A Mathematical Model of a Fishery with Variable Market Price: Sustainable Fishery/Over-exploitation

REGULAR A RTI CLE

A Mathematical Model of a Fishery with VariableMarket Price: Sustainable Fishery/Over-exploitation

Fulgence Mansal • Tri Nguyen-Huu •

Pierre Auger • Moussa Balde

Received: 12 December 2013 / Accepted: 4 June 2014

� Springer Science+Business Media Dordrecht 2014

Abstract We present a mathematical bioeconomic model of a fishery with a

variable price. The model describes the time evolution of the resource, the fishing

effort and the price which is assumed to vary with respect to supply and demand.

The supply is the instantaneous catch while the demand function is assumed to be a

monotone decreasing function of price. We show that a generic market price

equation (MPE) can be derived and has to be solved to calculate non trivial equi-

libria of the model. This MPE can have 1, 2 or 3 equilibria. We perform the analysis

of local and global stability of equilibria. The MPE is extended to two cases: an age-

structured fish population and a fishery with storage of the resource.

F. Mansal (&) � P. Auger � M. Balde

Departement de mathematiques et informatique, Faculte des Sciences et techniques, UMI IRD 209,

UMMISCO, IRD, Universite Cheikh Anta Diop, Dakar, Senegal

e-mail: [email protected]

P. Auger


M. Balde


T. Nguyen-Huu � P. Auger

UMI IRD 209, UMMISCO, Centre IRD de l’Ile de France, 32 avenue Henri Varagnat,

93143 Bondy Cedex, France


T. Nguyen-Huu

IXXI, ENS Lyon, Lyon, France

T. Nguyen-Huu � P. Auger

UPMC, Sorbonne University, Pierre et Marie Curie-Paris 6, Paris, France

123

Acta Biotheor

DOI 10.1007/s10441-014-9227-7

Keywords Dynamical systems � Fishery � Variable price � Market price equation �Demand function � Equilibrium � Stability � Sustainable exploitation/

overexploitation

1 Introduction

Fishery modelling aims at understanding the dynamics resulting from fishing

activities for ecological purpose, which aims at avoiding extinction of some species,

and economical purpose, which aims at providing a regular and optimal income.

Such models usually represent the evolution of the fish stock as well as economical

aspects such as changes of the fishing effort in response to higher or lower profits.

There was a lot of interest in bioeconomic modelling mainly from the point of view

of control theory (Clark 1990; Meuriot 1987) or optimization (Doyen et al. 2013).

We also refer to a book about management of renewable resources (Clark 1985,

2006; Lara and Doyen 2008).

Most mathematical models consider economical aspects of open-access fisheries:

boats can join or leave the fishery depending on the profit generated. However, they

ignore another important economical aspect related to free market: balance between

supply and demand set prices for the resource and therefore influence profits. As a

consequence, those models consider the price of the resource as a constant (Prellezo

et al. 2012), and demand is assumed to match supply. To our knowledge, few

contributions considered a variable price or a price depending on the catch (Smith

1968, 1969; Barbier et al. 2002). The aim of this work is to present a fishery

bioeconomic model which improves a class of classical fishery models by adding

market effects and price variation. Indeed, according to classical economic theory

(Walras 1874), the price variation depends on the difference between demand and

supply. So some questions in the elementary theory of supply and demand are

studied in renewable resource exploitation (see Clark 1990 section 5.2). The

originality of our work is to take into account explicitly the variation of the price

due to the law of supply and demand, the price being a variable of a dynamical

system. The main point of this model is thus to add an extra equation for the market

price to a classical fishery model. We assume that the demand is a linear function of

price such as in Lafrance (1985). Such a linear function with a maximum value

A and a maximum price (reserve price) over which demand is null is common (see

Mankiw 2011). The supply is given by the instantaneous catch.

In (Auger et al. 2010) some of the authors investigated a fishery model with a

variable price with time scales. In this previous work, one assume that the price

was varying at a fast time scale while the fish growth and the catch varied at a

slow time scale. Using aggregation of variables methods (Auger and Bravo de la

Parra 2000; Auger et al. 2008; Iwasa et al. 1987, 1989), the initial model has

been reduced. The aim of this work is to generalize the previous study to a

model without time scales.

Moreover, in Auger et al. (2010), the demand function DðpÞ was a linear

monotone decreasing function of price p with slope equal to -1, i.e. DðpÞ ¼ A� p

F. Mansal et al.

123

where A is the maximum demand. In the present paper, we consider a more general

case with a slope �a, i.e. DðpÞ ¼ A� ap. The study will show that this parameter a,

which represents how much an increase of the price decreases demand, plays an

important role in the dynamics of the system. We also extend the model to new

cases such as an age-structured fish population and to a fishery with storage.

This paper is organised as follows : In Sect. 2, we present the mathematical

model of a fishery with a variable price. In this part we study analytically the model

and we give a theorem with proof. We show phase portraits corresponding to the

different cases. In Sect. 3, we extend our model to an age structured population

model with juveniles and reproductive adults. In Sect. 4, we then extend the model

to a fishery model with storage of the resource. The work ends with a conclusion

and some perspectives.

2 Mathematical Model of a Fishery with a Variable Price of the Resource

We introduce a model of a coastal fishery and we consider the total coastline as a

single site. Let nðtÞ be the fish stock and EðtÞ the fishing effort at time t. The

following system describes the time evolution of the fishery:

dn

dt¼ rn 1� n

k

� �� qnE

dE

dt¼ pqnE � cE

dp

dt¼ u DðpÞ � qnEð Þ

8>>>>>><>>>>>>:

ð1Þ

Without any fishing activity, the fish population grows logistically, r [ 0 being the

fish growth rate and k [ 0 the carrying capacity (first equation). It is exploited

according to a classical Schaefer function where q is the fish catchability per fishing

effort unit. The quantity of fish harvested per time unit qnE is then proportional to q,

the fishing effort E and the fish stock size n.

c [ 0 is the maintenance cost per fishing effort unit and time unit. The profit is

then the difference between the revenue provided by selling harvested fish (pqnE)

and the costs of the fleet. The second equation reflects that the fishing fleet expands

when making profits, and decreases when the fishery is losing money.

The third equation describes the evolution of market price, which increases when

there is more demand than offer, according to classical economic theory (Walras

1874). It takes into account the demand function which is assumed to be a

decreasing linear function of the price given by DðpÞ ¼ A� apðtÞ, where A and aare positive constants which represent the maximal demand and the rate at which

the demand decreases with price. The variation of price is proportional to the

difference between demand and supply, with a coefficient of proportionality u [ 0.

We now perform a mathematical analysis of model (1) by determining possible

equilibria and their stability.

A mathematical model of a fishery with variable market price

123

2.1 Existence of Equilibria

Model ð1Þ has the following nullclines:

• The n-nullclines correspond to n ¼ 0 and E ¼ rqð1� n

kÞ;

• The E-nullclines correspond to E ¼ 0 and n ¼ cpq

;

• The p-nullclines correspond to p ¼ A�qnEa .

We determine three kind of equilibria from those nullclines: equilibrium

n0 ¼ ð0; 0; AaÞ, which corresponds to the extinction of fish population; equilibrium

nk ¼ ðk; 0; AaÞ, which is attained when there is no fishing; and a positive equilibria of

the general form n� ¼ ðn�;E�; p�Þ, where n� ¼ cp�q and E� ¼ r

q1� c

p�qk

� �both

depend on the price p�.Third equation gives that non-trivial equilibria verify

Dðp�Þ ¼ rc

p�q1� c

p�qk

� �ð2Þ

Equation (2) is called the Market Price Equation (or MPE). There can be up to three

positive equilibria (see Appendix 1).

Theorem 1 System (1) may have up to three positive equilibria:

• If a[ qkA=c, there is no positive equilibrium.

• If a\qkA=c and if k r\3A, there is exactly one positive equilibrium.

• If a\qkA=c and if kr [ 3A, there are three cases:

1. if a\a�: there is one and only one positive equilibrium;

2. if a�\a\aþ: there are three positive equilibria;

3. if aþ\a: there is one and only one positive equilibrium.

where

a� ¼ q�2r2k2 þ 9rAk � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27kr kr

3� A

� �3q

27rc

ð3Þ

and

aþ ¼ q�2r2k2 þ 9rAk þ 2


3� A

� �3q

27rc

ð4Þ

Proof The proof is detailed in Appendix 1.

2.2 Analysis of Local Stability

The Jacobian matrix associated to system (1) reads:

F. Mansal et al.

123

J ¼r 1� 2n

k

� �� qE � qn 0

pqE pqn� c qnE

�uqE � uqn � ua

26664

37775

We now determine the local stability at each equilibrium point.

1. For n0 ¼ ð0; 0; AaÞ, the Jacobian matrix reads:

J0 ¼r 0 0

0 � c 0

0 0 � ua

264

375

J0 has one positive and two negative eigenvalues. Equilibrium n0 is a saddle point

(unstable).

2. nk ¼ ðk; 0; AaÞ, the Jacobian matrix reads:

Jk ¼

�r � qk 0

0Aqk

a� c 0

0 � uqk � ua

2664

3775

• If a[ qkA=c, nk is a stable equilibrium;

• If a\qkA=c, then nk is a saddle point (unstable).

3. At equilibria n�, the Jacobian matrix reads:

J ¼� rn�

k� qn� 0

p�qE� 0 qn�E�

�uqE� � uqn� � ua

2664

3775

Theorem 2 There are two cases for stability of positive equilibria of system (1):

• if A [ rk=3, there exists a unique positive equilibrium n� which is locally

asymptotically stable.

• if A\rk=3, a positive equilibrium n� is locally asymptotically stable if and only

if p�\p� or p�[ pþ, where

pþ ¼kr þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � 3AÞ3

q

Akqc and p� ¼

kr �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � 3AÞ3

q

Akqc:

There are three different cases:

1. if a\a�, there is one positive equilibrium which is stable;


123

2. if a�\a\aþ, there are three equilibria n�1, n�2 and n�3 (ordered by

increasing values of p�i ). Equilibria n�1 and n�3 are locally asymptotically

stable, while n�2 is unstable;

3. if a[ aþ, there is one positive equilibrium which is stable.

Proof The proof is given in Appendix 2.

2.3 Typology of Dynamics

Theorem 3 There exists a bounded set Xþþ1 such that any trajectory with a

positive initial condition has its x-limit in Xþþ1 .

Proof We present here the main lines, details are provided in Appendix 3. We

introduce a Lyapunov function V and divide the space into two parts: a set X on

which V admits a maximum (see Lemma 5 and 6) and its complementary set, on

which _V � 0. From those two sets, we define a new set X1 which includes X and

which is forward invariant according to Lyapunov theory (Lemma 7). Furthermore,

any trajectory enters Xþ1 the intersection of X1 with the set defined by p� 0

(Lemma 7 again). Finally, we show that for any initial condition in Xþ1, the

trajectory stays bounded in a compact set Xþþ1 , which ends the proof.

This theorem implies that all trajectories enter a compact set Xþþ1 , which means

that they are positively bounded in a domain containing the different equilibria. We

now summarize the different cases obtained for local stability inside this domain.

For the following results, we checked numerically that there were no limit cycles

nor chaotic behavior.

• Case 1: a [ qkA=c: there is one saddle point (n0) and one stable equilibrium

(nk). When a [ qkA=c, there is no positive equilibria, and the system tends

toward equilibrium nk. The case is illustrated in Fig. 1.

• Case 2: a\qkA=c: equilibrium nk is unstable. There exists at least one positive

equilibrium. There are two subcases:

1. A [ rk=3: there is only one positive equilibrium, which is locally

asymptotically stable. The dynamics is represented in Fig. 2.

2. A\rk=3: there can be one to three positive equilibrium, depending on the

value of a. The case with three equilibria is represented in Fig. 3, while the

case with one equilibria, which is similar to the previous case, is not

represented.

The phase portrait in the general case (three equilibria) is represented in Fig. 4.

2.4 Interpretation and Comparison of Fish Price in the Case of Two Stable

Positive Equilibria

In case 1 (a[ qkA=c), the system tends toward and equilibrium composed of a fish

population at carrying capacity and no fishing activity. The demand decreases too

F. Mansal et al.

123

fast with price, and fisheries cannot be profitable. Condition for case 1 can be

rewritten qkA=a\c and can be interpreted in the following way: at maximum

harvest rate (fish population at carrying capacity, n ¼ k), and maximum price

(p ¼ A=a), the cost is greater than income. Then the fishery will never be profitable.

In case 2 (a\qkA=c), the fishery would be profitable if the fish stock could be

maintained at carrying capacity and price at it maximum. This is not possible to

keep the system in this state, but there are equilibria for which fishing effort is

Fig. 1 a[ qkA=c. Left demand (black) and offer (f ðpÞ for n ¼ c=pq and E ¼ rð1� c=pqkÞ=q) (grey)depending on p. The two curves do not intersect, hence there is no non-trivial positive equilibrium. RightTime series of the dynamics: fish stock (black), fishing effort (grey) and price (dotted). The initialsconditions are: 1, 2, 2 Parameters are r ¼ 0:9, k ¼ 3, q ¼ 0:5, c ¼ 0:6, A ¼ 0:29375, a ¼ 0:775 andu ¼ 1

Fig. 2 a\qkA=c and A [ rk=3. Left demand (black) and offer (grey) (f ðpÞfor n ¼ c=pq andE ¼ rð1� c=pqkÞ=q), depending on p. The two curves intersect at p�, corresponding to equilibrium n�.Right Time series of the dynamics: fish stock (black), fishing effort (grey) and price (dotted). The initialsconditions are: 3, 0:1, 3 Parameters are r ¼ 0:9, k ¼ 3, q ¼ 0:5, c ¼ 0:6, A ¼ 1:1, a ¼ 0:35 and u ¼ 0:5


123

positive and the fishery profitable. The number of equilibria depends either on the

maximum demand A and the rate a at which the demand decreases with price.

In the case with three positive equilibria that we denote n�1, n�2, n�3, with

n�i ¼ ðn�i ;E�i ; p�i Þ, and ordered by increasing value of p�i . The middle equilibrium n�2(p�1\p�2\p�3) being a saddle node while the two other equilibria are locally

asymptotically stable. Since a[ 0, we have n�3\n�1. A straightforward calculation

gives the following set of inequalities:

n�3\n�1;

E�3 [ E�1;

p�3 [ p�1

8><>:

ð5Þ

In other words, at equilibrium, the larger is the fish stock, the smaller is the fishing

effort and the smaller is the market fish price. As a consequence, the model predicts

that we can have two kinds of fishery:

• An over-exploited fishery n�3: there is a large fishing effort and an important

economic activity with a satisfying market price (Ekouala 2013). However, the

resource is maintained at a low level and due to some environmental changes,

there exists a risk of fish extinction.

• A traditional fishery n�1: the fishery maintains the fish stock at a desirable and

large level which is far from extinction. This is a sustainable equilibrium

(Ekouala 2013). Artisanal fisheries would correspond to such a case where the

resource is not over-exploited and allows local fishery activity. However, it does

Fig. 3 a\qkA=c and A\rk=3. Left Demand (black) and offer (grey) (f ðpÞfor n ¼ c=pq and

E ¼ rð1� c=pqkÞ=q), depending on p. The two dotted lines represent the demand for a� and aþ.Between the two black dotted lines, the two curves intersect 3 times, and only 1 outside. The black dotted

lines correspond to p ¼ p� and p ¼ pþ. When p�\p�\pþ, corresponding equilibrium n� is unstable,

while outside this area, n� is locally asymptotically stable. If a [ aþ or a\a�, there is only oneequilibrium. The dynamics is similar to the one in Fig. 2. Right Time series of the dynamics: fish stock(black), fishing effort (grey) and price (dotted). The initials conditions are: 3, 3:10�4, 0:4. Parameters arer ¼ 0:9, k ¼ 3, q ¼ 0:5, c ¼ 0:6, A ¼ 0:775, a ¼ 0:146 and u ¼ 1

F. Mansal et al.

123

not permit an important economic activity and can only support a rather small

fishing effort with a relatively small market price.

This model predicts that two types of fisheries are possible. An interesting concern

relates to the possibility to control the system and to switch from an over-

exploitation situation to a sustainable (artisanal) fishery. This question was

investigated in a paper to appear in the case of a multisite fishery (Ly et al. 2014).

3 Generalisation to a Population Model Structured in Age Classes

The following model describes the time evolution of population structured in two

age classes, juveniles (age class 1) and reproductive adults (age class 2). The model

reads as follows:

_n1 ¼ bn2 � vn1 � l1n1;

_n2 ¼ vn1 � l2n2 � bn22 � qn2E;

_E ¼ Eðpqn2 � cÞ;_p ¼ uðDðpÞ � qn2EÞ

8>>><>>>:

ð6Þ

Fig. 4 a\qkA=c and A\rk=3. Phase portrait of the dynamics. Equilibria are represented as grey circles,and heteroclines as grey curves. The black curves represent trajectories tending asymptotically toward thelocally asymptotically stable equilibria. The initials conditions are: (0, 0, 2) and (0, 3, 0). Parameters arer ¼ 0:9, k ¼ 3, q ¼ 0:5, c ¼ 0:6, A ¼ 0:775, a ¼ 0:146 and u ¼ 1


123

where b is the adult reproduction rate, v the juvenile aging rate, li is the mortality

rate for age class i. b is a Verhulst parameter for adults competing for some

resource. Other parameters are the same as in the previous model.

The E-isoclines are E ¼ 0 and pqn2 � c ¼ 0. We deduce that n�2 ¼ cpq

, and then

that n�1 ¼ bcpqðvþl1Þ. The p isoclines are given by

DðpÞ ¼ qn2E ¼ vn1 � l2n2 � bn22

At equilibrium, we have

Dðp�Þ ¼ Rn�2 1� n�2k

� �

where R ¼ bvðvþl1Þ

� l2 and k ¼ Rb. Dðp�Þ is positive when n�2\k. Substituting value

of n�2 in expression of demand function then: Dðp�Þ ¼ Rcp�q 1� c

p�qk

� �¼ f ðp�Þ which

is the same MPE as previously (Eq. 2) with differents values of k and R.

4 Generalisation to Auger–Ducrot Model

In Auger and Ducrot model (Auger and Ducrot 2009), fish can be stored in order to

be sold later. Therefore, a new variable SðtÞ is introduced in order to represent the

amount of fish in stock at time t. However, in Auger-Ducrot model, the price was

assumed to remain constant. In the following model, we extend this model to a

variable price. Thus, the model reads as follows:

_n ¼ rn 1� n

k

� �� qnE;

_E ¼ pð1� gÞqnE þ prS� cE;

_S ¼ gqnE � rS;

_p ¼ uðDðpÞ � ð1� gÞqnE � rSÞ

8>>>>><>>>>>:

ð7Þ

where g is the proportion of the catch which is not sold and is stored, while ð1� gÞis the proportion immediately sold on market. Parameter r is the return rate of

stored fish to the market. _S ¼ 0 implies that gqnE ¼ rS. When substituting rS in the

second equation for _E ¼ 0, we obtain Eðpqn� cÞ ¼ 0, in other words n� ¼ cpq

. In the

fourth equation, _p ¼ 0 implies that DðpÞ ¼ qnE. Then the first equation gives

rn 1� nk

� �¼ qnE ¼ DðpÞ.

Replacing n� by its expression, we find the expression of demand function as

follows: DðpÞ ¼ rcpq

1� cpqK

� �¼ f ðpÞ and it provides the same MPE that was studied

in the previous sections.

F. Mansal et al.

123

5 Conclusion and Perspectives

In this work, we presented a bioeconomic fishery model in which the price of the

resource is not constant, but varies with respect to the difference between the

demand and the supply. As a consequence, we deal with a model in dimension 3 that

we have handled analytically. Our results have shown that taking into account the

variation of the price has important consequences. The analysis of the model shows

that, according to parameters values, one, two or three strictly positive equilibria can

exist.

A condition of viability is given for an open-access fishery: if the income that

would be obtained for a fish population maintained at carrying capacity with the

higher possible price, then there exist equilibria for which the fishery is profitable. It

is easy to see that this is a necessary condition, however it is interesting to notice

that it is also a sufficient condition.

In the case of three equilibria, two kinds of fisheries are possible: a sustainable

artisanal fishery with a fish density far from extinction, and an over-exploited fishery

with a very low resource density and a large fishing effort. Since the profit is equal

to cE at equilibria, it is sadly more interesting for economical purpose to be in the

state of an over-exploited fishery, while conservation policies should try to maintain

a high stock level by trying to keep the fishery in the sustainable artisanal state.

There are some reasons to think that the later case with three positive equilibria

could be observed in some real commercial fisheries. Some resources which were

very abundant in the past, are now over-exploited with the risk to an irreversible

collapse in the near future. As an example, in Senegal, the thiof is a fish species that

has been over-exploited for several years (Sow et al. 2011). Nowadays, the resource

becomes very scarce and the price has increased a lot. Therefore, the example of the

thiof could correspond in our model to the case of over-exploitation that was found

when three equilibria can exist.

In the present work, we also extended our fishery model with variable price to a

set of models, such as age structured fish population and fishery with resource

storage. Our results illustrate that the MPE obtained can be generalized to different

kind of fishery models, which will then present equivalent typologies of equilibria

and dynamics. Preliminary results have shown that the MPE could also be extended

to more general catch functions different from the Schaefer function that we used

here, for example catch with saturation at large fish density, i.e. a holling type II

function.

As a perspective, it would also be important to take into account the

heterogeneity of the fishery, such as Marine Protected Areas (MPA) (Boudouresque

et al. 2005) as well as fish aggregating devices, (Robert 2013; Robert et al. 2013),

artificial reefs (Randall 1963). In Senegal, there are 5 MPAs that have been created

recently and there is no doubt that this will have important consequences on the

dynamics of fisheries. Therefore, it is important to deal with models of multi-site

fisheries, (Auger et al. 2010; Moussaoui et al. 2011). In the future we expect to

develop contributions in order to take into account the spatial heterogeneity coupled

to variable price in a bioeconomic model.


123

Appendix 1: Existence Domains for Non-trivial Equilibria (Positive Equilibria)

We determine existence domains for positive equilibria of system (1). Non-trivial

equilibria correspond to the solutions of the equation

Dðp�Þ ¼ qn�E� ð8Þ

which can be rewritten

A� ap� ¼ f ðp�Þ ð9Þ

where f ðpÞ ¼ rcpq

1� cpqk

� �. Solutions correspond to the roots of third degree

polynomial

PaðpÞ ¼ p3ðaq2kÞ � p2ðAq2kÞ þ pðrcqkÞ � rc2 ð10Þ

Because two consecutive coefficients of Pa have opposite signs, real roots are all

positive. An equilibrium n� is then positive if and only if p�qk [ c, because p�qk\c

implies E�\0.

Lemma 1 There is a positive equilibrium n� such that p�qk\c if and only if

a[ qkA=c. If p� exists, it is the unique real root of (10).

Proof D is decreasing, so for p\c=qk, DðpÞ[ Dðc=qkÞ. If a� qkA=c,

Dðc=qkÞ[ 0. Then for p\c=qk, DðpÞ[ 0 and f ðpÞ\0. There is no root p� such

that p�qk\c. On the other hand, if a [ qkA=c, then Dðc=qkÞ\0. We have the

following properties:

• Dð0Þ[ 0 [ limp!0

f ðpÞ;

• if p [ c=qk, DðpÞ\0\f ðpÞ;• D is monotonously decreasing, while f is monotonously increasing on ð0; c=qk�.

We deduce that there exists a unique p� which verifies Dðp�Þ ¼ f ðp�Þ. It also verifies

p�\c=qk.h

As a consequence, when a [ qkA=c, there is no positive equilibrium.

We now determine the existence domains of real roots of polynomial Pa:

Lemma 2 If kr\3A, there is always one real root. If kr [ 3A, there are three

domains:

• a\a�: there is one and only one real root;

• a�\a\aþ: there are three real roots;

• aþ\a: there is one and only one real root;

a� and aþ correspond to values for which two real roots merge and vanish, and

verify

F. Mansal et al.

123

a� ¼ q�2r2k2 þ 9rAk � 2


3� A

� �3q

27rc

ð11Þ

aþ ¼ q�2r2k2 þ 9rAk þ 2


3� A

� �3q

27rc

ð12Þ

Proof The discriminant Da of polynomial Pa (10) is given by

Da ¼ �R Pa;P

0a

� �aq2k

ð13Þ

where the resultant R Pa;P0a

� �of polynomials Pa and its derivated polynomial reads:

R Pa;P0a

� �¼ q6arc2

k2gðaÞ ð14Þ

where gðaÞ ¼ 27a2rc2 � 18aqrcAk � q2A2k2r þ 4q2A3k þ 4r2ck2qa. g is a degree 2

polynomial with two roots, a� and aþ.

• If kr\3A, g has no real roots. We deduce that 8a[ 0, gðaÞ[ 0, and Da\0. Pa

has exactly one real root.

• If kr [ 3A, for a\a� or a[ aþ, gðaÞ[ 0. Pa has exactly one real root. for

a�\a\aþ, gðaÞ\0. Pa has exactly three real root. For a ¼ a� or a ¼ aþ,

Da ¼ 0, Pa has real roots with order of multiplicity larger than 1.

h

Appendix 2: Local Stability Positive Equilibria n�

We now determine the stability of positive equilibria of system (1). Let us denote

pþ ¼kr þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � 3AÞ3

q

Akqc and p� ¼

kr �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � 3AÞ3

q

Akqc:

Lemma 3 If A [ rk=3, the positive equilibrium n� is locally asymptotically stable.

If A\rk=3, the positive equilibrium n� is locally asymptotically stable if and only if

p�\p� or p�[ pþ.

Proof The jacobian matrix of system corresponding to positive equilibria reads:

J� ¼� rn�

k� qn� 0

p�qE� 0 qn�E�

�uqE� � uqn� � ua

2664

3775 ð15Þ

The characteristic polynomial is:


123

vðkÞ ¼ detðJ� � kI3Þ ¼� rn�

k� k � qn� 0

p�qE� � k qn�E�

�uqE� � uqn� � ua� k

¼ k3 � k2 uaþ r

kn�

� �� k u

rak

n� þ uq2n�2E� þ p�q2n�E��

� uq2n�E�r

kn�2 þ ap� � qn�E�

� �

We now determine the local stability by using Routh-Hurwitz criterion. Let us

denote

a3 ¼ 1

a2 ¼ uaþ r

kn�

a1 ¼rak

n� þ uq2n�2E� þ p�q2n�E�

a0 ¼ uq2n�E�ðrk

n�2 þ ap� � qn�E�Þ

8>>>>>>><>>>>>>>:

ð16Þ

Equilibrium n� is stable if and only if ðiÞ ai [ 0 for i 2 f0; . . .; 3g and ðiiÞa2a1 [ a3a0. If n� is positive, conditions a3 [ 0, a2 [ 0, a1 [ 0 are always verified.

We now determine if condition ðiiÞ is satisfied:

a2a1 � a3a0 ¼ uaþ r

kn�

� � rak

n� þ q2n�E� un� þ p�ð Þ� �

� uq2n�E�r

kn�2 þ ap� � qn�E�

� �

¼ uaþ r

kn�

� � rak

n� þ q2n�E� u2an� þ uap� þ ur

kn�2 þ r

kn�p

� �

� q2n�E� ur

kn�2 þ uap� � uqn�E�

� �

¼ uaþ r

kn�

� � rak

n� þ q2n�E� u2an� þ r

kn�p� þ uqn�E�

� �

[ 0

If n� is positive, condition ðiiÞ is always verified. We now determine the sign of a0.

By replacing n� and E� by their values, we obtain

a0 ¼urcðp�qk � cÞðaq2kp�3 � rcqkp� þ 2rc2Þ

p�3q3k2ð17Þ

Since p� is a root of polynomial (10), we have

a0 ¼urcðp�qk � cÞ ðAq2kÞp�2 � 2ðrcqkÞp� þ 3rc2ð Þ

p�3q3k2ð18Þ

If A [ rk=3, polynomial ðAq2kÞp�2 � 2ðrcqkÞp� þ 3rc2 has no real roots and is

always positive. If A\rk=3, polynomial ðAq2kÞp�2 � 2ðrcqkÞp� þ 3rc2 has two

F. Mansal et al.

123

roots: p� and pþ. Since n� is positive, p�qk [ c. We deduce that a0 [ 0 if and only

if p\p� or p [ pþ. h

Lemma 4 p� (resp. pþ) is the double root of polynomial Paþ (resp. Pa�).

Proof From Cardano’s formula, we find that double root of polynomial Paþ reads:

3c r2k2 � 3rAk �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � 3AÞ3

q� �

qk 9A2 � 9rAk þ 2r2k2 � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirkðrk � AÞ3

q� � ð19Þ

By simplifying the expression, we obtain that the double root is equal to p�. The

same results holds for Pa� and pþ. h

Appendix 3: Bounded Attractor

We now show that there exists a bounded set in which every trajectories (for system

(1)) with a positive initial condition end. It is clear that the set X0 of the phase space

ðn;E; pÞ defined by

X0 ¼ fðn;E; pÞ j 0� n� k; p�A=ag ð20Þ

is a forward invariant set for system (1). Furthermore, any trajectory with a positive

initial condition has its x-limit in X0.

Let us consider the candidate Lyapunov function defined for n 2 R�þ, E 2 R

�þ,

p 2 R:

Vðn;E; pÞ ¼ pqnþ qE � c ln n� r 1� n

k� au

r

� �ln E ð21Þ

Along the trajectories of system (1), we have

_Vðn;E; pÞ ¼ �aucþ n uqAþ r

kr 1� n

k

� �� qE

� �ln E � uq2nE

� �ð22Þ

Note that _Vðn;E; pÞ does not depend on p.

Lemma 5 The set X ¼ fðn;E; pÞ j ðn;E; pÞ 2 X0; _Vðn;E; pÞ� 0g is included in

the set R� ð�1;A=a�, where R is a compact subset of ð0; k� � R�þ [ fðk; 0Þg.

Proof For E [ 1, _Vðn;E; pÞ\n uqAþ rk

r � qEð Þ ln E� �

. The right term tends

toward �1 when E tends toward þ1. We denote

R0 ¼ fðn;EÞ j 0� n� k;Emin�Eg ð23Þ

where Emin is such that uqAþ rk

r � qEminð Þ ln Emin

� �\0. We deduce that

8ðn;EÞ 2 R0 ¼) _Vðn;E; pÞ\0 ð24Þ


123

For E\1, _Vðn;E; pÞ\n uqAþ rk

r 1� nk

� �� qE

� �ln E

� �. We have

n\f ðEÞ ¼) _Vðn;E; pÞ\0 ð25Þ

where f ðEÞ ¼ k2uqAr2 ln E

þ kðr�qEÞr

. It is easy to see that f is defined on ð0; 1Þ and

monotonously decreasing, with limE!0

f ðEÞ ¼ k and limE!1

f ðEÞ ¼ �1. We denote

R1 ¼ fðn;EÞ j 0� n� k; 0�E\f�1ðnÞg. Equation (25) now reads

ðn;EÞ 2 R1 ¼) _Vðn;E; pÞ\0 ð26Þ

Let us consider E0min ¼ f�1ðk=2Þ. On the compact set ½0; k=2� � ½E0min;Emin�, term

uqAþ rk

r 1� nk

� �� qE

� �ln E � uq2nE has a maximum M.

Let us denote R2 ¼ ½0; auc=MÞ � ½E0min;Emin�. From Eq. (22), we deduce that

ðn;EÞ 2 R2 ¼) _Vðn;E; pÞ\0 ð27Þ

We now define R ¼ ð½0; k� � RþÞnðR1 [ R2 [ R3Þ. R, R0, R1 and R2 are represented

in Fig. 5. R is a compact subset of ð0; k� � R�þ [ fðk; 0Þg. Furthermore,

_Vðn;E; pÞ� 0) ðn;EÞ 2 R. We deduce that X is included in R� ð�1;A=a�. h

Lemma 6 In set X, V admits a maximum V0.

Proof Since R is a compact set, Vðn;E;A=aÞ admits a maximum V0 on R. From

Eq. (21), we deduce that 8ðn;E; pÞ 2 X, Vðn;E; pÞ�Vðn;E;A=aÞ, hence the result.

h

Let be V 00�V0, X1 ¼ fðn;E; pÞ j Vðn;E; pÞ�V 00g, and Xþ1 ¼ fðn;E; pÞ jðn;E; pÞ 2 X1; p� 0g. We now consider the flow / associated to system (1).

Lemma 7 X1 is forward invariant, and for all ðn;E; pÞ 2 X0, there exists t� 0

such that /tðn;E; pÞ 2 Xþ1.

Proof For ðn;E; pÞ 2 X0nX1, Vðn;E; pÞ[ V0 and _Vðn;E; pÞ\0, which means

that X1 is forward invariant. Furthermore, it is clear that limt!þ1

Vð/tðn;E; pÞÞ�V0,

which means that there exists t0 such that /t0ðn;E; pÞ 2 X1.

We now show that there exists t00 such that /t00ðn;E; pÞ 2 Xþ1. From system (1),

we deduce that if p\0, _E� � cE, and if p\0 and E\A=ð2kqÞ, _p [ A=2. Let us

consider ðn;E; pÞ 2 X1, and the solution ðnðtÞ;EðtÞ; pðtÞÞ ¼ /tðn;E; pÞ. We sup-

pose that 8t [ 0, pðtÞ\0. There exists t1 such that 8t [ t1, Eðt1Þ\A=ð2kqÞ. Then

for t [ t1, _pðtÞ[ A=2, and limt!þ1

pðtÞ[ 0, hence the contradiction. We deduce that

there exists t00� 0 such that p� 0. Since ðn;E; pÞ 2 X1 and X1 is forward

invariant, /t00ðn;E; pÞ 2 Xþ1, which ends the proof. h

F. Mansal et al.

123

We can now prove Theorem 3.

Theorem 3 There exists a bounded set Xþþ1 included in X0 which is forward

invariant and such that 8ðn;E; pÞ; t� 0 j f/tðn;E; pÞg \ Xþþ1 6¼ ; �

.

Fig. 5 _Vðn;E; pÞ (black wireframe surface). The white part of the plan _V ¼ 0 represents the compact set

which encompasses the set fðn;EÞ j _V [ 0g. Sets R and Ri for i ¼ 1. . .3 are represented as grey areas.Parameters are r ¼ 0:9, k ¼ 3, q ¼ 0:1, c ¼ 2, A ¼ 2, a ¼ 0:1 and u ¼ 0:1

Fig. 6 On the surface shown, Vðn;E; pÞ is constant. The space under the surface corresponds to X1. The

compact set Xþ1 corresponds to the space under the surface and over the plan p ¼ 0. Parameters are

r ¼ 0:9, k ¼ 3, q ¼ 0:1, c ¼ 2, A ¼ 2, a ¼ 0:1 and u ¼ 0:1


123

Proof It is easy to deduce from Eq. (21) that Xþ1 is a compact set. This is illus-

trated on Fig. 6.

Let us denote EM ¼ maxfE j ðn;E; 0Þ 2 X1g.For ðn;E; pÞ 2 Xþ1, we consider the solution ðnðtÞ;EðtÞ; pðtÞÞ ¼ /tðn;E; pÞ. Let

us define tm ¼ infft [ 0 j pðtÞ\0g and tM ¼ infft [ tm j pðtÞ[ 0g (tm and tM can

be equal to þ1). If tm ¼ þ1, we denote pinf ðn;E; pÞ ¼ 0, else we denote

pinf ðn;EÞ ¼ inft2ðtm;tMÞ

pðtÞ. This represents the minimal value of p that is reachable

when crossing the plan p ¼ 0 before returning to Xþ1. If tm\þ1, then _pðtmÞ� 0.

For all t 2 ðtm; tMÞ, _EðtÞ� � cEðtÞ, and so EðtÞ�EðtmÞe�ct�EMe�ct. Then we

have _pðtÞ�A� qkEðtÞ�A� qkEMe�ct. We deduce that pðtÞ reaches its minimum

before t0 ¼ ln A=qkEMð Þ=c. If we denote pm ¼R t0

0A� qkEMe�ctð Þdt, then

pinf ðn;E; pÞ� pm.

We now define Xþþ1 ¼ fðn;E; pÞ 2 X1jp� pmg. It is clear that Xþþ1 is bounded,

and from the previous demonstration, we deduce that it is forward invariant. since

Xþ1 is included in Xþþ1 , we deduce from Lemma (7) that 8ðn;E; pÞ;t� 0 j f/tðn;E; pÞg \ Xþþ1 6¼ ; �

. h

References

Auger P, Mchich R, Raıssi N, Kooi B (2010) Effects of market price on the dynamics of a spatial fishery

model: over-exploited fishery/traditional fishery. Ecol Complex 7:13–20

Auger P, Bravo de la Parra R (2000) Methods of aggregation of variables in population dynamics. C R

Acad Sci 323:665–674

Auger P, Ducrot A (2009) A model of fishery with fish stock involving delay equations. Philos Trans R

Soc A 367:4907–4922

Auger P, Bravo de la Parra R, Poggiale JC, Sanchez E, Nguyen-Huu T (2008) Aggregation of variables

and applications to population dynamics. In: Magal P, Ruan S (eds) Structured population models in

biology and epidemiology. Lecture notes in mathematics, Vol. 1936, Mathematical Biosciences

Subseries, Springer, Berlin, pp 209–263

Auger P, Lett C, Moussaoui A, Pioch S (2010) Optimal number of sites in artificial pelagic multi-site

fisheries. Can J Fish Aquat Sci 67:296–303

Barbier EB, Strand I, Sathirathai S (2002) Do open access conditions affect the valuation of an

externality? Estimating the welfare effects of mangrove-fishery linkages. Env Resour Econ

21:343–367

Boudouresque CF, Gadiou G, Le Direac’h L (2005) Marine protected areas: a tool for costal areas

management. In: Levner E, Linkov I, Proth JM (eds) Strategic management of marine Ecosystems.

Springer, Dordrecht, pp 29–52

Clark CW (1990) Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn.

Wiley, New York

Clark CW (1985) Bioeconomic modelling and fisheries management. Wiley, New York

Clark CW (2006) Fisheries bioeconomics: why is it so widely misunderstood? Popul Ecol 48(2):95–98

De Lara M, Doyen L (2008) Sustainable management of renewable resources: mathematical models and

methods. Springer, Berlin

Doyen L, Cisse A, Gourguet S, Mouysset L, Hardy PY, Bene C, Blanchard F, Jiguet F, Pereau JC,

Thebaud O (2013) Ecological-economic modelling for the sustainable management of biodiversity.

Comput Manag Sci 10(4):353–364

Ekouala L (2013) Le developpement durable et le secteur des peches et de l’aquaculture au Gabon: une

etude de la gestion durable des ressources halieutiques et de leur ecosysteme dans les provinces de

l’Estuaire et de l’Ogoue Maritime. PhD Thesis, Universite du Littoral Cote d’Opale

F. Mansal et al.

123

Iwasa Y, Andreasen V, Levin SA (1987) Aggregation in model ecosystems. I. Perfect aggregation. Ecol

Model 37:287–302

Iwasa Y, Levin SA, Andreasen V (1989) Aggregation in model ecosystems. II. Approximate aggregation.

IMA J Math Appl Med Biol 6:1–23

Lafrance JT (1985) Linear demand functions in theory and practice. J Econ Theory 37:147–166

Mankiw NG (2011) Principles of economics, 5th edn. South-Western Cengage Learning, Boston

Ly S, Mansal F, Balde M, Nguyen-Huu T, Auger P (2014) A model of a multi-site fishery with variable

price: from over-exploitation to sustainable fisheries. Mathematical Modelling of Natural

Phenomena (in press)

Meuriot E (1987) Les modeles bio-economiques d’exploitation des pecheries. Demarches et enseign-

ements. Rapports economiques et juridiques de l’IFREMER N 4

Moussaoui A, Auger P, Lett C (2011) Optimal number of sites in multi-site fisheries with fish stock

dependent migrations. Math Biosci Eng 8:769–783

Prellezo R, Accadia P, Andersen JL, Andersen BS, Buisman E, Little A, Nielsen JR, Poos JJ, Powell J,

Rockmann C (2012) A review of EU bio-economic models for fisheries: the value of a diversity of

models. Mar Policy 36:423–431

Robert M, Dagorn L, Filmalter JD, Deneubourg JL, Itano D, Holland K (2013) Intra-individual behavioral

variability displayed by tuna at fish aggregating devices (FADs). Mar Ecol-Prog Ser (in press)

Randall JE (1963) An analysis of the fish populations of artificial and natural reefs in the virgin islands.

Caribb J Sci 3(1):31–47

Robert M, Dagorn L, Lopez J, Moreno G, Deneubourg JLA (2013) Does social behavior influence the

dynamics of aggregations formed by tropical tunas around floating objects ? An experimental

approach. J Exp Mar Biol Ecol 440:238–243

Smith VL (1968) Economics of production from natural resources. Am Econ Rev 58(3):409–431

Smith VL (1969) On models of commercial fishing. J Polit Econ 77(2):181–198

Sow FN, Thiam N, Samb B (2011) Diagnostic de l’etat d’exploitation du stock de merou Epinephelus

aeneus(Geoffroy St. Hilaire, 1809) au Senegal par l’utilisation des frequences des tailles. J Sci Hal

Aquat 3:82–88

Walras L (1874) Elements d’economie Politique Pure. Corbaz, Lausanne


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A Mathematical Model of a Fishery with Variable Market Price: Sustainable Fishery/Over-exploitation

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