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Acta Biotheor (2014) 62:305–323 DOI 10.1007/s10441-014-9227-7 REGULAR ARTICLE

A Mathematical Model of a Fishery with Variable Market Price: Sustainable Fishery/Over-exploitation Fulgence Mansal • Tri Nguyen-Huu Pierre Auger • Moussa Balde

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Received: 12 December 2013 / Accepted: 4 June 2014 / Published online: 21 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 equilibria 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 agestructured fish population and a fishery with storage of the resource.

F. Mansal (&) P. Auger M. Balde De´partement de mathe´matiques 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

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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

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A mathematical model of a fishery with variable market price

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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: 8 dn n > > ¼ rn 1 qnE > > > dt k > < dE ð1Þ ¼ pqnE cE > dt > > > > dp > : ¼ uðDðpÞ qnEÞ dt 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 a are 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.

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2.1 Existence of Equilibria Model ð1Þ has the following nullclines: • •

The n-nullclines correspond to n ¼ 0 and E ¼ qr ð1 nkÞ; c The E-nullclines correspond to E ¼ 0 and n ¼ pq ;

•

The p-nullclines correspond to p ¼ AqnE a .

We determine three kind of equilibria from those nullclines: equilibrium n0 ¼ ð0; 0; Aa Þ, which corresponds to the extinction of fish population; equilibrium nk ¼ ðk; 0; Aa Þ, 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 pc q and E ¼ qr 1 pcqk both depend on the price p . Third equation gives that non-trivial equilibria verify rc c Dðp Þ ¼ 1 p q 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. 2. 3.

if a\a : there is one and only one positive equilibrium; if a \a\aþ : there are three positive equilibria; if aþ \a: there is one and only one positive equilibrium.

where a ¼ q

2r 2 k2 þ 9rAk 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 27kr kr3 A

ð3Þ

27rc

and

aþ ¼ q

Proof

2r 2 k2 þ 9rAk þ 2


27rc

The proof is detailed in Appendix 1.

2.2 Analysis of Local Stability The Jacobian matrix associated to system (1) reads:

123

ð4Þ


2 2n r 1 qE 6 k 6 J¼6 pqE 4 uqE

309

3 qn

0

7 7 7 qnE 5 ua

pqn c uqn

We now determine the local stability at each equilibrium point. 1.

For n0 ¼ ð0; 0; Aa Þ, the Jacobian matrix reads: 2 3 r 0 0 6 7 J0 ¼ 4 0 c 0 5 0 0 ua

J0 has one positive and two negative eigenvalues. Equilibrium n0 is a saddle point (unstable). 2. nk ¼ ðk; 0; Aa Þ, the Jacobian matrix reads: 2 3 r qk 0 6 7 Aqk Jk ¼ 6 c 0 7 4 0 5 a 0 uqk ua

• • 3.

If a [ qkA=c, nk is a stable equilibrium; If a\qkA=c, then nk is a saddle point (unstable).

At equilibria n , the Jacobian matrix reads: 2 rn qn 6 k J¼6 4 p qE 0 uqE

Theorem 2 • •

uqn

0

3

7 7 qn E 5 ua

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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 kr þ rkðrk 3AÞ kr rkðrk 3AÞ3 c and p ¼ c: pþ ¼ Akq Akq

There are three different cases: 1.

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

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2.

3. Proof

if a \a\aþ , there are three equilibria n1 , n2 and n3 (ordered by increasing values of pi ). Equilibria n1 and n3 are locally asymptotically stable, while n2 is unstable; if a [ aþ , there is one positive equilibrium which is stable. The proof is given in Appendix 2.

2.3 Typology of Dynamics Theorem 3 There exists a bounded set Xþþ 1 such that any trajectory with a þþ positive initial condition has its x-limit in X1 . 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 Xþ 1 the intersection of X1 with the set defined by p 0 (Lemma 7 again). Finally, we show that for any initial condition in Xþ 1 , the trajectory stays bounded in a compact set Xþþ , which ends the proof. 1 This theorem implies that all trajectories enter a compact set Xþþ 1 , 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. 2.

A [ rk=3: there is only one positive equilibrium, which is locally asymptotically stable. The dynamics is represented in Fig. 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

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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. Right Time series of the dynamics: fish stock (black), fishing effort (grey) and price (dotted). The initials conditions 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 and u¼1

Fig. 2 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 curves intersect at p , corresponding to equilibrium n . Right Time series of the dynamics: fish stock (black), fishing effort (grey) and price (dotted). The initials conditions 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

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

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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 one equilibrium. 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 are r 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

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 n1 , n2 , n3 , with ni ¼ ðni ; Ei ; pi Þ, and ordered by increasing value of pi . The middle equilibrium n2 (p1 \p2 \p3 ) being a saddle node while the two other equilibria are locally asymptotically stable. Since a [ 0, we have n3 \n1 . A straightforward calculation gives the following set of inequalities: 8 > < n3 \n1 ; E3 [ E1 ; ð5Þ > : p3 [ p1 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 n3 : 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 n1 : 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

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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 the locally asymptotically stable equilibria. The initials conditions are: (0, 0, 2) and (0, 3, 0). Parameters are r 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

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 overexploitation 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: 8 n_1 ¼ bn2 vn1 l1 n1 ; > > > < n_ ¼ vn l n bn2 qn E; 2 1 2 2 2 2 ð6Þ _ > E ¼ Eðpqn cÞ; 2 > > : p_ ¼ uðDðpÞ qn2 EÞ

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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. c The E-isoclines are E ¼ 0 and pqn2 c ¼ 0. We deduce that n2 ¼ pq , and then bc that n1 ¼ pqðvþl Þ. The p isoclines are given by 1

DðpÞ ¼ qn2 E ¼ vn1 l2 n2 bn22 At equilibrium, we have n Dðp Þ ¼ Rn2 1 2 k bv l2 and k ¼ Rb. Dðp Þ is positive when n2 \k. Substituting value where R ¼ ðvþl 1Þ of n2 in expression of demand function then: Dðp Þ ¼ pRc 1 pcqk ¼ f ðp Þ which q

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: 8 n > > _ n ¼ rn 1 qnE; > > k > < E_ ¼ pð1 gÞqnE þ prS cE; ð7Þ > > S_ ¼ gqnE rS; > > > : p_ ¼ uðDðpÞ ð1 gÞqnE rSÞ where g is the proportion of the catch which is not sold and is stored, while ð1 gÞ is 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 c second equation for E_ ¼ 0, we obtain Eðpqn cÞ ¼ 0, in other words n ¼ pq . In the _ fourth equation, p ¼ 0 implies that DðpÞ ¼ qnE. Then the first equation gives n rn 1 k ¼ qnE ¼ DðpÞ. Replacing n byits expression, we find the expression of demand function as

rc c 1 pqK ¼ f ðpÞ and it provides the same MPE that was studied follows: DðpÞ ¼ pq

in the previous sections.

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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.

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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 rc c where f ðpÞ ¼ pq 1 pqk . Solutions correspond to the roots of third degree polynomial Pa ðpÞ ¼ p3 ðaq2 kÞ p2 ðAq2 kÞ 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 [ lim f ðpÞ;

• •

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

p!0

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 domains: • • •

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

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

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a ¼ q

aþ ¼ q

317


ð11Þ

27rc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2r 2 k2 þ 9rAk þ 2 27kr kr3 A

ð12Þ

2r 2 k2 þ 9rAk 2

27rc

The discriminant Da of polynomial Pa (10) is given by R Pa ; P0a ð13Þ Da ¼ aq2 k where the resultant R Pa ; P0a of polynomials Pa and its derivated polynomial reads:

Proof

q6 arc2 gðaÞ R Pa ; P0a ¼ k2

ð14Þ

where gðaÞ ¼ 27a2 rc2 18aqrcAk q2 A2 k2 r þ 4q2 A3 k þ 4r 2 ck2 qa. 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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 kr þ rkðrk 3AÞ kr rkðrk 3AÞ3 c and p ¼ c: pþ ¼ Akq Akq 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: 2 3 rn qn 0 6 7 k 7 J ¼ 6 ð15Þ 4 p qE 0 qn E 5 uqE

uqn

ua

The characteristic polynomial is:

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rn k qn 0 k vðkÞ ¼ detðJ kI3 Þ ¼ p qE k qn E uqE uqn ua k ra r ¼ k3 k2 ua þ n k u n þ uq2 n2 E þ p q2 n E k k 2 r 2 n þ ap qn E uq n E k We now determine the local stability by using Routh-Hurwitz criterion. Let us denote 8 a3 ¼ 1 > > > > r > > > < a2 ¼ ua þ k n ra ð16Þ n þ uq2 n2 E þ p q2 n E a ¼ > > > 1 k > > > r > : a0 ¼ uq2 n E ð n2 þ ap qn E Þ k Equilibrium n is stable if and only if ðiÞ ai [ 0 for i 2 f0; . . .; 3g and ðiiÞ a2 a1 [ a3 a0 . If n is positive, conditions a3 [ 0, a2 [ 0, a1 [ 0 are always verified. We now determine if condition ðiiÞ is satisfied: r ra n þ q2 n E ðun þ p Þ a2 a1 a3 a0 ¼ ua þ n k k 2 r 2 n þ ap qn E uq n E k r ra r r n þ q2 n E u2 an þ uap þ u n2 þ n p ¼ ua þ n k k k r k 2 2 q n E u n þ uap uqn E k r ra r n þ q2 n E u2 an þ n p þ uqn E ¼ ua þ n k k k [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 ¼

urcðp qk cÞðaq2 kp3 rcqkp þ 2rc2 Þ p3 q3 k2

ð17Þ

Since p is a root of polynomial (10), we have a0 ¼

urcðp qk cÞððAq2 kÞp2 2ðrcqkÞp þ 3rc2 Þ p3 q3 k2

ð18Þ

If A [ rk=3, polynomial ðAq2 kÞp2 2ðrcqkÞp þ 3rc2 has no real roots and is always positive. If A\rk=3, polynomial ðAq2 kÞp2 2ðrcqkÞp þ 3rc2 has two

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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 Proof

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

From Cardano’s formula, we find that double root of polynomial Paþ reads: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3c r k 3rAk rkðrk 3AÞ3 ð19Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 2 qk 9A 9rAk þ 2r k 2 rkðrk AÞ

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 Rþ , E 2 Rþ , p 2 R: n au Vðn; E; pÞ ¼ pqn þ qE c ln n r 1 ln E ð21Þ k r Along the trajectories of system (1), we have _ E; pÞ ¼ auc þ n uqA þ r r 1 n qE ln E uq2 nE Vðn; k k

ð22Þ

_ E; pÞ does not depend on p. Note that Vðn; _ E; pÞ 0g is included in Lemma 5 The set X ¼ fðn; E; pÞ j ðn; E; pÞ 2 X0 ; Vðn; the set R ð1; A=a, where R is a compact subset of ð0; k Rþ [ fðk; 0Þg. _ E; pÞ\n uqA þ r ðr qEÞ ln E . The right term tends Proof For E [ 1, Vðn; k toward 1 when E tends toward þ1. We denote

where Emin

R0 ¼ fðn; EÞ j 0 n k; Emin Eg is such that uqA þ kr ðr qEmin Þ ln Emin \0. We deduce that _ E; pÞ\0 8ðn; EÞ 2 R0 ¼) Vðn;

ð23Þ

ð24Þ

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_ E; pÞ\n uqA þ r r 1 n qE ln E . We have For E\1, Vðn; k k _ E; pÞ\0 n\f ðEÞ ¼) Vðn;

ð25Þ

2

kðrqEÞ where f ðEÞ ¼ kr2 uqA . It is easy to see that f is defined on ð0; 1Þ and r ln E þ monotonously decreasing, with lim f ðEÞ ¼ k and lim f ðEÞ ¼ 1. We denote E!0

E!1

R1 ¼ fðn; EÞ j 0 n k; 0 E\f 1 ðnÞg. Equation (25) now reads _ E; pÞ\0 ðn; EÞ 2 R1 ¼) Vðn;

ð26Þ

0 0 Let us consider Emin ¼ f 1 ðk=2Þ. On the compact set ½0; k=2 ½Emin ; Emin , term r n 2 uqA þ k r 1 k qE ln E uq nE has a maximum M. 0 ; Emin . From Eq. (22), we deduce that Let us denote R2 ¼ ½0; auc=MÞ ½Emin

_ E; pÞ\0 ðn; EÞ 2 R2 ¼) Vðn;

ð27Þ

We now define R ¼ ð½0; k Rþ ÞnðR1 [ R2 [ R3 Þ. R, R0 , R1 and R2 are represented in Fig. 5. R is a compact subset of ð0; k Rþ [ fðk; 0Þg. Furthermore, _ E; pÞ 0 ) ðn; EÞ 2 R. We deduce that X is included in R ð1; A=a. h Vðn;

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 V00 V0 , X1 ¼ fðn; E; pÞ j Vðn; E; pÞ V00 g, and Xþ 1 ¼ fðn; E; pÞ j ðn; 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 /t ðn; E; pÞ 2 Xþ 1. _ E; pÞ\0, which means Proof For ðn; E; pÞ 2 X0 nX1 , Vðn; E; pÞ [ V0 and Vðn; that X1 is forward invariant. Furthermore, it is clear that lim Vð/t ðn; E; pÞÞ V0 , t!þ1

which means that there exists t0 such that /t0 ðn; E; pÞ 2 X1 . We now show that there exists t00 such that /t00 ðn; E; pÞ 2 Xþ 1 . 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 /t ðn; E; pÞ. We suppose that 8t [ 0, pðtÞ\0. There exists t1 such that 8t [ t1 , Eðt1 Þ\A=ð2kqÞ. Then _ [ A=2, and lim pðtÞ [ 0, hence the contradiction. We deduce that for t [ t1 , pðtÞ t!þ1

there exists t00 0 such that p 0. Since ðn; E; pÞ 2 X1 and X1 is forward invariant, /t00 ðn; E; pÞ 2 Xþ h 1 , which ends the proof.

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_ E; pÞ (black wireframe surface). The white part of the plan V_ ¼ 0 represents the compact set Fig. 5 Vðn; 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 Xþ 1 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

We can now prove Theorem 3. Theorem 3 There exists a bounded set Xþþ is forward 1 included in X0 which

þþ invariant and such that 8ðn; E; pÞ; t 0 j f/t ðn; E; pÞg \ X1 6¼ ; .

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Proof It is easy to deduce from Eq. (21) that Xþ 1 is a compact set. This is illustrated on Fig. 6. Let us denote EM ¼ maxfE j ðn; E; 0Þ 2 X1 g. For ðn; E; pÞ 2 Xþ 1 , we consider the solution ðnðtÞ; EðtÞ; pðtÞÞ ¼ /t ðn; 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 inf pðtÞ. This represents the minimal value of p that is reachable t2ðtm ;tM Þ

_ m Þ 0. when crossing the plan p ¼ 0 before returning to Xþ 1 . If tm \ þ 1, then pðt ct ct _ For all t 2 ðtm ; tM Þ, EðtÞ cEðtÞ, and so EðtÞ Eðtm Þe EM e . Then we _ A qkEðtÞ A qkEM ect . We deduce that pðtÞ reaches its minimum have pðtÞ R t0 before t0 ¼ lnð A=qkEM Þ=c. If we denote pm ¼ 0 ð A qkEM ect Þdt, then pinf ðn; E; pÞ pm . þþ We now define Xþþ 1 ¼ fðn; E; pÞ 2 X1 jp pm g. It is clear that X1 is bounded, and from the previous demonstration, we deduce that it is forward invariant. since deduce from Lemma (7) that 8ðn; E; pÞ; Xþ is included in Xþþ 1 , we

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