COMPETITION FOR A SINGLE RESOURCE AND COEXISTENCE OF SEVERAL SPECIES IN THE CHEMOSTAT

. We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-speciﬁc interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and suﬃcient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the eﬀect of the intra-speciﬁc competition on the coexistence region of the species.


(Communicated by Patrick De Leenheer)
Abstract. We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.
it is done in [11]. We give a necessary and sufficient condition for the existence of a positive equilibrium, not only a sufficient condition as it is done in [11]. In particular, we show that if the death rate parameters of all species are positive, then the existence of a stable positive equilibrium is possible for certain values of the operating parameters. Hence, the sufficient condition (the death rate parameters of all species are large enough) given by [11] is not necessary. These results were obtained in [3]. We generalize the Lyapunov function proposed by Wolkowicz and Lu [24], in the case of two species, to prove the global stability of the equilibrium, corresponding to the extinction of all species except the one with the lowest breakeven concentration. Last but not least we determine the operating diagram of the model, that is to say, we determine the regions of the operating parameters (dilution rate and input concentration of the nutrient), which depict the existence and the stability of each equilibrium. This question was not considered in [11]. We show that the operating diagram is independent of the intra-specific competition parameter of the weaker species.
In this paper, we may allow that there is no intra-specific competition of the weaker species contrary to the literature where it is apparently always assumed that all intra-specific competition parameters are positive [3,5,6,7,11,14,15,24]. Ruan et al. [17] considered the issue of the coexistence in a two-competitors/oneprey (biotic resource) model with a density-dependent mortality rate of only one of the competitors. Similarly to what is shown by [17] for a biotic resource, our findings demonstrate that in the case of an abiotic resource, two competitors can coexist when only one of the competitors exhibits intra-specific competition, namely the stronger one. The possible coexistence when the weaker species has no intraspecific competition is not surprising since the lack of intra-specific competition for the weaker species has a beneficial effect on this species' ability to survive. What is much more interesting is that, the operating diagram being independent of the intra-specific competition parameter of the weaker species, coexistence is possible. This occurs even for arbitrarily high levels of intra-specific competition of the weaker species and also for suitable given dilution rate and input concentration of the nutrient. This paper is organized as follows: in Section 2 we present an intra-specific competition model of n species and give some preliminary results. Section 3 is devoted to analyse this model in the case of two species. Using the concept of steady-state characteristics, we give a geometric characterization of the existence and stability of all equilibria. We prove that only one equilibrium is stable. A global asymptotic stability result is given. In Section 4, this approach is extended to the study of the multi-species model. In Section 5, we present the operating diagrams which depict the existence and the stability of each equilibrium according to control parameters. In Section 6, numerical simulations with realistic growth functions (of Monod-type) illustrate either the coexistence or the competitive exclusion in different cases. Finally, some conclusions are drawn in Section 7.
For convenience, we use the abbreviations LES for Locally Exponentially Stable equilibria and GAS for Globally Asymptotically Stable equilibria, in all what follows.
2. Mathematical model. We consider the chemostat model of n species competing for a single nutrient with intra-specific linear interactions, introduced by De Leenheer et al. [11]. This model reads: where S denotes the concentration of the substrate; x i denotes the concentration of the ith population of micro-organisms; S in and D denote, respectively, the concentration of substrate in the feed bottle and the dilution rate of the chemostat; D i denotes the removal rate of the species i which is the sum of the death rate of species i and the dilution rate D, (D i are not necessarily equal); a i is a non-negative parameter giving rise to death rate a i x i which is due to intra-specific interaction and f i (·) denotes the per-capita growth rate of the ith population. We first make the following assumption on the growth functions: Hypothesis H1 means that the growth can take place if and only if the substrate is present. Moreover, the growth rate of each species increases with the concentration of substrate. In the following, we prove that system (1) is behaving as well as one would expect for any reasonable chemostat model.

Proposition 1.
For any non-negative initial condition, the solutions of (1) remain non-negative and are positively bounded. The set where D * = min(D, D 1 , . . . , D n ), is positively invariant and a global attractor for (1).
Proof. From (1), we havė From Gronwall's Lemma, we obtain It is easy to see from (1) that the cone R n+1 + is positively invariant. Thus solutions are non-negative for all t 0 and from (2) we deduce that solutions are positively bounded and that the set Ω is positively invariant and is a global attractor for (1).

3.
Analysis of the competition model with two species. For a better understanding of the qualitative behavior of solutions of the model (1), we begin with the case n = 2. In this particular case, we can describe precisely the solutions and give stability results for all equilibria. Moreover, we can show that a necessary and sufficient condition of coexistence of both species is the inhibition of the strongest species by intra-specific competition. System (1), in the case of two species competing for a single nutrient, reads We assume that H1 is satisfied for n = 2. Now, we shall discuss the existence of equilibria of system (3) and then their asymptotic stability for a 1 > 0 and a 2 0.
We assume that the populations x i are labeled such that λ 1 < λ 2 . The equilibria are solutions of system (4): (4) By solving system (4), we will prove the existence of four equilibria, according to the concentration S in : a washout equilibrium, two equilibria corresponding to the extinction of the first and the second species, respectively, and a positive equilibrium corresponding to the coexistence of both species. For the description of the equilibria, we need the following notations Notice that the function h 2 (·) is defined only when a 2 > 0. In this case, we define the function H(·) by Then, we can state: Proposition 2. System (3) admits the following equilibria: 1. The washout equilibrium E 0 = (S in , 0, 0), that always exists. 2. The equilibrium E 1 = (S 1 ,x 1 , 0), of extinction of species x 2 , where S 1 is the if a 2 = 0 and S 2 is the unique solution of H 2 (S) = 0, x * 2 = H 1 (λ 2 )/D 2 if a 2 = 0 and S * is the unique solution of equation H(S) = 0,

ABDELLATIF, FEKIH-SALEM AND SARI
Proof. We first note that if x 1 = x 2 = 0, we obtain the washout equilibrium E 0 = (S in , 0, 0) which always exists. For the proof of item 2, if x 2 = 0 and x 1 = 0, then from the second equation of (4), we deduce that If a 2 = 0 then f 2 (S) = D 2 which implies that S 2 = λ 2 and then For the proof of item 4, if x 1 = 0 and x 2 = 0, then two cases must be distinguished.
If a 2 > 0 then, from the second and the third equation of (4), we obtain which is positive if and only if S > λ i . From the first equation, we deduce that H(S) = 0. Since H is decreasing on [λ 2 , +∞[, there exists a unique solution S * > λ 2 of equation H(S) = 0 if and only if H(λ 2 ) > 0, that is, S in >λ 2 . If a 2 = 0 then the third equation of system (4) implies that S * = λ 2 and from the second one we have x * 1 = (f 1 (S) − D 1 )/a 1 . The first equation leads to In the next section, we will show that the positive equilibrium E * is stable as long as it exists. We explain here what are the consequences of the existence of this equilibrium on the coexistence of the species. More precisely, we justify the claim of the introduction that coexistence may be possible, even though the weaker competitor exhibits arbitrarily high levels of intra-specific competition. According to Proposition 2, the existence of a positive equilibrium holds if S in >λ 2 or equivalently, using (5) and (6), if This is the necessary and sufficient condition for the existence of the positive equilibrium. This condition does not involve the intra-specific competition parameter a 2 of the weaker competitor. In particular, the weaker competitor may exhibit arbitrarily high levels of intra-specific competition a 2 . However, since the density of the weaker competitor at the equilibrium is given by it tends to 0 as a 2 tends to +∞. There is another interesting observation which can be obtained from (7). If for example S in > λ 2 (which expresses that species 2 could survive in the absence of the stronger competitor 1, and the absence of intraspecific competition with itself), then (7) holds for all sufficiently large a 1 , i.e. for all sufficiently large levels of intra-specific competition of the stronger competitor 1. This relates to the findings of De Leenheer et al. [11] who established the existence of a positive equilibrium if the parameters a 1 and a 2 are large enough.

Stability of equilibria.
For a 2 0, the local asymptotic stability of equilibria of (3) is given in the following result: Proof. Let J be the Jacobian matrix of (3) at (S, x 1 , x 2 ), that is given by The eigenvalues are on the diagonal. They are negative, that is, For the proof of item 2, the Jacobian matrix of (3) at E 1 = (S 1 ,x 1 , 0), is given by: The other eigenvalues of J E1 are the eigenvalues of the matrix We can see that det(A 1 ) > 0 and tr(A 1 ) < 0, then the two eigenvalues of A 1 have negative real parts. The equilibrium E 1 is then LES if and only if S 1 < λ 2 or equivalently S in <λ 2 .
For item 3, we use similar arguments to check the stability of E 2 = (S 2 , 0,x 2 ), which exists if and only if λ 2 < S in . Since the Jacobian matrix at E 2 is .
The two other eigenvalues of J E2 are the eigenvalues of the matrix Since det(A 2 ) > 0 and tr(A 2 ) < 0, the two eigenvalues of A 2 have negative real parts. Consequently, E 2 is LES if and only if S 2 < λ 1 . Since we have S 2 > λ 2 > λ 1 , E 2 is a saddle point when it exists. For the proof of item 4, we can write the Jacobian matrix at E * = (S * , x * 1 , x * 2 ) in the form: , which are positive. The characteristic polynomial is given by Since m ij 0, for all i, j = 1, . . . , 3, it follows that c i < 0. Then, a straightforward calculation gives which is positive. Thus all the conditions of the Routh-Hurwitz criterion are satisfied and so E * is LES when it exists. Table 1 summarizes the previous results. Fig. 1 shows that the stable equilibrium is the one on the red arc and all other equilibria (in blue) are unstable. More precisely, the equilibria are given by the intersection between the line ∆ of equation y = D(S in −S) and either the curve of the function h 12 (·) = h 1 (·)+h 2 (·) defined for S > λ 2 , or the curve of the function h i (·) defined respectively for S > λ i , i = 1, 2, or the line of equation y = 0 which represents the washout equilibrium E 0 . If S in >λ 2 , the intersection between the line ∆ and the curve of the function h 12 (·) represents the solution S * of the equation H(S) = 0 satisfying S * > λ 2 . Therefore, the condition x * i > 0, i = 1, 2, is satisfied, and there exists a unique positive equilibrium  (3). Figure 1. Steady-state characteristics with S in >λ 2 : (a) equilibria of (3) when a 2 > 0, (b) equilibria of (3) when a 2 = 0. We use red color for LES equilibria and blue color for unstable equilibria.
E * . If λ 2 < S in <λ 2 , there exist three equilibria E 1 , E 2 and E 0 given by the intersection between the line ∆ and either the function h i (·), i = 1, 2, respectively, or the line of equation y = 0. If λ 1 < S in < λ 2 , there exist two equilibria E 1 and E 0 given by the intersection between the line ∆ and either the function h 1 (·) or the line of equation y = 0. If S in < λ 1 , there exists one equilibrium E 0 given by the intersection between the line ∆ and the line of equation y = 0. Notice that when S in =λ 2 , E 1 coalesces with E * where E 1 is a saddle-node equilibrium since it has a zero eigenvalue and it is due to a saddle-node bifurcation of E 1 (saddle point) and E * (stable node), (see Fig. 1(a)). Similarly, when S in = λ 2 , E 0 coalesces with E 2 and when S in = λ 1 , E 0 coalesces with E 1 . Fig. 1(b) illustrates the steady-state characteristics for a 2 = 0 and the existence of the positive equilibrium E * which is obtained by the intersection between the line ∆ and the vertical line that corresponds to the curve of function h 12 (·). Since E * and E 2 have the same S component, S = λ 2 , they appear at the same point of the (S, y) plane. However, these equilibria do not coincide, since they have distinct x 1 and x 2 components.
Thus, if λ 1 < λ 2 then the first species has a competitive advantage over the second species and so this second species need not to inhibit its growth in order to coexist with the other species. The absence of intra-specific inhibition for the weaker species may allow it to survive. Hence, the coexistence is due to the fact that the most efficient species sees its growth inhibited by its own intra-specific competition.
We can derive, now, the global asymptotic behavior of (3) according to S in . More specifically, we have the following result: Proposition 4. Under assumption H1 in the case n = 2 and for a i > 0, i = 1, 2, the following cases occur: 1. If S in < λ 1 , there exists a unique equilibrium E 0 = (S in , 0, 0) which is GAS. 2. If λ 1 < S in < λ 2 , then there exist two equilibria: E 0 is unstable and E 1 = (S 1 ,x 1 , 0) is GAS. 3. If λ 2 < S in <λ 2 , then there exist three equilibria: E 0 and E 2 = (S 2 , 0,x 2 ) are unstable while E 1 is LES. Moreover, if there exists a constant α > 0 which satisfies: then E 1 is GAS with respect to all solutions with x 1 (0) > 0. 4. If S in >λ 2 , then there exist four equilibria: E 0 , E 1 and E 2 are unstable while Proof. First, the global stability of the washout equilibrium follows from part 5. of Lemma 1.1 of [24]. For item 2, the global stability of E 1 derives from part 4. of Theorem 2.2 of [24]. For item 3, it derives from part 5. of Theorem 2.2 of [24]. We conclude by using Proposition 3. Item 4 is a consequence of Proposition 3.

4.
Study of the competition model with several species. Now, we consider the case of n species competing for a same limiting resource. We determine the equilibria of (1) under assumption H1 and we specify the asymptotic stability according to the control parameter S in . For that, we use the concept of steady-state characteristics, to describe geometrically the equilibria. The steady-state characteristic is a curve which is associated with each species. It permits if the dynamic of the renewal of the resource is known to give sufficient conditions for coexistence and to predict the issue of the competition.

4.1.
Existence of equilibria. In the following, we study the existence of equilibria of system (1) under assumption H1 and for all a i > 0, i = 1, . . . , n. If equation f i (S) = D i has a solution, then we denote λ i = f −1 i (D i ). Otherwise, λ i = +∞. We assume that the populations x i are labeled such that To find the equilibria of (1), we solve the following system: For convenience, we introduce the following functions, for i = 1, . . . , n: and If x i = 0 for all i = 1, . . . , n, then S = S in from the first equation of (8). This corresponds to the washout equilibrium E 0 = (S in , 0, . . . , 0), which always exists. If x i = 0, for all i = 1, . . . , n, we deduce from equation i + 1 of (8) that, which is positive if and only if S > λ i . From the first equation of (8), we deduce that H(S) = 0. Since H is decreasing on [λ n , +∞[ and h n (λ n ) = 0, there exists a unique solution S * > λ n of equation H(S) = 0 if and only if H(λ n ) > 0, that is, Hence, one has the following result: In order to identify the other equilibria corresponding to the extinction of one or many species, we first introduce the following definition: In Fig. 2, we illustrate the case of three species competing for a single nutrient. It shows the number of equilibria of (1) according to S in . We use the following notationsλ Then, we can see that • If S in >λ 3 , there exist 2 3 equilibria: A washout equilibrium E 0 , a positive equilibrium E * , which is the intersection of ∆ and the curve h 123 := 3 i=1 h i . Three equilibria E 1 , E 2 and E 3 , where one species survives, three equilibria E 12 , E 13 and E 23 , where two species coexist while the third species is excluded. • Ifλ 13 < S in <λ 3 , then there exist seven equilibria: E 0 , E 1 , E 2 , E 3 , E 12 , E 13 and E 23 . • Ifλ 23 < S in <λ 13 , then there exist six equilibria: E 0 , E 1 , E 2 , E 3 , E 12 and E 23 . • If λ 3 < S in <λ 23 , then there exist five equilibria: E 0 , E 1 , E 2 , E 3 and E 12 .
• If S in < λ 1 , then there exists a unique equilibrium E 0 .

4.2.
Stability of equilibria. We are interested, now, in the asymptotic behavior of (1). As explained in the previous section, the system can have at most 2 n equilibria obtained by taking the intersections of the line ∆ with the steady-state characteristics. We show that in each case, only one equilibrium is stable, namely the leftmost one (with the smallest S component), corresponding to the intersection of ∆ with the red steady-state characteristic y = k i=1 h i (S). All other equilibria corresponding to the intersections of ∆ with the blue steady-state characteristics are unstable. In order to do this, we calculate the Jacobian matrix and we use mainly Lemma 6.3 of [4], which we recall here: Assume that D > 0 and for i = 1, . . . , n, α i 0, b i > 0 and c i b i . Then, all the eigenvalues of A have negative real parts.
We have the following result: Proposition 6. Let D and S in be fixed. The equilibrium with the smallest S component is LES. All other equilibria are unstable.
Proof. Let E = (S, x 1 , . . . , x n ) an equilibrium. The Jacobian matrix at E has the form (11) with From (10), we easily see that λ 1 =λ 1 <λ 2 < · · · <λ n . Assume first that S in >λ n . The leftmost equilibrium is the positive equilibrium E * = (S * , x * 1 , . . . , x * n ), which satisfies f i (S * ) − a i x * i − D i = 0. Therefore the Jacobian matrix terms at E * satisfy: Using H1, the positivity of the coefficients a i and Lemma 4.2, we conclude that E * is LES. If we denote byĒ = (S, x 1 , . . . , x n ) an equilibrium point corresponding to the intersection of ∆ with the blue steady-state characteristic, we have x k = 0 for at least one k = 1, . . . , n. The Jacobian matrix atĒ has the form (11) with α k = 0 and has −b k = f k (S) − D k as an eigenvalue. Because S in >λ n , we haveS > λ n λ k and the eigenvalue −b k is positive. ThusĒ is unstable.
Assume now that S in < λ 1 . Then the washout E 0 = (S in , 0, . . . , 0) is the only equilibrium. It corresponds to the intersection of ∆ with the red steady-state characteristic y = 0. The Jacobian matrix at E 0 has the form (11) with α k = 0 for all k = 1, . . . , n and has −D and −b k = f k (S in ) − D k , k = 1, . . . , n, as eigenvalues. Because S in < λ 1 , all eigenvalues −b k are negative. Therefore E 0 is LES.
Assume now thatλ k < S in <λ k+1 for some k = 1, . . . , n − 1. We denote by E k = (S k , x k 1 , . . . , x k n ) the leftmost equilibrium, that is to say, the intersection of ∆ with the steady-state characteristic y = k i=1 h i (S). Then we have λ k < S k < λ k+1 .
From definition (9) we see that, h k+1 (S) = . . . = h n (S) = 0, for λ k < S < λ k+1 and then x k k+1 = . . . = x k n = 0. The Jacobian matrix at E k has the form (11) with α j = 0 for j = k + 1, . . . , n. We deduce that −b j = f j (S k ) − D j , j = k + 1, . . . , n are eigenvalues of the Jacobian matrix at E k . Because S k < λ k+1 , these eigenvalues are negative. The upper-left k × k square matrix of the Jacobian matrix has also the form (11) with Using Lemma 4.2, the eigenvalues of this matrix have negative real parts. Therefore, the equilibrium E k is LES. If we denote byĒ = (S, x 1 , . . . , x n ) an equilibrium point corresponding to the intersection of ∆ with a blue steady-state characteristic, we have x i = 0 for at least one i = 1, . . . , k. The Jacobian matrix atĒ has the form (11) with α i = 0 and hence −b i = f i (S) − D i is an eigenvalue. Because S in >λ k , we haveS > λ k λ i and then, the eigenvalue −b i is positive. ThusĒ is unstable.
In Fig. 2, we stained in red the part of the characteristics which correspond to LES equilibria, and in blue the unstable equilibria. Table 2 summarizes the previous results where the letter S (resp. U) means stable (resp. unstable). The absence of letter means that the corresponding equilibrium does not exist. Table 2. Existence and local stability of equilibria of (1) with n = 3.
We aim, now, to prove in the case of the multi-species model the global stability of the equilibrium E 1 = (S 1 ,x 1 , 0, . . . , 0) corresponding to the extinction of all species except the one which has the lowest break-even concentration.
Proposition 7. Assume that λ 1 < S in <λ 2 and that there exist constants α i > 0, for each i 2 satisfying λ i < S in such that where Then, the equilibrium E 1 is GAS for system (1) with respect to all solutions with Proof. Consider the Lyapunov function V = V (S, x 1 , . . . , x n ) defined as follows: where α i > 0 are the positive constants satisfying (12) if S < S in and α i > 0 are arbitrary if S > S in . The function V is continuously differentiable for 0 < S < S in and x i > 0, i = 1, . . . , n and positive except at the point E 1 , where it is equal to 0. The time derivative of V computed along the trajectories of (1) is given by: Note that, the first term of the above sum is always non-positive for 0 < S < S in and equals 0 for S ∈]0, S in [ if and only if S = S 1 or x 1 = 0. The second and the third term are obviously non-positive and vanish only if x 1 =x 1 and x i = 0 for i = 2, . . . , n. Finally, the last term of the above sum is always non-positive for every S ∈]0, S in [ and is equal to zero if and only if x i = 0 for i = 2, . . . , n. Then,V 0 and V = 0 if and only if S = S 1 , x 1 =x 1 and x i = 0 for i = 2, . . . , n. Hence, the result follows by applying the LaSalle extension theorem (see [21]).
5. Operating diagram. The operating diagram describes the asymptotic behavior of (1) when the concentration of the substrate in the feed bottle S in and the dilution rate D vary. In model (1), each parameter D i , i = 1, · · · , n, can be written as D i = D + A i , A i 0 where A i can be interpreted as the specific natural death rate of species i.
We first denotem i = sup S 0 f i (S) − A i and we assume thatm i > 0. For the description of the steady states and their stability, with respect to control parameters S in and D, we define the inverse function F i of the increasing functions f i , i = 1, · · · , n, so that: Note that the inverse function F i can be calculated explicitly in the case of the Monod growth functions considered in Section 6. In the following, we assume, without loss of generality, that: To illustrate the operating diagram, we also define the following functions: where J is a non-empty subset of {1, . . . , n}, andj = max{j : j ∈ J}. Let Γ J be the curve of equation S in = F J (D). Thus, there exist 2 n−1 curves that separate the plane (D, S in ) into regions of existence and stability of the different equilibria. Since the functions F J depend only on a 1 , . . . , a n−1 , the regions of the operating diagram are independent of a n . Hence, the intra-specific competition of the least competitive species has no effect on the coexistence region. For a better understanding, we illustrate the previous results in the cases n = 2 and 3, with the parameter values provided in Table 5.
In the case n = 2, the curves Γ i , i = 1, 2 and Γ 12 separate the operating plane (D, S in ) in four regions, as shown in Fig. 3(b-c), labeled as I k , k = 0, . . . , 3. Table 3 shows the existence and local stability of equilibria in the regions I k , k = 0, . . . , 3, when the curves Γ 1 and Γ 2 do not intersect. Note that the case where the curves Γ i intersect can be treated similarly. Table 3. Existence and local stability of steady states according to (D, S in ), in the case n = 2 and Γ 1 ∩ Γ 2 = ∅.
The transition from the region I 0 to the region I 1 by the curve Γ 1 (the red curve) corresponds to a saddle-node bifurcation making the equilibrium E 0 unstable (saddle point) with the appearance of an LES equilibrium E 1 . The transition from the region I 1 to the region I 2 by the curve Γ 2 (the blue curve) corresponds to the appearance of a saddle point E 2 by a bifurcation with a saddle point E 0 . The transition from the region I 2 to the region I 3 by the curve Γ 12 (the magenta curve) corresponds to a saddle-node bifurcation making the equilibrium E 1 unstable (saddle point) with the appearance of an LES equilibrium E * . When a 1 = 0, the operating diagram corresponds to the classical chemostat model (see Fig. 3(a)). Increasing a 1 leads to the emergence of the coexistence region I 3 and to the reduction of the region I 2 corresponding to the competitive exclusion of the second species (see Fig. 3(b-c)). Thus, the intra-specific competition of the most competitive species leads to changes in the size and emergence of coexistence regions.
Notice that the function F i (·) is not defined if sup S 0 f i (S) A i and we let F i (0) = +∞. In this case, the regions I 1 , I 2 and I 3 are empty. In addition, if A i > 0, then lim D−→0 + F 12 (D) = +∞, and since h 1 (F 2 (D)) > 0, we have F 2 (D) < F 12 (D) for all D ∈]0,m 2 [. For n = 3, the seven curves Γ i , Γ ij , i, j = 1, 2, 3 with i < j and Γ 123 determine the regions where various equilibria exist, and indicate their stability properties. These curves separate the operating plane (D, S in ) in ten regions, as shown in Fig.  4(a), labeled I k , k = 0, . . . , 9. Table 4 shows the existence and local stability of equilibria in the regions I k , k = 0, . . . , 9, when the curves Γ i , i = 1, 2, 3, do not intersect. Table 4. Existence and local stability of steady states of three species model.
The curve Γ 12 does intersect the curves Γ 23 and Γ 3 in D * 1 and D * 2 , respectively. According to the value of dilution rate D with respect to D * 1 and D * 2 , we obtain the three steady-state characteristics illustrated in Figs. 4(b) and 5(a-b).   Figure 5. Steady-state characteristics for (a) D * 1 < D < D * 2 and (b) D * 2 < D.

6.
Simulations. In the following, we illustrate the results obtained for system (1) in the case n = 3 and the functions f i (·) are of Monod-type, defined by: where m i is the maximum specific growth rate and K i is the Michaelis-Menten (or half-saturation) constant. Straightforward calculation shows that For the numerical simulations, we use the parameters provided in Table 5.  Table 5. Parameter values of model (1) with the Monod function (13) for n = 2 and n = 3.
For these parameter values, the curves S in = F i (D) do not intersect and we obtain the operating diagram in Fig. 4(a). Note that The steady-state characteristics are depicted in Fig. 4(b) for D < D * 1 , (D, S in ) = (0.6, 60) ∈ I 9 or even S in >λ 3 34.443, where there exist 2 3 equilibria for system (1). In this case, Fig. 6(a) shows the coexistence of the three species and the convergence towards the positive equilibrium E * (18.214, 9.021, 2.732, 1.199) for several positive initial conditions. Fig. 6(b) shows the competitive exclusion of the third species for (D, S in ) = (0.6, 32) ∈ I 8 or evenλ 13 29.840 < S in <λ 3 .
For several positive initial conditions, the solutions of (1) converge towards the equilibrium E 0 (0.6, 0, 0, 0). 7. Conclusion. In this paper, we considered the mathematical model describing multi-species competition for a single growth-limiting resource in a chemostat proposed by De Leenheer et al. [11]. For monotonic growth functions and different dilution rates, we proved that the outcome of competition does not always satisfy the CEP which predicts that only one species can exist in the long term. Indeed, we proved that according to the operating parameters (D the dilution rate and S in the concentration of substrate in the feed bottle) several species can coexist: the system has one and only one LES equilibrium for which a certain number p n of the species are present. In the other equilibria, the present species are less than p and all these equilibria are unstable.
If the intra-specific parameters values of all species are positive, we proved the existence of a stable positive equilibrium for certain values of the operating parameters. We determined precisely the region of the operating parameters for which the coexistence of all species holds. This region is one of the regions of the operating diagram which depicts the existence and the stability of each equilibrium. We generalize the Lyapunov function proposed by Wolkowicz and Lu [24], in the case of two species, to prove the global stability of the unique LES equilibrium, corresponding to the extinction of all species except the one with the lowest breakeven concentration. This result shows that even if the species exhibits intra-specific competition, then the CEP may hold. Indeed, if λ 1 < S in <λ 2 then all species except the one with the lowest break-even concentration are excluded. Moreover, if S in < λ 1 then as in the classical chemostat, all species are washed out. If S in >λ 2 then the coexistence of two species or more is possible.
The GAS result of the equilibrium where only one species is present, in the case where it is LES, imposes some assumptions on the growth functions, which do not necessarily hold for all monotone growth functions. We conjecture that the result is true for all monotone growth function. The problem is certainly as difficult as the similar classical one with no intra-specific competition which is still open. Another interesting and challenging open problem for further work is to find Lyapunov functions showing the global stability of the coexistence equilibrium under some assumptions on the growth functions.
In the case of two species, we showed that coexistence occurs also if the death rate parameter of the weaker (less competitive) species is non-negative (the zero value is allowed) and the death rate parameter of the stronger species is positive. The coexistence is not unexpected, since the lack of intra-specific competition for the weaker species has a beneficial effect on its ability to survive. What is much more interesting is that coexistence may be possible, even though the weaker competitor exhibits arbitrarily high levels of intra-specific competition. A surprising and unexpected result is that the operating diagram does not depend on the intraspecific competition parameter a n of the weaker species. Obviously, the values of some equilibrium components will depend on a n , but the conditions of existence and stability of all equilibria do not involve the parameter a n .
The operating diagram depicts regions in the (D, S in ) plane in which the various outcomes occur. To maintain the coexistence of species in the chemostat, the parameter values of D and S in should be chosen in the coexistence region, not in the other regions where exclusion of at least one species occurs. The operating diagram is of great importance in the applied literature since it permits to determine critical limits for the good functioning of the chemostat and the protection of the weaker competing species in the microbial ecosystems. We have showed how the intra-specific competition of the n − 1 most efficient species introduces a coexistence region of n species, while the death rate parameter of the least competitive species has no effects on the regions of the operating diagram. When the intra-specific competition terms of the n − 1 stronger species are zero, we find the operating diagram of the classical chemostat model. Increasing these terms reduces the regions of competitive exclusion and increases the region of coexistence. The simulations illustrate the mathematical results demonstrated in the case where the growth rates are of Monod-type.