the assumptions of Lemma 2.3 we have limT-1 Ai ðt; TÞ ¼ Ai for i ¼ 1; 2 independent of t. Lemma 2.4 is proved in exactly the same way as Lemma 2.2 and we omit the proof at this place. 3. Sensitivity analysis In this section, we examine how each of the various parameters affects the optimal utility and harvesting rate. The mathematical derivation of the results is lengthy and in parts non-instructive, they are included in an earlier version of the paper and available on request. Here we will simply present the effects in the form of two tables, and give an interpretation, where we focus in particular on the ecologically relevant parameters as well as the parameters that relate to uncertainty in the model. At the beginning let us consider the case of a competitive ecological system. In the first column we indicate the parameter, and we denote a plus resp. a minus in the adjacent columns if the corresponding variable is increasing resp. decreasing in this parameter, given that all other parameters are fixed. Not included in the table is the parameter aj . It is obvious that A1 and A2 do not depend on aj , A3 however does. In fact, A3 is increasing as either a1 or a2 increases. As we have indicated before, aj is related to the birth rate of species j and it can be seen that a higher birth rate leads to a larger optimal utility. The impact of the number of agents on individuals is that equilibrium utility is reduced when N increases. Furthermore, lim uj i ðxj Þ ¼ lim

N-1

N-1 p

ð2N1Þ2 xj jN

3 ð2c þA Þ2 j j

¼ 0;

j ¼ 1; 2

while aggregate harvest is increasing in N. Note that Aj also depends on N but it is possible to show that it is bounded as a function of N which then trivially implies the result above. From an ecological point of view most important are the effects of the parameters gj which measure the ecological interaction. An increase in the parameter gj means that species j suffers more from the presence of population j0 . This induces the fisheries to reduce their level of harvesting of species j to allow for sufficient growth in species j. On the other side the fishery increases the harvest rate of species j0 , which stabilizes the overall population. The existence of the fisheries have in fact a stabilizing character. The effect of uncertainty on the harvest rate is more surprising. An increase in sj , which means that species j’s inherent fluctuation rate becomes larger, has the effect that the fisheries increase the harvesting rate of the same species j. This is in parts counter-intuitive, if the fishery is risk averse. Essentially the fishery makes a larger investment into an even riskier asset. The explanation is that because of the ecological interaction, part of the uncertainty in species j is transferred to species j0 , even when the two Brownian motions inherent in the model are uncorrelated. It appears that here, this effect is dominant, and therefore the fishery reduces harvesting in species j0 which is indirectly effected by more uncertainty in terms of sj and increases the harvest

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Table 1 Effects of parameters in the competitive case. Parameters

Aj

A j0

uij

uij0

r, N pj , bj , sj cj , gj

 

þ þ

þ þ

 

þ





þ

A1

A2

ui1

ui2

    þ

  þ þ  

þ þ þ þ 

þ þ   þ þ

Table 2 Effects of parameters in the predator–prey case. Parameters r, N pj g1

g2 s1 s2

rate in species j. By keeping the population size of the species that causes the uncertainty in terms of sj down, the fisheries once more have a stabilizing effect (Table 1). Let us now come to the case of a predator prey system with species 1 preying on species 2. Again we omit all proofs and computations and simply summarize the results in Table 2. As before we denote the effects of the parameters with simple þ and  signs. The relationship between r, N and A1 is more subtle and diffuse, we leave this entry blank. Also note that in the predator–prey case g1 is negative, and we present the effect of g1 in the table in terms of g1 . The effect of gj can be easily understood intuitively. A larger g1 leads the biomass of species 1 to increase perhaps due to a better metabolism of the predator species which in effect causes species 2 to decrease due to an increase in numbers of predators. This then raises the fisheries incentives to harvest more of species 1 and less of species 2. Similarly, a larger g2 causes a lower biomass of species 2 perhaps due to a more efficient preying technique of the predatory species, which again gives a higher incentive for the fisheries to harvest more of species 1 and less of species 2. Let us now consider the effect of the uncertainty parameters sj on the optimal harvest rate. An increase in the volatility s1 of the predator population causes the fisheries to increase harvesting of both species. This effect is not present in the case of a competitive ecological system. It is potentially very dangerous as an increased rate of harvest in both species together with an increased level of uncertainty may cause one of the populations, most likely the predator, to fall below a certain critical level. The consequences could be extinction and loss of biodiversity. The reason for this is that we have a transfer effect of volatility between both species as in the case of a competitive system, but that this time they work in the same direction. On the other side, an increase in the volatility s2 of the prey species causes the fisheries to reduce harvesting the predator species and shift towards the prey species. Note that this shift is in the same direction as in the case of a competitive ecological system, but more dangerous for the ecosystem, as the fisheries implicitly support the predator species (by less harvesting), which in effect can bring the prey species down to extinction. We see in this analysis that the effects of parameters on the optimal harvest rate can have very different and in parts contrary implications on the optimal behavior of the fisheries, depending on the nature of the ecological interaction. None of the models that have appeared in the literature so far and that we referred to in the Introduction, were able to indicate these aspects.

4. The cases of one species restricted fisheries and cooperation In the first part of this section, we assume that the fisheries are restricted to harvest only one of the two species. This is a realistic setup, if either a policy maker is able to impose this, or if fishing for the two species demands specific and different fishing vessels. Datta and Mirman (1999) also consider this case and argue that this situation may occur, in the case where fisheries represent different countries, which have different tastes in fish. We focus on the case of a competitive ecological system. The results in the predator–prey case are more diffuse and subtle. Without loss of generality we only consider the case of two fisheries where fishery i is only allowed to harvest species i. This can be easily generalized to the case of M fisheries harvesting species 1 and N fisheries harvesting species 2, leading to the same results, but with weight coefficients that depend on M and N. Nevertheless we now face a non-symmetric game model. The objective functional for fishery i is defined by (Z ) "pffiffiffiffiffiffiffiffiffiffi # 1 ui ðtÞ ci ui ðtÞ rt ð30Þ e maxE pffiffiffiffi  pffiffiffiffiffiffiffiffiffi dtjx1 ð0Þ ¼ x10 ; x2 ð0Þ ¼ x20 ui pi xi ðtÞ 0

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and state equations are given by sffiffiffiffiffiffiffiffiffiffi# " ( ) ai xi0 ðtÞ ui ðtÞ dt þ si xi ðtÞ dW i ðtÞ bi gi dxi ðtÞ ¼ xi ðtÞ pffiffiffiffiffiffiffiffiffi xi ðtÞ xi ðtÞ "

(

a0

i bi0 gi0 xi0 ðtÞ pffiffiffiffiffiffiffiffiffiffi xi0 ðtÞ

dxi0 ðtÞ ¼

sffiffiffiffiffiffiffiffiffiffi# ) xi ðtÞ ui0 ðtÞ dt þ si0 xi0 ðtÞ dW i0 ðtÞ xi0 ðtÞ

ð31Þ

As before, we can apply the finite horizon approximation approach to solve this problem. In an analogous way as in Section 2 we are able to show that under the assumption of 4cj2 k1 k2 pj cj2 pj g1 g2 2kj0 Z 0 the value functions are given by pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi V1r ðx1 ; x2 Þ ¼ Ar1 x1 þAr2 x2 þAr3 resp. V2r ðx1 ; x2 Þ ¼ Br1 x1 þBr2 x2 þ Br3 and the optimal harvesting rates are determined as x1

ur 1 ¼

p1 ð2c1 þAr1 Þ2

ur 2 ¼

;

x2 p2 ð2c2 þBr2 Þ2

with Ar1 and Br2 the unique positive pair that solves     g f ðB2 Þ 3 1 c1 k1 þ 2 þ 2c12 g2 f ðB2 Þ A1  ¼0 A1 þ ½4c1 k1 þ 2c1 g2 f ðB2 ÞA21 þ 4c12 k1  p1 2 2p1     g gðA1 Þ 3 1 c2 k2 þ 1 þ 2c22 g1 gðA1 Þ B2  ¼0 B2 þ ½4c2 k2 þ 2c2 g1 gðA1 ÞB22 þ 4c22 k2  p2 2 2p2

ð32Þ

where f ðB2 Þ ¼ 

gðA1 Þ ¼ 

g1 p2 ð2c2 þ B2 Þ2 2k2 p2 ð2c2 þB2 Þ2 þ 1

g2 p1 ð2c1 þA1 Þ2 2k1 p1 ð2c1 þ A1 Þ2 þ 1

and the other parameters are obtained from " # g Ar 1 Ar2 þ 1 1 ¼ 0 k2 þ 2 2 2p2 ð2c2 þ B2 Þ rAr3 

a1 Ar1 2



" k1 þ

rBr3 

a2 Ar2 2

¼0 #

1 2p1 ð2c1 þ Ar1 Þ2

a1 Br1 2



a2 Br2 2

Br1 þ

g2 Br2 2

¼0

¼0

ð33Þ

Let us now compare the optimal utility and optimal control obtained in the setup of this section to the optimal utility from the previous section. Note that we have used the upper index r in the coefficients and harvesting rates to indicate that they relate to the case where fisheries are restricted. To distinguish them even better from the previous case of non-restricted fisheries, we now use the upper index nr to indicate the corresponding parameters and harvesting rates for the case of nonrestricted fisheries, that we studied in Section 2. In Eq. (11) and the first equation in (32) we have cubic polynomials. Moreover, the coefficients of A31 , A21 and A1 in Eq. (11) are greater than those in the first equation of (32). On the other hand, 0 4 2c12 g2 A2 

3c1 c1 c1 4 2c12 g2 A2  4 8p1 p1 p1

ð34Þ

Similar properties hold for Eq. (12) and the second equation in (32). It can then be proved that the solutions for Eqs. (11) r nr r and (12) are less in value than the solutions of Eq. (32), in particular Anr 1 r A1 and A2 rB2 . Since the optimal harvesting strategies (restricted and non-restricted) are ur 1 ¼

x1 p1 ð2c1 þAr1 Þ2

;

ur 2 ¼

x2 p2 ð2c2 þBr2 Þ2

;

unr ¼ i

9xi 8pi ð2ci þ Anr Þ2 i

; i ¼ 1; 2

we obtain that under one species restriction fisheries harvest less than without restriction. On the other hand, it can be seen that if the following inequality holds: Ar1 Anr Ar Anr pffiffiffiffiffi Br Anr pffiffiffiffiffi Br Anr 1 pffiffiffiffiffi x1 þ 3nr 3r Z x2 Z 1nr 1r x1 þ 3nr 3r nr r A2 A2 A2 A2 A2 B2 A2 B2

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853

then agents obtain more utilities in the case of one species restriction than in the case of competition. The last equation indicates a very important point: under the perspective of social welfare of the fisheries, one species restriction is sometimes advantage, while at other times it is not, according to the current state ðx1 ; x2 Þ of the ecological system. At this point one may well think about a fishery policy maker who actively changes the policy from restriction to non-restriction and vice versa (for example by selling licenses), every time a certain threshold level is reached. We do not consider this point here any further, but would like to point out that this aspect indicates a very important direction in future research in fisheries economics. Let us now consider the case where fisheries cooperate and are not restricted to one species only. The idea is to derive the optimal joint utility function and compare it, with the non-cooperative case. Since our model is symmetric, it is naturally to assume that fisheries share the catch equally among all members. The joint objective functional is given by 9 8 "pffiffiffiffiffiffiffiffiffiffi # =
where vj ðtÞ ¼

N X

uji ðtÞ;

j ¼ 1; 2

i¼1

and the state equations are given by sffiffiffiffiffiffiffiffiffiffi# " ( ) a1 x2 ðtÞ v1 ðtÞ dt þ s1 x1 ðtÞ dW 1 ðtÞ dx1 ðtÞ ¼ x1 ðtÞ pffiffiffiffiffiffiffiffiffiffib1 g1 x1 ðtÞ x1 ðtÞ ( dx2 ðtÞ ¼

"

a

2 x2 ðtÞ pffiffiffiffiffiffiffiffiffiffib2 g2 x2 ðtÞ

sffiffiffiffiffiffiffiffiffiffi# ) x1 ðtÞ v2 ðtÞ dt þ s2 x2 ðtÞ dW 2 ðtÞ x2 ðtÞ

ð36Þ

Obviously this model is consistent with the case N ¼ 1 and therefore, the aggregate optimal harvest rate and joint utility function are given by (28) and (29) under the respective conditions for ecological competition (condition (1b) in Lemma 2.1) and predator–prey (condition (1b) in Lemma 2.3) together with 4cj2 k1 k2 pj cj2 pj g1 g2 2kj0 Z0. Let ðAc1 ; Ac2 Þ denote the nc fixed points obtained from Eqs. (11) and (12) for the case of cooperation and ðAnc 1 ; A2 Þ the fixed point for the case of noncooperation. We have xj xj ð2N1Þ2 xj r rN pj ð2cj þ Acj Þ pj ð2cj þ Anc Þ pj N 3 ð2cj þAnc Þ2 j

ð37Þ

j

since ð2N1Þ2 =N 2 Z1. The second inequality above shows that the aggregate amount of harvest for the case of cooperation is less than in the case of competition. Clark has mentioned several methods to reduce the direct impact of the ecological system, as measured in aggregate harvest rates, and one of them is cooperation of agents, see Clark (2006, Chapter 1). The impact on the ecological system can also be measured in terms of the effect on population size and potential loss of biodiversity, and in fact we reflect on this in Section 5. There, we also provide an example that this result depends crucially on the type of ecological interaction, and that if the underlying ecological system is predator–prey, the relationship may actually become reversed, i.e. non-cooperation between the fisheries may actually be better for a predator prey ecological system. 5. Numerical results In this section, we present some numerical results. To derive a positive pair of solutions to Eqs. (11) and (12) as well as Eq. (32), we apply an iterative method. The idea is to start with a sufficiently large discretization of the set of potential values for A1 i.e. fAi1 ji ¼ 0; . . . ; ng. Substituting each Ai1 into Eq. (12) and solving it numerically by Newton’s method we obtain fAi2 ji ¼ 0; . . . ; ntg. We then examine each pair ðAi1 ; Ai2 Þ via Eq. (11) and find out which Am 1 minimizes the left hand side m of Eq. (11). We then choose ðAm 1 ; A2 Þ as an approximation of the positive pair of solutions. Let us start with the case of a two species competitive system and assume that the coefficients for the ecological system are given by a1 ¼ 1:3, a2 ¼ 1, b1 ¼ 0:6, b2 ¼ 0:55, g1 ¼ 0:9, g2 ¼ 0:7, s1 ¼ 0:15 and s2 ¼ 0:55. For the fisheries we assume that r ¼ 0:2, p1 ¼ 1:85, p2 ¼ 1:65, c1 ¼ 1:45, c2 ¼ 1:3 and N ¼ 2. For this case, we obtain the following set of Figs. 2–5. Note that the parameters have been chosen in compliance with the conditions in Lemma 2.1. Because of limitations in space, we restricted ourself to a particular set of parameters, but qualitatively similar figures are obtained when using other sets of parameters. Figs. 2 and 3 display the difference between the value functions, i.e. optimal utilities, under restriction and non-restriction. They indicate, what we pointed out in Section 4, that depending on the ecological state, fisheries would sometimes prefer to be restricted, and sometimes prefer not to be restricted. In this particular case, with this particular set of parameters, if pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 0:9585 x1 0:61575Z x2 Z0:82154 x1 0:16998

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Fig. 2. Difference of optimal utility functions between restriction and non-restriction for agent 1.

Fig. 3. Difference of optimal utility functions between restriction and non-restriction for agent 2.

The expectations with T = 50

3

Mean for x 1 Mean for x 2

2.5 2 Mean

854

1.5 1 0.5 0

0

10

20

30

40

50

Time Fig. 4. Means of states for competitive ecology and non-cooperating fisheries.

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The expectations with T = 50

3.5

Mean for x1 Mean for x2

3

Mean

2.5 2 1.5 1 0.5 0

0

10

20

30

40

50

Time Fig. 5. Means of states for competitive ecology and cooperating fisheries.

The expectations with T = 500

40

Mean for x 1

35

Mean for x 2

30

Mean

25 20 15 10 5 0

0

100

200

300

400

500

Time Fig. 6. Means of states for predator-prey ecology and non-cooperating fisheries.

fisheries prefer being restricted. Figs. 4 and 5 show the population mean of species 1 and 2 evolving over time with initial conditions x10 ¼ 0:38028 and x20 ¼ 1:0678 the fixed points of the corresponding deterministic population dynamic. In Fig. 4 it is assumed that fisheries do not cooperate, while in Fig. 5 they cooperate. As it can be observed clearly from the two figures, while in the non-cooperative case the population mean of species 2 is driven to a dangerously low level, this is not the case in the cooperative case, Fig. 5. This observation underlines our claim that a competitive ecological system thrives better, under cooperative fisheries management. Let us now discuss the case of a predator prey ecological system, with species 1 as predator and species 2 as prey. We use the following parameters: a1 ¼ 1:3, a2 ¼ 4, g1 ¼ 0:9, g2 ¼ 0:7, b1 ¼ b2 ¼ 0, p2 ¼ 3, c2 ¼ 2 and the remaining parameters similar to the competitive case. In analogy to Figs. 4 and 5, we obtain Figs. 6 and 7. We can now observe, that in the limit, both species are essentially driven to extinction, but that in the case where fisheries do not cooperate, this process is much slower, and significant population numbers of both species exists under non-cooperative fishery management about twice as long as under cooperative fisheries management. This observation underlines our claim from Section 4, that a predator–prey ecology thrives better under non-cooperative fisheries

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The expectations with T = 500

40

Mean for x 1

35

Mean for x 2

30

Mean

25 20 15 10 5 0

0

100

200

300

400

500

Time Fig. 7. Means of states for predator–prey ecology and cooperating fisheries.

management. We would also like to mention that in both cases, competition and predator–prey, the results for one-species restriction are very similar to the case of cooperation. 6. Conclusions We have presented a continuous time game theoretic model of a two species fishery with ecological interaction and uncertainty. This line of research extends and combines previous studies of two species fishery models in discrete time with no uncertainty and the line of stochastic differential game models developed which focuses on one species fisheries. Enough motivation for either of the two research lines has been given in the corresponding literature, but it is only by combining them, that a more realistic model containing uncertainty, continuous time and the ecological aspect can be obtained. The model is richer, but also mathematically more complex. We have shown that under appropriate assumptions our model has a stationary feedback Nash-equilibrium and derived various formulas and equations which characterize it. We also discussed the comparative statics of the model and presented numerical results. We studied the different cases of competitive, restricted and cooperative fisheries management and the differences in their impacts on the ecological system.

Acknowledgments Christian-Oliver Ewald gratefully acknowledges funding under the ARC Discovery scheme as well as support from the research grant ‘‘Dependable adaptive systems and mathematical modeling’’, Rheinland-Pfalz Excellence Cluster and the Royal Society (UK). Both authors are thankful to Prof. David Ulph and Prof. John Harwood for useful comments and suggestions, as well as to two anonymous referees and an Associate Editor for excellent comments and suggestions which greatly helped us to improve the previous versions of this article. References Anderson, L.G., 1975. An analysis of open-access commercial exploitation and maximum economic yield in biologically and technologically interdependent fisheries. Journal of the Fisheries Research Board of Canada 32, 1825–1842. Clark, C.W., 1980. Restricted access to common-property fishery resources: a game-theoretic analysis. In: Liu, P.T. (Ed.), Dynamic Optimization and Mathematical Economics. Plenum Press, New York, pp. 117–132. Clark, C.W., 2006. The Worldwide Crisis in Fisheries: Economic Models and Human Behavior. Cambridge University Press, Cambridge. Datta, M., Mirman, L.J., 1999. Externalities, market power and resource extraction. Journal of Environmental Economics and Management 37. Dockner, E., Feichtinger, G., Mehlmann, A., 1989. Non cooperative solutions for a differential game model of fishery. Journal of Economic Dynamics and Control 13, 1–20. Dockner, E., Jorgensen, S., Long, N.V., Sorger, G., 2000. Differential Games in Economics and Management Science. Cambridge University Press, Cambridge. Ewald, C.-O., Wang, W.-K., 2009. Analytic solutions for infinite horizon stochastic optimal control problems via finite horizon approximation: a practical guide. Available on /http://ssrn.com/abstract=1374216S. Fisher, R.D., Mirman, L.J., 1996. The compleat fish wars: biological and dynamic interactions. Journal of Environmental Economics and Management 30. Haurie, A., Krawczyk, J.B., Roche, M., 1994. Monitoring cooperative equilibria in a stochastic differential game. Journal of Optimization Theory and Applications 81, 73–95.

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