EFFECT ON PERSISTENCE OF INTRA-SPECIFIC COMPETITION IN COMPETITION MODELS

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Electronic Journal of Differential Equation, Vol. 2007(2007, No. 25, pp. 0. ISSN: 072-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu ftp ejde.math.txtate.edu (login: ftp EFFECT ON PERSISTENCE OF INTRA-SPECIFIC COMPETITION IN COMPETITION MODELS CLAUDE LOBRY, FRÉDÉRIC MAZENC Abtract. An ecological model decribing the competition for a ingle ubtrate of an arbitrary number of pecie i conidered. The mortality rate of the pecie are not uppoed to have all the ame value and the growth function of the ubtrate i not uppoed to be linear or decreaing. Intra-pecific competition i taken into account. Under additional technical aumption, we etablih that the model admit a globally aymptotically table poitive equilibrium point. Thi enure peritence of the pecie. Our proof relie on a Lyapunov function.. Introduction Current reearch effort focu on the analyi of the olution of model of chemotat with everal pecie competing for one growth-limiting nutrient and undergoing an extra competition, which reult from the difficulty of acce to the ubtrate encountered by the micro-organim. Thee model belong to a general cla of ytem of the form ṡ = f( h i (, x Y i x i, ẋ = [h (, x d ]x, ẋ n = [h n (, x d n ]x n, (. evolving on E = (0, + n+. In thee ytem, i the concentration of the nutrient, the x i are the concentration of pecie of organim, x = (x,, x n and the Y i are poitive contant called yield coefficient. The function h i atify h i (0, x = 0 for all x becaue, no growth of the pecie i poible in the abence of ubtrate. In addition, the function h i are increaing with repect to and decreaing with repect to each component x j of the vector x to take into account the fact that the more there are micro-organim, the more difficult i their acce to the nutrient. Recent work [7,, 4, 2] are devoted to tability analyi problem for (. in the particular cae where the function h i depend only on and x i. Thi property 2000 Mathematic Subject Claification. 92B05, 92D25. Key word and phrae. Specie coexitence; ecology; population dynamic. c 2007 Texa State Univerity - San Marco. Submitted May 4, 2007. Publihed September 24, 2007.

2 C. LOBRY, F. MAZENC EJDE-2007/25 expree the o-called intra-pecific competition: the tronget the concentration of a pecie i, the mallet i it growth. In other word, the acce of a micro-organim to the nutrient i uppoed be hampered only by the preence of micro-organim of it own pecie. The phenomenon of intra-pecific competition can be explained by the flocculation proce, which i of major importance in watewater treatment plant: the preence of flock limit the acce of the bioma to the ubtrate. In [3] an effective way to include flocculation in exiting model of chemotat, i propoed. It i hown that under certain condition, thi lead to denity-dependent growth function of the form h i (, x i. Thi etablihe the link between the limited acce to the ubtrate inide the flock, and the growth characteritic of the bioma on the level of the bioreactor. The work [7] and [] preent a tudy of the ytem (. in the particular cae where only intra-pecific competition occur, where f i a linear function of the form f( = D( in and where the mortality can be neglected, which correpond to the cae where d = = d n = D. The main meage conveyed by thee work i that intra-pecific competition may lead to the exitence of a globally aymptotically table poitive equilibrium point and therefore can explain coexitence of the pecie. Hence, thee work complement the literature (ee for intance [2], [5], [4], [6] devoted to the problem of explaining why coexitence i oberved in real-world application, in pite of the prediction of the Competitive Excluion Principle, which, generally peaking, claim that when there i a ingle nutrient, aymptotically only one pecie urvive and the other tend to extinction. However, in more complex ecological context, the growth of the ubtrate i not linear, not necearily decreaing, and the mortality term cannot be neglected. Firt attempt to cope with the correponding model are made in [9] and [4]. In [9], for a general model of chemotat with two pecie which take into account intra-pecific effect, the peritence of the two pecie i etablihed. The technique of proof i baed on a comparion principle. In [4], it i hown that a general ytem (. with different removal rate d i admit a globally aymptotically table poitive equilibrium point, provided that only intra-pecific competition occur and f i decreaing and it decay i ufficiently fat. The main advantage of thi reult i that it applie to ytem (. for which no explicit expreion for the growth function f and h i i available. However, numerical imulation ugget that the tability property hold even when f i not decreaing. The objective of the preent paper i to how that when f belong to a family of function which contain function which have a poitive, but mall, firt derivative and when the growth function h i (, x i admit a decompoition of the form h i (, x i = µ i (θ i (x i where the function θ i are decreaing and where the function µ i belong to a family lightly larger than the family of the Michaeli-Menten function, then global aymptotic tability of a poitive equilibrium point can be etablihed. Our technique of proof relie on a Lyapunov approach which i ignificantly different from the one ued in [4] but i reminicent of the one preented firt in [8] and i incorporated in [6, Chapter 2]. Thi Lyapunov function allow to prove the Competitive Excluion Principle in the particular cae where the growth and there are different removal rate d i. The fact that the function µ i are of the Michaeli-Menten (or Monod type i crucial in thi Lyapunov approach. For general growth function and ditinct removal rate d i, the Lyapunov function approach of [8] doe not apply function are f( = D( in and h i (, x i = µ i ( = Ki L i+

EJDE-2007/25 EFFECT ON PERSISTENCE 3 and the problem of proving the Competitive Excluion Principle in that cae i till open. To undertand the difficulty of thi problem, it i worth reading for intance the paper [7], [0], [8] where, through elegant and ophiticated proof, partial olution to thi problem are etablihed, in more general context. Oberve that, in contrat to the Lyapunov function propoed in [8], the Lyapunov function we exhibit i a trict Lyapunov function i.e. it derivative along the trajectorie of the ytem i a negative definite function of the tate variable. Thi property make it poible to quantify the effect of diturbance or error of modeling (a illutrated for intance by [], [5], [3]. In particular, it follow that the tability reult we will etablih till hold when, intead of being Michaeli- Menten function, the growth function are almot Michaeli-Menten function, in a ene which can be made precie by mean of the Lyapunov function. Finally, we wih to point out that we conjecture that the global tability reult we will etablih can be extended to ytem with general growth function, but we alo preume that proving thi extenion i a difficult a proving the general verion of the Competitive Excluion Principle. The paper i organized a follow. In Section 2, we introduce the family of ytem we tudy a long a baic aumption, accompanied with preliminary reult. The main reult i tated and proved in Section 3. Section 4 i dedicated to imple particular cae. 2. Sytem decription, preliminary reult and comment Preliminarie. Throughout the paper, the function are uppoed to be of cla C. The argument of the function will be omitted or implified whenever no confuion can arie from the context. Conider a differential equation ẋ = F (x (2. with x R p where F i continuouly differentiable on R p. An equilibrium point of thi ytem i called poitive equilibrium point if all it component are poitive. Let G c be cloed and poitively invariant for (2. and let u aume that the origin i an equilibrium point of (2.. A function V i called a Lyapunov function for (2. on an open et G G c if (i V i continuouly differentiable on G, (ii For each x G, the cloure of G, the limit lim x x, x G V (x exit a either a real number or +, (iii V x (xf (x 0 on G. (iv A function V i called a trict Lyapunov function for (2. if V x (xf (x < 0 for all x G, x 0. (v A function V i aid to be proper if for each x b G\G, the boundary of G, lim x xb, x G V (x = +.

4 C. LOBRY, F. MAZENC EJDE-2007/25 The Model and the Baic aumption. We conider the ytem ṡ = f( µ i (θ i (x i x i, ẋ = [µ (θ (x d ] x, ẋ n = [µ n (θ n (x n d n ] x n, (2.2 evolving on the tate domain D f = [0, + [0, + [0, + where the d i are poitive contant. We introduce the aumption: (H The function f i uch that f(0 0. (H2 The function θ i (x i are poitive, decreaing and θ i (0 =. The function θ i (x i x i are increaing. (H3 There exit (, x,, x n (0, in (0, + (0, + uch that and, for all i {,, n}, f( = d j x j (2.3 j= µ i ( θ i (x i = d i. (2.4 (H4 The function µ i are bounded, zero at zero, increaing and µ i (0 > 0. There i a poitive function Ω and poitive contant c i uch that, for all i {,, n}, µ i ( c i µ i (0 = Ω(0 (2.5 and, for all > 0,, c i µ i ( µ i ( µ i ( = Ω(. (2.6 Dicuion of the aumption Aumption (H enure that the domain D f i poitively invariant. In the ytem (2.2 the yield coefficient are equal to. Without lo of generality, thi aumption can be made becaue thee parameter can be eliminated by a imple linear change of coordinate. The function f( = D( in (which i preent in model of chemotat atifie Aumption (H. Oberve that the requirement (2.6 i equivalent to µ i ( = c i µ i ( c i + ( Ω(. (2.7 We hall ee in Section 4 that thi requirement i atified in the particular cae where the function µ i are of Monod type. Moreover, oberve that the function Ω i continuou on [0, +. The function θ i expre the intra-pecific competition: the growth of a pecie i inhibited by it own concentration. Auming that the function θ i (x i x i are increaing i relevant from a biological point of view. Aumption (H3 i not retrictive: if a poitive equilibrium point exit, then necearily thi aumption i atified.

EJDE-2007/25 EFFECT ON PERSISTENCE 5 Equilibrium point. Under the aumption we have introduced, we can eaily etablih the exitence and unicity of a poitive equilibrium point for the ytem (2.2: Lemma 2.. Aume that the ytem (2.2 atifie Aumption (H (H4. Then the point E = (, x,, x n i a poitive equilibrium point. Proof. From Aumption (H3, it follow that E i a poitive equilibrium point of the ytem (2.2. Lemma 2. allow u to introduce the aumption: (H5 The function i poitive. Γ( = f( f( + n [µ i( µ i (]θ i (x i x i (2.8 Remark. One can check that Aumption (H5, in combination with Aumption (H2 and the fact that the function µ i are increaing enure that the ytem (2.2 admit only one poitive equilibrium point. 3. Main reult In thi ection, we tate and prove the main reult of the work. Theorem 3.. Aume that the ytem (2.2 atifie Aumption (H (H5. Then the poitive equilibrium E = (, x,, x n i a globally aymptotically and a locally exponentially table equilibrium point of the ytem (2.2 on D o = (0, + (0, + (0, +. Proof of Theorem 3.. Attractive invariant domain. Lemma 3.2. The domain D f and D o are a poitively invariant domain. Proof. The ign propertie of the function f and the fact that each function µ i i zero at zero imply that D f and D o are a poitively invariant domain. Lyapunov contruction. Let u ue the variable =, x i = x i x i. Then from Lemma 2., we deduce that = f( f( µ i (θ i (x i x i + µ i ( θ i (x i x i, x = [µ (θ (x µ ( θ (x ]x, x n = [µ n (θ n (x n µ n ( θ n (x n ] x n. From the definition of Γ in (2.8 and the equality (3. µ i (θ i (x i µ i ( θ i (x i = µ i ( [θ i (x i θ i (x i ] + [µ i ( µ i ( ]θ i (x i (3.2

6 C. LOBRY, F. MAZENC EJDE-2007/25 it follow that = Γ( + µ i ( [θ i (x i x i θ i (x i x i ], x x = µ ( [θ (x θ (x ] + [µ ( µ ( ]θ (x, (3.3 x n x n = µ n ( [θ n (x n θ n (x n ] + [µ n ( µ n ( ]θ n (x n. Let u introduce the implifying notation: α i (x i = µ i ( θ i(x i θ i (x i x i x i, β i (x i = θ i(x i x i θ i (x i x i x i x i. (3.4 Aumption (H2 enure that the function α i and β i are poitive. The ytem (3.3 rewrite = Γ( µ i ( β i (x i x i, x x = α (x x + [µ ( µ ( ]θ (x, (3.5 x n = α n (x n x n + [µ n ( µ n ( ]θ n (x n. x n From Aumption (H4, we deduce that Ω( = Ω(Γ( 2 c i [µ i ( µ i ( ]β i (x i x i, β (x α (x β (x c x x = c x 2 + c [µ ( µ ( ]β (x x, θ (x x θ (x β n (x n α n (x n β n (x n c n x n x n = c n x 2 n + c n [µ n ( µ n ( ]β n (x n x n. θ n (x n x n θ n (x n 0 0 (3.6 Thee equalitie lead u to conider the function l xi β i (x i + l U(, x,, x n = Ω(l + dl + c i ldl (3.7 l + θ i (x i + l(x i + l which i poitive definite on D t = (, + ( x, + ( x n, + becaue the contant c i and the function Ω, β i, θ i are poitive. From (3.6, we deduce that it derivative along the trajectorie of (3. atifie U = W (, x,, x n (3.8 with W (, x,, x n = Ω(Γ( 2 + α i (x i β i (x i c i x 2 i. (3.9 θ i (x i

EJDE-2007/25 EFFECT ON PERSISTENCE 7 Stability analyi. Let u firt prove the following reult. Lemma 3.3. The function U defined in (3.7 i poitive definite and proper on D t. Proof. We have already hown that U i poitive definite. Next, oberve that µ ( µ ( ( Ω( = c = c µ (. (3.0 µ ( µ ( Therefore, ince µ i increaing, for all 2, ( Ω( c µ ( > 0. (3. µ (2 We deduce eaily that lim + 0 l Ω(l + dl = +. (3.2 l + Since the function Ω i poitive and continuou on [0, +, we deduce that Next, oberve that l lim Ω(l + dl = +. (3.3 l + 0 β i (x i θ i (x i = θ i(x i x i θ i (x i x i θ i (x i (x i x i = θi(xi x i θ i(x ix i. (3.4 xi x i According to Aumption (H2, θ i i decreaing and θ i (x i x i i increaing. deduce that, for all x i 2x i, We β i (x i θ i (x i θ i(x i x i > 0. (3.5 θ i (2x i 2x i It follow that xi β i (x i + l lim ldl = +. (3.6 x i + 0 θ i (x i + l(x i + l Since the function β i and θ i are poitive and continuou on [0, +, we deduce that xi β i (x i + l lim ldl = +. (3.7 x i x i 0 θ i (x i + l(x i + l At lat, from (3.3, (3.6, (3.7 we deduce that U i proper. Next, by taking advantage of Aumption (H5, one can eaily prove that the function W i poitive definite on D t. Thi property and the reult of Lemma 3.3 enure that the Lyapunov theorem applie and therefore lim (t = 0, t + lim x i(t = 0, i =,, n. (3.8 t + Moreover, the local exponential tability of the origin of the ytem (3. can be proved by verifying that both U and W are, on a neighborhood of the origin, lower bounded by a poitive definite quadratic function. By returning to the original coordinate, we deduce that E i a globally aymptotically and a locally exponentially table equilibrium point of the ytem (2.2 on D o.

8 C. LOBRY, F. MAZENC EJDE-2007/25 4. Particular cae, example Familie of function µ i atifying Aumption (H4. In thi ection, we exhibit familie of function which fullfill Aumption (H4. Lemma 4.. Let u conider n linear function: µ i ( = K i (4. with K i > 0. Thee function atify Aumption (H4 with Ω( = and, for i =,, n, c i =. The proof of the above lemma i trivial and i omitted. Lemma 4.2. Let u conider n function µ i ( = K i A( L i B( + A( (4.2 with K i > 0, L i > 0 and where A i increaing and atifie A(0 = 0, A (0 > 0 and B i poitive and nondecreaing. Thee function atify Aumption (H4 with Ω( = and, for i =,, n, c i = LiB( +A( L i. A(B( A( B( A( We remark that when A( = and B( =, the function (4.2 belong to the family of the Michaeli-Menten function and the correponding function Ω and contant c i are Ω( =, c i = Li+ L i. Proof of Lemma 4.2. The reult i a conequence of the imple calculation: µ i ( µ i ( µ i ( = L ib( + A( [ K i A( K i A( L i B( + A( K i A( ] L i B( + A( = [A((L i B( + A( A( (L i B( + A( ] A( L i B( + A( L i B( + A( = [A((L i B( + A( A( (L i B( + A( ] A( L i B( + A( L i A(B( A( B( =. L i B( + A( A( (4.3 Since A i increaing, atifie A(0 = 0, A (0 > 0 and B i poitive and nondecreaing, it follow that the function Ω( = A(B( A( B( A( i well-defined and poitive on [0, +. Familie of function θ i atifying Aumption (H2. In thi ection, we exhibit familie of function which fulfill Aumption (H2. Lemma 4.3. Conider a function θ(x = a (a + x ν (4.4 with a > 0 and ν (0, ]. Then thi function i poitive, decreaing and xθ(x i increaing.

EJDE-2007/25 EFFECT ON PERSISTENCE 9 Proof. One can check eaily that θ i poitive with a negative firt derivative. Moreover, d[θ(xx] = a (a + xν νx(a + x ν a + ( νx dx (a + x 2ν = a (a + x ν+ > 0 (4.5 and therefore xθ(x i increaing. Example. We illutrate Theorem 3. by applying it to the ytem ṡ = 7 6 x 4 x 2, + + x 2 + + x 2 ẋ = [ ] x, + + x 2 ẋ 2 = [ 4 ] x 2. 2 + + x 2 (4.6 With our general notation, we have f( = 7 6, µ ( = +, µ 2( = 4 2+, θ (x = +x, θ 2 (x 2 = +x 2. Oberve that the growth function of the ubtrate f( = 7 6 i a contant. Therefore the reult of [], [7] or [4] cannot be ued to etablih the global aymptotic tability of an equilibrium point of (4.6. Let u verify that the ytem (4.6 atifie the aumption (H (H5. ( Since f(0 = 7 6 > 0, Aumption (H i atified. (2 We deduce from Lemma 4.3 that Aumption (H2 i atified. (3 Aumption (H3 i atified: the poitive point E = (2, 3, i an equilibrium point of (4.6. (4 We deduce from Lemma 4.2 that Aumption (H4 i atified. Since µ and µ 2 are Monod function, one can chooe Ω( =, c = 3, c 2 = 2. (5 Simple calculation yield 7 6 7 6 + [ 2 Γ( = 3 + ] 3 + 3 2 + [ 2 4 2+] 2 = 2( + + 2 +. Therefore, the function Γ i poitive and Aumption (H5 i atified. We conclude that Theorem 3. applie. It follow that E i a globally aymptotically and a locally exponentially table equilibrium point of (4.6. Moreover, the derivative of the Lyapunov function U(, x, x 2 x l β (x + l = dl + c 0 l + 0 θ (x + l(x + l ldl + c β 2 (x 2 + l 2 0 θ 2 (x 2 + l(x 2 + l ldl = ln ( + + c x l + x 0 x + l dl + c x2 2 l + x 2 0 x 2 + l dl = ln ( + + 9 [ x x ln ( + x ] + x 2 x 2 ln ( + x 2 4 x x 2 (4.7 along the trajectorie of (4.6 atifie with ( W (, x, x 2 = 2( + + 2 2 + x2 U = W (, x, x 2 (4.8 + 9 8( + x x2 + ( + x 2 x2 2. (4.9

0 C. LOBRY, F. MAZENC EJDE-2007/25 Reference [] D. Angeli, E. Sontag, Y. Wang. A characterization of integral input-to-tate tability. IEEE Tran. Autom. Control, vol. 45, no. 6, pp. 082-097, 2000. [2] G.J. Butler, S.B. Hu, P. Waltman. A mathematical model of a chemotat with periodic wahout rate. SIAM J. Appl. Math., vol. 45, pp. 435-449, 985. [3] B. Haegeman, C. Lobry, J. Harmand., An effective model for flocculating bacteria with denity-dependent growth dynamic. American Intitute of Chemical Engineering Journal, to appear. [4] P. De Leenheer, H.L. Smith. Feedback control for the chemotat. J. Math. Bio. vol. 46, pp. 48-70, 2003. [5] P. De Leenheer, D. Angeli, E. Sontag. A feedback perpective for chemotat model with crowding effect. In Lecture Note in Control and Inform. Sci., vol. 294, pp. 67-74 pp. 2003. [6] H.I. Freedman, J. So, P. Waltman. Coexitence in a model of competition in the chemotat incorporating dicrete delay. SIAM J. Appl. Math., vol.49, No. 3, pp. 859-870, June 989. [7] F. Grognard, F. Mazenc, A. Rapaport. Polytopic Lyapunov function for the tability analyi of peritence of competing pecie. 44th IEEE Conference on deciion and control and European Control Conference, Seville, Spain 2005 and Dicrete and Continuou Dynamical Sytem-Serie B, vol. 8, No, July 2007. [8] S.B. Hu. Limiting behavior for competing pecie. SIAM Journal on Applied Mathematic, 34:760-3, (978. [9] L. Imhof, S. Walcher. Excluion and peritence in determinitic and tochatic chemotat model. J. Differential Equation 27, pp. 26-53, 2005. [0] B. Li. Global aymptotic behavior of the chemotat: general repone function and different removal rate. SIAM J. Appl. Math. 59, pp. 4-422, 998. [] C. Lobry, F. Mazenc, A. Rapaport. Peritence in ecological model of competition for a ingle reource. C.R. Acad. Sci. Pari, Ser. I 340, 99-204, 2005. [2] C. Lobry, F. Mazenc, A. Rapaport. Peritence in Ratio-dependent Model for Conumer- Reource Dynamic. Sixth Miiippi State Conference on Differential Equation and Computational Simulation. Electron. J. Diff. Eqn., Conference 5 (2007, pp. 2-220. [3] M. Malioff, F. Mazenc. Further Remark on Strict Input-to-State Stable Lyapunov Function for Time-Varying Sytem. Automatica, Vol. 4, pp. 973-978, 2005. [4] F. Mazenc, C. Lobry, A. Rapaport. Stability Analyi of an Ecological Model. Springer- Verlag, ISSN:070-8643, Volume 34/2006 Title: Poitive Sytem ISBN: 978-3-540-3477-2, pp. 3-20. [5] L. Praly, Y. Wang. Stabilization in pite of matched unmodeled dynamic and an equivalent definition of input-to-tate tability. Math. Control Signal Syt., vol. 9, pp. -33, 996. [6] H.L. Smith, P. Waltman. The Theory of the Chemotat. Cambridge Univerity Pre, 995. [7] L. Wang, G.S.K. Wolkowicz. A delayed chemotat model with general nonmonotome repone function and differential removal rate. Journal of Mathematical Analyi and Application, Volume 32, Iue, September 2006, Page 452-468. [8] G.S.K. Wolkowicz, Z. Lu. Global Dynamic of a Mathematical Model of Competition in the Chemotat: General Repone Function and Differential Death Rate. SIAM J. Appl. Math., Volume 52, No, pp. 222-233, February 992. Claude Lobry Projet MERE INRIA-INRA, UMR Analye de Sytème et Biométrie, INRA 2, pl. Viala, 34060 Montpellier, France E-mail addre: claude.lobry@inria.fr Frédéric Mazenc Projet MERE INRIA-INRA, UMR Analye de Sytème et Biométrie, INRA 2, pl. Viala, 34060 Montpellier, France E-mail addre: mazenc@upagro.inra.fr

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