GRETHA UMR CNRS 5113

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Optimal allocation of wetlands: Study on conflict between agriculture and fishery

Natacha LASKOWSKI

GREThA, CNRS, UMR 5113

Université de Bordeaux

Cahiers du GREThA

n° 2013-07

February

Cahiers du GREThA 2013 – 01

GRETHA UMR CNRS 5113

Univers i té Montesquieu Bordeaux IV Avenue Léon Dugui t - 33608 PESSAC - FRANCE

Te l : +33 (0 )5 .56 .84.25 .75 - Fax : +33 (0 )5 .56 .84.86 .47 - www.gretha.f r

Allocation optimale d'une zone humide :

Etude du conflit d'usage entre agriculture et pêcherie

Résumé

Le modèle que nous développons s'intéresse à la question du conflit d'usage sur une zone humide exploitée par l'activité agricole et la pêcherie. A travers la modification de la fonction de croissance logistique et l'introduction de l'activité agricole dans les modèles traditionnels de pêcherie, nous nous intéressons à la taille optimale de la zone humide. Celle-ci sera fonction des profits des agents exploitants les ressources piscicoles d'une part, et les ressources agricoles d'autre part. L'étude statique puis dynamique de ce modèle permettra de tirer des enseignements sur le partage optimal des zones humides puis pourra servir de base l'élaboration de politiques de préservation des ressources.

Mots-clés : Zones humides, optimisation dynamique, capacité de charge, agriculture,

pêcherie, conflit d'usage

Optimal allocation of wetlands:

Study on conflict between agriculture and fishery

Abstract

The model developed here addresses the question of wetland conflict between agricultural production and fishery. Using the modification of the logistic function of growth and the introduction of agricultural activity into traditional fishing models, we consider the optimum allocation of wetlands. This depends on the profits of agents exploiting the fishing resources on the one hand and the agricultural resources on the other. Static followed by dynamic analysis of the model enable us to determine how best to allocate the use of wetlands, and then subsequently using that as a basis for developing sustainable conservation policies for resources

Keywords: Wetlands, dynamic optimisation, carrying capacity, agriculture, fishery, conflict

JEL: Q15, R52

Reference to this paper: LASKOWSKI Natacha (2013) Optimal allocation of wetlands: Study on

conflict between agriculture and fishery, Cahiers du GREThA, n°2013-07.

http://ideas.repec.org/p/grt/wpegrt/2013-07.html.

Introduction

Wetlands constitute some of the most vital environmental resources on our planet .

Once deemed unproductive, many of them have been converted to satisfy the needs

of agriculture for both water and land (Mitsch and Gosselink, 2000). Our insu�cient

understanding of wetlands and of their ecological and economic importance has led to

their being neglected, so that today it has now become urgent to protect and restore

them (Costanza et al., 1989 ; Costanza et al., 1998 ; Mitsch et Gosselink, 2000; de Groot,

Wilson and Boumans, 2002).

Ensuring their preservation is, however, a�ected by the underlying question of use

con�ict, when di�erent forms of activity are in competition within the boundaries of a

single site. This question of use con�ict has already been studied extensively. Bouba-

Olga et al. (2008) suggest addressing it as an issue of social competition concerning the

use of an environmental resource, or as a question of negative externalities which have

to be internalised. Cases of con�ict between aquaculture and �shery have been studied

by Hoagland, Jin and Kite-Powell (2003) and also by Mikkelsen (2007), who highlight

the competition that exists for the use of the area and its biological resources. Their

research attempted to resolve the question of the optimum amount of space required

by each activity within a given wetland area, by using simulations and modi�cations

of the parameters logistic function. This led to fresh light being shed on the negative

externalities that aquaculture can trigger for wild species and their farming.

These case studies, on which our approach is based, address the question of con�ict

between aquaculture and �shery. Here, however, we consider con�ict between �shery and

agriculture. Consequently, we do not study con�ict in terms of resources, but examine

con�ict focused on space, and its use for two di�erent activities. We explore these issues

via the pro�t functions of two agents, a farmer and a �sherman, considering the �sh

species from a biological angle. Using the logistic function of growth enables biological

parameters relating to resources to be integrated, thereby allowing us to examine the

question of con�ict in terms of the sustainability of the species in question.

This research lies at the crossroads of two main bodies of literature. On the one hand,

we examine use con�ict based on the case studies of �sh farming con�icts examined by

Hoagland, Jin and Kite-Powell (2003) and Mikkelsen (2007), but analysed here from

the perspective of the con�ict between �shery and agriculture by means of static and

dynamic modelling. Our static approach di�ers only marginally from that adopted

in the preceding research, even if the parameters it addresses are not the same. Our

dynamic modelling, however, innovates by allowing us to study the evolution of the

variables identi�ed and thus to de�ne an optimum allocation of wetlands between the

two competing commercial activities in place. In addition, from the perspective of

3

formalisation, our research develops and enriches that carried out by Barbier (2000;

2003), which stresses the consequences of a reduction in wetlands surface on the optimum

amount of �sh harvested. Barbier sheds light on the relation that exists between the size

of the zone in which �sh species live and the stock levels of each species. He concludes

that zone size reduction exercises a negative impact on the species. We reach similar

conclusions, in what concerns the link between agricultural activity and �shery, via the

notion of the area's carrying capacity.

The importance of this research is related to its impact in the �eld of policy mak-

ing, as the analysis of each agent's pro�ts and the optimum size of each agents zone,

along with the formulation of a shadow price, could be used to calculate a tax in order

to internalise externalities. The arbitrations highlighted here could then prove useful

for concrete applications, enabling policy makers to establish rules for dividing up the

available space, reconciling each agent's pro�ts with the biological sustainability of the

species. These elements will be examined from a static perspective in Section II, and

then from a dynamic perspective in Section III. Conclusions are then proposed concern-

ing the practical applications of this analysis for environmental policy making.

1 A Static Approach to Economic Modelling of the

Allocation of Wetlands

Our study of the con�ict between the area of the zone available for �shery and agricul-

tural activity is based on the logistic function of growth outlined by Verhulst (1838) and

on the models of Schaefer (1954) and Gordon-Schaefer (1954).

Let us consider wetlands in which agricultural activity occupies a part of the available

space, α (0 ≤ α < 1). The wetlands have a �xed surface area of S. Agriculture therefore

occupies a space which may be expressed as αS , in which S = γK. γ is a positive, �xed

parameter of spatialisation dependent on the characteristics of the natural environment.

K is the carrying capacity of that natural environment. The remaining area (1 - α)

is presumed to be non-anthropised and functioning in its original state. We suppose

the presence of �sh species and an ensuing commercial �shing activity, but without any

agricultural activity. When, however, the two activities are present, the question of

con�icting surface usage arises.

Agriculture, depending on the area it occupies, has a direct impact on wetlands.

We do not consider here any other direct or indirect e�ects, such as pollution of the

land or waterways due to the use of fertilisers or pesticides. Any increase in the total

area occupied by agricultural activity causes clear changes to the nature of the land

itself, making wetlands dry up, thereby reducing the area available for �sh species. We

4

suppose that the relationship between area and �sh production is linear.

Wetland con�ict arises from the fact that a single asset must be shared between

di�erent parties - on the one hand, the land may be exploited in the form of agri-

cultural production and, on the other, the water may be used for �sh farming. This

type of con�ict raises problems, which have not often been studied that of sharing the

same physical area but not the same resource, unlike more typical forms of competition

between di�erent agents for a single resource, as seen in �shery, for instance.

If we apply Verhulst's model of logistic growth, the logistic function expresses the

growth function of the �sh stock x as a function of the intrinsic growth rate r of the

population (r > 0) and the carrying capacity of the natural environment K (K > 0).

Given the negative impact of extension of the agricultural zone on the carrying capacity

of the natural environment, we replace K by K (1 - α) in the logistic growth function.

The logistic function now integrates the presence of agricultural activity in wetlands:

F (x) = rx

(1− x

K(1− α)

)(1)

The intrinsic growth rate of the population r coupled with the environmental resis-

tance factor(1− x

K(1−α)

)indicates the growth rate of the population.As the population

grows in its natural environment, con�ned by K(1−α), F(x) approaches 0. Conversely,

the smaller the �sh stock, the more population growth approaches r. Intuitively, we

understand that the more α approaches 1, the more the �sh population growth func-

tion approaches 0. Thus, if α is close to 1, this implies strong agricultural presence in

wetlands, triggering a fall in the �sh population and impoverishment of the wetlands.

Once the biological equilibrium of the resource at hand had been determined, we

studied the harvest function h de�ned by Schaefer (1954):

h = qEx (2)

with E representing the �shing e�ort mobilised and q the �shing e�ciency. The

�shing e�ort in relation to time t, depends on the number of boats deployed, the crew

size, number of days spent at sea, etc. The catchability q in relation to time t depends on

the given species (some being more di�cult to catch than others). The use of Schaefer's

harvest function posits that catch per unit of e�ort hE is proportional to �sh stock size,

for every level of stock and every level of �shing e�ort, thus presupposing that the

�sh population is evenly distributed throughout its natural habitat (Clark, 1990). The

predatory species here is the �sherman, whose harvest rate is directly linked to �shing

e�ort E, which allows this model to be used to analyse the paired evolution of the �shing

sector and the �sh population from a dynamic perspective and within a context in which

5

the two are inter-dependent. We suppose a reasonable �sh harvest, i.e h ≤ F (x). The

evolution of the �sh stock may then be expressed:

.x= rx(t)

(1− x(t)

K(1− α(t))

)− qE(t)x(t) (3)

At the point of biological equilibrium, the resource in steady state may be expressed

as:

F (x) = h(x) (4)

xSS(E) = K(1− α)(1− qE

r

)

The biological equilibrium of the resource xSS(E) in steady state, with modi�ed

carrying capacity, is lower than the point of equilibrium of the initial logistic function

xSSInitial(E) = (1 − qEr )K. Increased agricultural production, which raises the value of

the variable α, causes the level of the �sh stock in steady state to diminish, for a given

�shing e�ort.

Let us suppose a constant unitary cost of �shing e�ort c, and a constant price market

of the resrouce harvested p. The �sherman's pro�t may therefore be expressed as:

π(x,E) = pqEx− cE (5)

We have seen that the �sh population falls as the size of the agricultural area in-

creases. Given that(∂π∂x

), the �sherman's pro�t will necessarily fall in direct proportion

to the fall in �sh stock.

1.1 The open access situation

In the open access situation, the �sherman's return is equivalent to zero if we posit the

hypothesis of ownership of property rights by the farmer, and inversely if we posit that

all property rights are held by the �sherman. Economic equilibrium is de�ned by a

pro�t equivalent to zero. The Gordon-Schaefer (1954) model serves to analyse the long-

term consequences of �shing e�ort levels. It enables us to express the level of economic

equilibrium of the �shing sector. EOA is obtained by equalising total proceeds against

costs π = 0. This gives us:

EOA =r

q

(1− c

pqK(1− α)

)

EOA is an expression of �shing e�ort in the open access situation. We see that

6

∂EOA

∂α < 0 . This means that, as the area given over to agriculture increases, the �shing

e�ort decreases.

On the basis of biological and economic points of equilibrium, we may express the bio-

economic equilibrium of the resource as:

xOA = xSS(EOA) =c

pq(6)

The result of the Gordon-Schaefer model does not change, despite the introduction

of a variability parameter relative to carrying capacity. The �sh stock level in the open

access situation is not de�ned in relation to the area the �sh may colonise, but exclusively

in relation to the economic parameters and harvesting model. Indeed, xOA is a positive

function of �shing costs and a negative function of �sh prices and catchability. xOA

progression depends exclusively on its correlation with EOA, where these parameters

are �xed.

1.2 Individual optima and social optimum

Let us now look separately at the speci�c programmes of the �sherman and the farmer,

and then that of the social planner:

1.2.1 The Fisherman

The �sherman's aim is to maximise pro�t in relation to �shing e�ort and the amount

of �sh available. We will determine the optimal level of �shing e�ort and �sh stock. To

do so, we must maximise the function of the �sherman's pro�t:

maxE

π(xSS(E), E

)= pqK(1− α)

(1− qE

r

)E − cE (7)

We obtain

E∗ =r

2q

(1− c

pqK(1− α)

)x∗ =

1

2

((1− α)K +

c

pq

)π∗ =

r

4q

((pqK(1− α)− c)2

pqK(1− α)

)

For expressions E* and x*, we note that an increase in carrying capacity has a

positive e�ect on both �shing e�ort and �sh stock, and that conversely, over-present

agricultural activity weighs heavily onK and therefore also on E* and x*. The maximum

7

pro�t made by the �sherman decreases with α.

1.2.2 The Farmer

Let us now consider the pro�ts, expressed as ϕ(α,m), made by the farmer who cultivates

wetlands:

ϕ(α,m) = mαS (8)

wherem is de�ned as the margin or net revenue the farmer makes from his production(∂ϕ∂m

)and where S = γK. We pose the hypothesis of production being homogeneous

over the entire area of land cultivated. As the farmer's pro�t is a positive linear function

of the area cultivated αS, he will seek to maximise this area (optimal situation for the

farmer when α approaches 1). We should note, however, that although the hypothesis of

a variation in α on each occasion is certainly debateable, it is nonetheless indispensable

to our model at this stage.

1.2.3 The Social Planner

The main aims of the social planner are to guarantee ecological sustainability and to

maximise social well-being W. In the present case, we suppose that the maximisation of

W will ful�l these objectives in that the �sherman practices his pro�t-seeking activity

on a sustainable and reasonable basis. We must determine the value of α which max-

imises W, determined by the consolidation of the pro�ts of both agents, the farmer and

�sherman:

W = π∗ + ϕ =r

4q

((pqK(1− α)− c)2

pqK(1− α)

)+mαγS (9)

Finally, we will determine the value of the optimal part of agricultural activity which

best maximises social well-being. The maximisation of social well-being in relation to α

enables us to express this variable optimally:

maxα

W = π∗ + ϕ (10)

s.c. 0 ≤ α ≤ 1

The condition of the �rst order de�ned by ∂π∗

∂α + ∂ϕ∂α = 0 enables us to express the

value of α∗ :

α∗ = 1−2√mγr

(c√p− p

√Kcq

)+ cpr

Kpq(pr − 4mγ)

From an individual perspective, the farmer's pro�t has been maximised thanks to

the area of land cultivated α∗ = 1. From a social perspective, the optimal area of

8

agricultural land will increase or decrease in relation to varying parameters. A study

of the comparative statics allows us to advance a certain number of intuitive results.

We may observe, for example, that the optimal area of agricultural land cultivated is

correlated positively with the carrying capacity of the natural environment, just as it

is for price or catchability, and is negatively correlated with the cost of exploiting the

resource at hand and with the intrinsic growth rate of the stock.

∂(α∗)

∂K

∂(α∗)

∂c

∂(α∗)

∂r

∂(α∗)

∂p

∂(α∗)

∂q

∂(α∗)

∂γ

∂(α∗)

∂m

>0 <0 <0 >0 >0 >0 >0

Table 1: Comparative statics of parameters in α∗

Analysis of the optimal value of the proportion of agricultural land cultivated ?*

enables us to observe the progression of the pro�t made by the farmer in relation to

the progression of the land available for cultivation (Fig. 1). This shows how the pro�t

margin progresses and enables us to calculate the amount of compensation which a

farmer must receive if he is allow a given portion of his land to lie fallow, so that it

may be restored to its initial state as a wetland (from the perspective of a study of

compensation speci�cally designated for restoration projects). These results can help

pose the bases of e�cient environmental conservation policy making from the economic

point of view.

Figure 1: Progression of the farmer's pro�t margin in relation to area of agriculural land cultivated

(Source: author)

As the structure of this model is similar to that set out by Mikkelsen (2007), we may

also therefore apply his results and arrive at the following conclusion: if ∂W∂α > 0, ∀α,

then the wetlands are reserved for agricultural production alone and �shery cannot take

9

place due to the draining e�ect on the submerged areas of land. If ∂W∂α < 0, ∀α then

the wetlands have not been converted into usable farming land and �sh species may

therefore colonise them. These conclusions are valid if we accept the hypothesis that

we are exclusively considering the direct production function of �shery or agriculture.

In the present case, we do not attribute any value to the ecosystem or to the functions

it performs. If we do, however, take this into account, then we might suppose that

the signi�cance of the added environmental value of the wetland ecosystem would tip

the balance in favour of conservation of the zone, its speci�c functions and related

biodiversity.

2 Study of the Economic Model of the Allocation of

Wetlands from the Perspective of Dynamic Optimisa-

tion

2.1 Optimisation

Let us now address the problem from the dynamic perspective. The programme of the

social planner may be expressed as:

maxα(t),E(t)

∫ ∞0

(π(t) + ϕ(t)) e−ρtdt

s.c.

.x= rx(t)

(1− x(t)

K(1− α(t))

)− qE(t)x(t)

x0 = x(0)

where ρ is the discount rate. Our aim here is to maximise well-being in relation to

the constraints of resource dynamics and initial stock. The aim is to establish optimal

control involving two control variables - α(t) and E(t), and one state variable - x(t).

According to the Pontryagin Maximum Principle, the necessary condition which must

be ful�lled if control variables are to maximise the objective function, in light of the

constraints imposed, is that a variable λ(t) also exists which expresses the shadow price

of the resource. On the left side of the following Hamiltonian, we �nd the well-being

function studied in our static analysis, by specifying a quadratic function of the cost for

the �sherman. The current Hamiltonian is associated with the equation as:

H(E(t), x(t), α(t), λ(t), t) = pqE(t)x(t)− cE(t)2 +mγα(t)K

+ λ(t)

(rx(t)

(1− x(t)

K(1− α(t))

)− qE(t)x(t)

)(11)

10

2.2 First Order Conditions of the Hamiltonian

This Hamiltonian should reach its maximum at each moment t for the control and

state variables. In order to do so, the following First Order Conditions (FOC) must be

veri�ed:

FOC 1:

∂H(·)∂α(t)

= 0

Let:

ϕ′

α − λ(t)F′

α = 0 (12)

This enables us to determine the following expression of α(t) along the optimal trajec-

tory:

α(t) = 1− x(t)

K

√λ(t)r

mγ(13)

FOC 2:

∂H(·)∂E(t)

= 0

Let:

π′

E − λ(t)h′

E = 0 (14)

This enables us to determine the following expression of E(t) along the optimal trajec-

tory:

E(t) =qx(t)(p− λ(t))

2c(15)

FOC 3:

−∂H(·)∂x(t)

+ ρλ(t) =.

λ

Let:.

λ +π′

x = λ(t)(F′

x − h′

x + ρλ(t) (16)

This enables us to determine the following expression of λ(t) along the optimal trajec-

tory:.

λ= ρλ(t)− rx(t)(1− x(t)

K(1− α(t))

)− qE(t)x(t)) (17)

Equations 14 and 16 indicate the marginal conditions which the decision variables

must satisfy. The �rst equation indicates that the farmer's pro�t margin depends on

the marginal evolution of his resource and on the price at which the latter is evaluated.

Equation 16 expresses the fact that the �sherman's pro�t margin is equal to the harvest

function evaluated at its shadow price - each supplementary unit of pro�t made from

�shing is determined by each supplementary unit of resource harvested at the shadow

price. Equation 18 indicates how the adjoint variable progresses along its optimal tra-

11

jectory - the term on the left hand side represents the marginal increase in net price

and the marginal increase in the future pro�t then made by the �sherman. The term

on the right hand side represents the value of future harvesting of the resource and the

updated shadow price.

The calculation of Equations 15 and 17 enables us to express the values α(t) and

E(t) in terms of the model's di�erent parameters. The expression in Equation 15 shows

that the area of wetland given over to agricultural activity is a decreasing function of

the ratio x(t)K , which expresses the resource in relation to the natural environment's

carrying capacity, along with the shadow price and natural growth rate of the resource.

On the other hand, Equation 15 progresses in positive manner with the farmer's pro�t

margin and the positive parameter γ. The �shing e�ort E(t)is a positive function of the

catchability of the resource, of stock levels and the price of the resource 1 and is a negative

function of the cost of the �shing e�ort. Equation 19 indicates that the marginal growth

of the net price progresses positively with the updated shadow price and stock level of

the resource, and also with the catchability and �shing e�ort, but progressives negatively

with the carrying capacity of the natural environment..

λ is negatively associated with

changes in the level of �sh stock. Finally, Equation 3 expresses the sustainability of

the �sh stock..x evolves negatively with the level of �shing e�ort and the catchability,

but positively with the rate of intrinsic �sh population growth and the environmental

resistance factor, the latter being negatively correlated with the variable α(t).

The results are a �rst step towards implementing e�cient policy making rooted in

optimal allocation of the wetlands available. Once appreciated at is implicit price λ(t),

the carrying capacity will be far more e�ciently conserved. The implicit price also

serves as a basis for taxing agricultural use of wetlands, thus enabling decision makers

to achieve the social optimum in terms of how such areas are allocated in cases of

con�icting use. The shadow price is the marginal cost of relaxing the constraint, or, put

otherwise, the marginal pro�t of strengthening the constraint. It can be interpreted as

being the gain in both current and future social well-being associated with a unitary

rise in the evolution of today's �sh stock.

2.3 General properties of the system

The existence of a steady state is proven by the concavity of the Hamiltonian ∂2H∂x2(t) < 0.

In order to determine the maxima of a constrained maximisation problem, we must

study the saddle point calculated from the Jacobian matrix and the calculus of the

determinant. The negative value of the determinant con�rms the existence of a saddle

point and, therefore, of a local maximum (Annex 1). Based on the expressions of.

λ

1We put p− λ(t) > 0 because E(t) > 0

12

and.x determined by the formulations of FOC n 3 and Equation 3 respectively, we may

proceed to determine the values of.

λ and.x :

.

λ = ρλ− ∂H(·)∂x

= ρλ−(pqE + λ

(r − 2rx

K(1− α)− qE

)).x = ∂H(·)

∂λ = rx

(1− x

K(1− α)

)− qEx

2.4 Expression of stady state variables

From the �rst FOC, in which we have λ =ϕ′α

F ′αwe may obtain the value of λ by replacing

this equality in (17) and solving:

.

λ= 0 : λ =mγK(1− α)

pqEx

(1

x

(qE

r− ρ− 1

)+ 2

)

We may also de�ne the value of x by solving.x= 0 (also present in (4)):

x =

(1− qE

r

)K(1− α)

From the values of E in expression(15) and α in expression (13), we may also deter-

mine the values of the �rst three steady state variables 2.

xSS =2Kcq

cr +Kq2(p− λ)

ESS =Kq2(p− λ)

Kq2(p− λ) + 2cr

αSS = 1− 2cp

Kq2(p− λ) + 2cr

√λr

m

λSS =

3

√C

2A+

√C2

4A+

B3

27A+

3

√C

2A−√C2

4A+

B3

27A

2

The table below sums up the study of the comparative statics of Equations xSS

and αSS . It enables us to observe the evolution of steady state variables relative to

the parameters composing them. This allows us to con�rm a number of intuitions:

carrying capacity is positively correlated with the stock variable in a steady state, while

an increase in its price triggers the opposite e�ect on both variables. Moreover, new

light is shed on a further interesting intuition, i.e. the negative relation between the

surface area of agricultural land in a steady state and the shadow price of the resource.

Indeed, the shadow price enables the regulator to regulate �sh stock resources in relation

2 The resolution of λSS is proven in Annex 2

13

to their initial state. A stock which is initially low may be bolstered by the regulator,

either by acting on the level of �shing e�ort ESS , or by increasing the shadow price

of the resource, envisaged as a tax on agricultural activity to discourage farmers from

farming potential wetlands.

∂(·)∂K

∂(·)∂c

∂(·)∂r

∂(·)∂p

∂(·)∂q

∂(·)∂m

∂(·)∂λ

xSS >0 >0 <0 <0 >0 - >0

αSS >0 <0 >0 <0 >0 >0 <0

Table 2: Comparative statics of variables x and α in a stady state

2.5 Phase Diagram

The phase diagram below illustrates the progression of stock and variables and the

shadow price, together with the presence of a steady state represented by the junction

of xSS and λSS . From Equations (13) and (15), we may re-write expressions of.

λ and.x

:

.x= rx

(1−

√mγ

λr

)− q2x2(p− λ)

2c

.

λ= ρλ− rx(1−

√mγ

λr− q2x2(p− λ)

2c

)

This in turn enables us to express the equations of the isoclines of the phase diagram

(cf. Annex 4) and to observe that the isocline equations are not common to the standard

framework. This leads to the elaboration of an inversed diagram. However, to simplify

the reading of this, we have maintained the standard axes while still allowing for our

speci�c case. The existence of a unique steady state is a function of the position of

the curves on the y-axis: if the curve.

λ= 0 is left of the curve.x= 0, the existence of

an unstable steady state can be envisaged. Such a situation is, however, unlikely and

would only be possible with speci�c values for some parameters. Indeed, an unstable

steady state is possible with excessively high values of p and m 3, which may be logically

explained by the predominance of one activity over the other and, therefore, the absence

of any system regulating con�icting uses of land.

The phase diagram below was built on a simulation of parameter values based on

studies carried out in the �eld of the logistic function (notably research carried out by

3 As simulation of the values of the parameters was carried out in sequential fashion, we cannot thereforeconclude about any combined e�ect of a rise or drop concerning several parameters simultaneously.

14

Conrad (1999) and Seijo et al. (1998)). Dynamic analysis of wetland con�ict proves

the existence of a steady state in which the shadow price could decentralise the optimal

state, from which we may then calculate the level of taxation necessary to internalise

externalities and guarantee the stability of the resource in its natural environment.

The phase diagram characterises the evolution of the dynamic system. Analysis of the

trajectories (Annex 3) enables us to trace the trajectories of our diagram and to observe

the regions which converge towards a steady state.

Figure 2: Optimal trajectories and steady state (Source: author)

The aim of the regulator is to tend towards a steady state and then to adopt a

position based on a point which tends towards this state. Indeed, depending on the

initial value of the stock, the regulator determines a shadow price which will allow this

to happen, either by acting on E(t) to limit or encourage �shing activity, or on α(t) to

limit or encourage agriculture. For example, if the initial stock level is x0 < xSS , the

regulator will choose a value of λ which will allow him to position himself on a trajectory

which converges towards a steady state. If the initial condition x0 is weak compared with

the point of equilibrium, the regulator will choose a shadow price which is su�ciently

high λ0 to discourage excessive exploitation of the resource, thereby providing favourable

conditions for natural stock replenishment. As λ is negatively correlated with the α(t)

and E (t)variables, any increase in the shadow price would cause a diminution of the

�shing e�ort and the area cultivated. We may not draw any conclusions here as to the

15

greater or lesser impact this might have on either activity. This would depend �rst and

foremost on the values of the parameters and the relative e�ciency of these commercial

activities.

If, however, the initial stock level is relatively high in comparison with the point of

equilibrium, x1 > xSS , then the regulator will aim at reducing the stock level. To do so,

he will choose a shadow price which is su�ciently weak λ1 to encourage either �shing

activity, through a variation in the �shing e�ort, or agricultural activity, through an

increase in the area of land available for cultivation. Therefore, depending on the initial

level of the stock, the regulator will determine a shadow price which will enable him

to position himself at A or B, thereby placing himself on a trajectory which converges

towards a steady state.

Conclusion

The main issue at stake in this research was to understand how best to allocate the use

of wetlands between competing commercial activities - �shing and agriculture. Although

our analysis of this question used previous research carried out by Barbier (2000 and

2003), Hoagland, Jin and Kite-Powell (2003) and Mikkelsen (2007), we have developed

and enriched their ideas by including a new element - agriculture - and a new method

- dynamic analysis. While the conclusions which may be drawn from our model are

subject to the supply and demand conditions of the market, the model, which accounts

for the biological elements of the species harvested, still remains highly original, thanks

to the logistic growth function. Indeed, the integration of this function adds a further

dimension to the model - that of sustainable development. The preservation or restora-

tion of wetlands can favour the development of numerous ecosystem goods and services,

notably in the �eld of �sh farming when it is managed on a sustainable basis.

The aim of this research was to study the impact of the reduction in surface area of

wetlands on �sh species and the ensuing economic consequences. In line with Barbier

(2000), our conclusions highlight the existence of a positive correlation between loss

of habitat and a decrease in the level of �sh stock. Arbitration and the de�nition

of an optimal size for each species is carried out with the economic interest of both

�shing and agricultural activity in mind. It would be possible to render this model

more complex in the future by integrating the question of contaminating substances

(pesticides, fertilizer?) used in agricultural production. To do so, we would need to

study in greater detail work carried out by Feunteun (2002) and Courrat et al. (2009)

dealing with the e�ect of polluting substances on �sh species in estuarine areas.

Finally, our paper is also relevant to the �eld of economic policies and is designed

16

to help decision making concerning the e�cient management of resources and highly

valuable ecological and economic zones. Our work would be useful in designing a tool

to encourage the restoration of wetlands thanks to the inclusion of a shadow price and

to the analysis of functions of pro�t margins of agents pro�ts.

References

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Annexes

Annex 1

Study of the sign of the determinant in the Jacobian matrix

.

λ = ρλ−(pqE + λ

(r − 2rx

K(1− α)− qE

))= 0

.x = rx

(1− x

K(1− α)

)− qEx = 0

J(x∗, λ∗) =

.xx (x∗, λ∗)

.xλ (x∗, λ∗)

.

λx (x∗, λ∗).

λλ (x∗, λ∗)

J(x∗, λ∗) =

< 0 = 0

> 0 > 0

⇒ detJ < 0

19

Annex 2

The resolution of λSS is more complex and requires the use of speci�c mathematical

resolution methods. After replacing the values of xSS , ESS and αSS in the expression

of λ below, we obtain a simpli�ed equation of the form:

λ3/2(Kq2√r − Eq4K

√mγ

)+ λ1/2

(Eq2√mγ

(2cr +Kpq)

)−(2cr3/2 +Kpq

√r)= 0

Let Z =√λ we obtain a cubic equation of the type AZ3 +BZ − C = 0 where

A =

(Kq2√r − Eq4K

√mγ

)B =

(Eq2

√mγ(2cr +Kpq)

)C =

(2cr3/2 +Kpq

√r)

The so-called Cardano method, set forward by Cardano in 1545, provides the means

for solving cubic polynomial equations of the type Z3 + BZ − C = 0 by using the

following formula:

Z =3

√C

2+

√C2

4+B3

27+

3

√C

2−√C2

4+B3

27

The solution to our problem is therefore: λSS =

3

√C

2A+

√C2

4A+

B3

27A+

3

√C

2A−√C2

4A+

B3

27A

2

Annex 3

Analysis of the trajectories was carried out using derivations of the isocline equations of

the diagram (x| .x=0 and x| .

λ=0). The results below enable us to mark out the arrows of

the diagram, thereby indicating the direction in which our diagram is evolving.

∂.x (x, λ)

∂x< 0

∂.

λ (x, λ)

∂x> 0

∂.x (x, λ)

∂λ> 0

∂.

λ (x, λ)

∂λ<

0

N-E N-W S-E S-W

(+,+) (+,-) (-,+) (-,-)

Table 3: Comparative statics in α∗

20

Annex 4

The equations of the isoclines in the phase diagram are as follows:

x| .x=0 =r(1−A)

B

x| .λ=0

=

(1−A)3B

3

√√√√√√√√√√√ A

9B3

(1−A+

A2

3− 1

3A

) +λ2ρ2

4B2r2− λρ

2Br

+3

√√√√√√√√√√√ A

9B3

(1−A+

A2

3− 1

3A

) +λ2ρ2

4B2r2− λρ

2Br

Where A =

√mγ

λrand B =

q2(p− λ)2c

, which enables us to re-write.

λ (x, λ) =

ρλ− rx(1−A−Bx2) and .x (x, λ) = rx(1−A)−Bx2.

21

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