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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
e-mail: [email protected]
M. Balde
e-mail: [email protected]
T. Nguyen-Huu � P. Auger
UMI IRD 209, UMMISCO, Centre IRD de l’Ile de France, 32 avenue Henri Varagnat,
93143 Bondy Cedex, France
e-mail: [email protected]
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27kr kr
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;
A mathematical model of a fishery with variable market price
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
A mathematical model of a fishery with variable market price
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
A mathematical model of a fishery with variable market price
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.
A mathematical model of a fishery with variable market price
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27kr kr
3� A
� �3q
27rc
ð11Þ
aþ ¼ q�2r2k2 þ 9rAk þ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27kr kr
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:
A mathematical model of a fishery with variable market price
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
A mathematical model of a fishery with variable market price
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
A mathematical model of a fishery with variable market price
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
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