c 2004 Society for Industrial and Applied Mathematics

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1 SIAM J. APPL. MATH. Vol. 65, No., pp c 24 Society for Indstrial and Applied Mathematics A MATHEMATICAL MODEL OF COMPETITION FOR TWO ESSENTIAL RESOURCES IN THE UNSTIRRED CHEMOSTAT JIANHUA WU, HUA NIE,, AND GAIL S. K. WOLKOWICZ Abstract. A mathematical model of competition between two species for two growth-limiting, essential (complementary) resorces in the nstirred chemostat is considered. The eistence of a positie steady-state soltion and some of its properties are established analytically. Techniqes inclde the maimm principle, the fied point inde, and nmerical simlations. The simlations also seem to indicate that there are regions in parameter space for which a globally stable positie eqilibrim occrs and that there are other regions for which the model admits bistability and een mltiple positie eqilibria. Key words. chemostat, essential or complementary resorces, steady-state soltion, fied point inde, nmerical simlation AMS sbject classifications. 35K55, 35J65, 92A7 DOI..37/S Introdction. An apparats called the chemostat, sed for the continos cltre of microorganisms, has played an important role in ecology. It has been thoght of as a lake in a laboratory. See [9, 25, 29] for a description of the apparats and the general theory. In the basic set p, the cltre essel is assmed to be well stirred. One or more poplations of microorganisms grow and/or compete eploitatiely for a single, nonreprodcing, growth-limiting ntrient that is spplied at a constant rate. The contents of the cltre essel are remoed at the same constant rate as the medim containing the ntrient is spplied, and ths the olme of the cltre essel remains constant. Species-specific parameters can be measred one species at a time, and based on these parameters the theory predicts the qalitatie otcome in adance of actal competition. In particlar, the theory predicts that the species with the lowest break-een concentration ecldes all other competitors (see [6, 4, 29]). Eperiments confirmed this prediction in the case of aotrophic bacterial strains competing for limiting tryptophan []. Mathematical analysis of chemostat models inoling two limiting resorces nder the assmption that the cltre essel is well stirred can be fond, for eample, in [2, 3, 7, 3, 2, 7, 8, 9, 28]. When more than one resorce is limiting, it is necessary to consider how these resorces promote growth. At one etreme are resorces that are sorces of different essential sbstances that mst be taken together, becase each sbstance flfills different physiological needs with respect to growth, for eample, a Receied by the editors Febrary 22, 23; accepted for pblication (in reised form) Febrary 6, 24; pblished electronically September 24, 24. This work was spported in part by the Natral Science Fondation of China, the Ecellent Yong Teachers Program of the Ministry of Edcation of China, the Fondation for Uniersity Key Teacher of the Ministry of Edcation of China, the Scholarship Fondation of CSC, the Institte for Mathematics and its Applications with fnds proided by the NSF, and by the Natral Sciences and Engineering Research Concil of Canada. Department of Mathematics, Shaani Normal Uniersity, Xi an, Shaani 762, People s Repblic of China (wjha@snn.ed.cn). Department of Mathematics and Statistics, McMaster Uniersity, Hamilton, Ontario, Canada L8S 4K (wolkowic@mcmaster.ca). 29

2 2 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ carbon sorce and a nitrogen sorce. Sch resorces are called complementary by Leon and Tmpson [7], Rapport [22], and Baltzis and Fredrickson [4]; essential by Tilman [28]; and heterologos by Harder and Dijkhizen [2]. The model of eploitatie competition for two essential resorces in the wellstirred case is gien by S t =(S S)D y s g (S, R) y s2 g 2 (S, R), R t =(R R)D y r g (S, R) y r2 g 2 (S, R), t = [ D + g (S, R)], t = [ D + g 2 (S, R)]. S(t), R(t) denote the ntrient concentrations at time t, and (t) and (t) denote the biomass of each poplation in the cltre essel. S > and R > are constants that represent the inpt concentrations of ntrients S and R, respectiely, D is the diltion rate, and y si and y ri,i=, 2, are the corresponding growth yield constants. The response fnctions are denoted g i (S, R) = min(p i (S),q i (R)), i =, 2, where p i (S) denotes the response fnction of the ith poplation when only resorce S is limiting and q i (R) denotes the response fnction of the ith poplation when only resorce R is limiting. We will consider the case that the Monod model for eploitatie competition for one resorce is generalized to the two essential resorces case, i.e., p i (S) = ms i S K si +S, q i (R) = mr i R K ri +R, i =, 2, where m s i, m ri, K si, K ri, are positie constants. In this paper, we stdy the nstirred chemostat and consider two species competition for two, growth-limiting, nonreprodcing essential resorces. Motiated by the work on the nstirred chemostat in the case of one limiting resorce (see [5, 8, 5, 6, 2, 23, 24, 25, 26, 3, 3] ) and in the case of two limiting resorces in [32], the model takes the form of the following reaction-diffsion eqations: S t = ds y s g (S, R) y s2 g 2 (S, R), R t = dr y r g (S, R) y r2 g 2 (S, R), t = d + g (S, R), t = d + g 2 (S, R), <<, t>, <<, t>, <<, t>, <<, t>, with bondary conditions S (,t)= S, S (,t)+γs(,t)=, (,t)+γ(,t)=, R (,t)= R, (,t)=, (,t)=, R (,t)+γr(,t)=, (,t)+γ(,t)=. The bondary conditions are ery intitie. Readers may refer to [5, 6, 26] for their deriation. These eqations can be simplified sing the nondimensional ariables and parameters defined as follows: S = S S, R = R R, α = S y s R y r, β = R y r2 S y s2,ḡ i ( S, R) = min( ms S i K si + S, m ri R ), i =, 2, ū = K ri + R y s S, = y r2 R, where K si = Ks i S, Kri = Kr i R, i =, 2. For more conenient notation, we drop the bars on the nondimensional

3 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 2 ariables and parameters, yielding the following model: () S t = ds g (S, R) βg 2 (S, R), R t = dr αg (S, R) g 2 (S, R), t = d + g (S, R), t = d + g 2 (S, R), <<, t>, <<, t>, <<, t>, <<, t>, with bondary conditions S (,t)=, R (,t)=, (,t)=, (,t)=, S (,t)+γs(,t)=, R (,t)+γr(,t)=, (,t)+γ(,t)=, (,t)+γ(,t)=, and initial conditions S(, ) = S (), R(, ) = R (), (, ) = (), (, ) = (). Denote ϕ = S + + β, ϕ 2 = R + α +, where ϕ i,i=, 2, is the soltion of ϕ it = dϕ i, ϕ i (,t)=, ϕ i (,t)+γϕ i (,t)=, ϕ i (, ) = ϕ i (). <<, t>, Then and satisfy ( ) t = d + g (ϕ β,ϕ 2 α ), <<, t>, t = d + g 2 (ϕ β,ϕ 2 α ), <<, t>, (,t)=, (,t)+γ(,t)=, t >, (,t)=, (,t)+γ(,t)=, t >. This paper is deoted to determining the positie soltion of this two-species model of eploitatie competition for two essential resorces in the nstirred chemostat. Since the reaction terms are Lipschitz continos, bt not C, many methods sed to analyze elliptic systems do not apply. This makes the analysis more difficlt. Some methods sed to proe the eistence of the positie eqilibrim in the region D = {(ˆλ, ˆλ 2 ):ˆλ >, ˆλ2 > } occpy a major portion of the paper, where ˆλ i,i=, 2, is defined in the net section. The main reslt is established in Theorem 3. The other related reslts are also obtained in section 2. Etensie nmerical stdies were rn, and some conclsions are smmarized in section 3. The simlations conince s that mch more comple dynamics can occr in region D. The paper is organized as follows. In section 2, the eistence of a positie steadystate soltion and some of its properties are established by sing the maimm principle and fied point inde theory, which is closely related to bonding the principal eigenales of certain differential operators. Some reslts on etensie nmerical stdies are reported in section 3, complementing the mathematical reslts in section 2, and a nmber of typical figres chosen from many simlations are also gien in this section.

4 22 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ 2. The positie steady-state soltion. First, we consider the steady state of system (): (2) ds g (S, R) βg 2 (S, R) =, dr αg (S, R) g 2 (S, R) =, d + g (S, R) =, d + g 2 (S, R) =, <<, <<, <<, <<, with bondary conditions S () =, R () =, ()=, ()=, S () + γs()=, R () + γr()=, () + γ()=, () + γ()=. It follows that S + + β = z, R + α + = z, where z = z() = +γ γ. Then and satisfy (3) d + g (z β,z α ) =, d + g 2 (z β,z α ) =, ()=, () + γ()=, ()=, () + γ()=. <<, <<, Let λ i be the principal eigenale and let φ i () > on[, ], i =, 2, be the corresponding eigenfnction, normalized as ma [,] φ i () =, of the following problem: (4) dφ i + λ i φ i g i (z,z) =, <<, φ i ()=, φ i () + γφ i ()=. Let U() be the soltion of (5) du + Ug (z U, z αu) =, U ()=, U () + γu()=, <<, and let U(, t) be the soltion of (6) U t = du + Ug (ϕ U, ϕ 2 αu), U (,t)=, U (,t)+γu(,t)=, U(, ) = U (). <<, t>, From Lemmas and Theorem 2.5 in [32] we hae the following lemma. Lemma. If λ <, then there eists a niqe positie soltion U() of (5), satisfying <U<min{, α }z on [, ]. If λ, the only nonnegatie soltion of (5) is U =. Frthermore, lim t U(, t) =U() if λ <, and lim t U(, t) = if λ >. Remark. If λ 2 <, a similar reslt holds for V (), where V () is the niqe positie soltion of dv + Vg 2 (z βv,z V )=, V ()=, V () + γv ()=. <<,

5 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 23 Since we are only concerned with the nonnegatie steady-state soltions of (3), there is no loss of generality if we redefine { msi S K p i (S) = si +S, S, { mri R K q i (R) = ri +R, R,, S <,, R <. Lemma 2. Sppose (, ) is the nonnegatie soltion of (3). Then (i) > or, and > or ; (ii) + β < z, α + <z; (iii) U, V. Moreoer, <U or U, and <V or V. Proof. (i) This part can be proed by the maimm principle, in the sal way, and the details are omitted here. (ii) Define w = + β z. Note that by (3) it follows that dw + g ( w, z α )+βg 2 ( w, z α ) =, w () = and w () + γw()=. First we show that w on[, ]. Sppose not. If w() >, then w () <. Therefore, there eists a [, ) so that for all (a, ], w() >, and either a = or w(a) =. Bt then for all [a, ], w = and so w () =w () <, i.e., w() is decreasing there. Since w ()=>, a. Bt a> is also impossible since then w(a) =,w() is decreasing in [a, ], and w() >. Therefore, w(). Net, assme there eists [, ) with w( ) >. Then there eist δ and δ 2 > sch that w() > for all ( δ, + δ 2 ) (, ), w( + δ 2 )=, and either δ = or w( δ )=. Bt then for all [ δ, + δ 2 ], w () = and so w () is constant. Since w( + δ 2 )=, it follows that w ( + δ 2 ), and so w() is nonincreasing on [ δ, + δ 2 ]. Then δ, since w ()=, and so w( δ )=. Therefore, w() on[ δ, + δ 2 ], a contradiction. Hence, + β z on [, ]. That α + z follows similarly. It is easy to see that + β z, α + z; otherwise we hae d =,d =, with the sal bondary condition, which gies,, a contradiction. Let w = z β, w 2 = z α. Then w i,, and w satisfies which leads to dw + g (w,w 2 )+βg 2 (w,w 2 )=, w () =, w () + γw ()=, ( ) ms dw + w K s +w + ms 2 β K s2 +w, w () =, w () + γw ()=. If w ( ) = for some point [, ], by applying the strong maimm principle (see [2]) we obtain a contradiction. Hence w > on[, ]. The proof that w 2 > on [, ] is similar. (iii) It follows by the monotone method and the niqeness of U that U min{, α }z. By the Lipschitz continity of g (S, R), there eists a constant L>, sch that g (z, z α) g (z U, z αu) L(U ). Let Û = U. Then Û satisfies dû + Û[g (z U, z αu) L], <<, Û ()=, Û () + γû()=. If Û, then the maimm principle leads to Û>. Ths either <U or U. The proof for is similar.

6 24 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ Remark 2. It follows from Lemmas and 2 that, for λ i, i=, 2, the only nonnegatie soltion of (3) is (, ). In order to garantee the eistence of a positie soltion of (3), we mst assme that λ i < for i =, 2. Let ˆλ i be the principal eigenales and let ˆφ i () >, [, ], i=, 2, be the corresponding eigenfnctions of the problem d ˆφ + ˆλ ˆφ g (z βv,z V )=, <<, ˆφ ()=, ˆφ () + γ ˆφ ()=, d ˆφ 2 + ˆλ 2 ˆφ2 g 2 (z U, z αu) =, <<, ˆφ2 ()=, ˆφ2 () + γ ˆφ 2 ()=. Theorem. Sppose ˆλ i < for i =, 2. Then there eists a positie steady-state soltion (, ) of (3) satisfying <() <U(), <() <V() for [, ]. Proof. It is easy to check that (U, V ) is the sp-soltion of (3). Let (,)= (δ ˆφ,δˆφ 2 )(δ >). Then for δ sfficiently small, we hae d + g (z βv,z α V ) =[g (z βv,z α V ) ˆλ g (z βv,z V )] = [( ˆλ )g (z βv,z V ) +(g (z βv,z α V ) g (z βv,z V ))] >. Hence there eists a soltion (, ) of (3) satisfying (δ ˆφ,δˆφ 2 ) (, ) (U, V ) for small δ. By Lemma 2 we obtain the strict ineqalities in Theorem. Now we consider the special case that g = g 2 = g, and we find that there eist infinitely many positie soltions of (3). Theorem 2. Sppose that λ i < for i =, 2 and g = g 2 = g. Then there eist infinitely many positie soltions ( ρ, ρ )(ρ>) of (3) satisfying < ρ min{ ρ+β, αρ+ }z, ρ = ρ ρ. Proof. Set ω =. Then ω satisfies dω 2d ω =, ω () = ω ()=. By the maimm principle it follows that ω ρ, a positie constant, i.e., = ρ. Ths satisfies d + g(z (ρ + β), z (αρ +)) =, ()=, () + γ ()=. For ρ> fied, arging as for the eistence of U or V, and noting that λ 2 <, it follows that there eists a niqe positie soltion of the aboe problem, say, ρ, satisfying < ρ min{ ρ+β, αρ+ }z. Ths ( ρ, ρ )(ρ>), where ρ = ρ ρ, is the positie soltion of (3). This completes the proof. Remark 3. Sppose that g g 2, g g 2 or g g 2, g g 2. Then there eists no positie soltion of (3). This conclsion is consistent with the analysis in [9] for the pre and simple competition model. In fact, sppose >, > satisfy (3). We consider the first case, since the second case can be proed similarly. Denoting ω =, we hae dω 2d ω + ω[g 2 (z β,z α ) g (z β,z α )]=, ω () = ω ()=. Then ω = constant, and hence ω =, a contradiction.

7 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 25 Theorem 3. Sppose λ i < and ˆλ i > for i =, 2. Then there eists a positie soltion (, ) of (3). Proof. Let C B [, ] = {() C[, ] : ()=, () + γ()=} be the Banach space, with the sal maimm norm, denoted by,x= C B [, ] C B [, ], K = C + B [, ] C+ B [, ], the positie cone of X. Let N =( d ), the inerse operator of d inc B [, ]. Then system (3) can be written as N(g (z β,z α ))=, N(g 2 (z β,z α ))=. Let T (, ) =(N(g (z β,z α )),N(g 2 (z β,z α ))). Then the fied points of T in K are the corresponding nonnegatie soltions of (3). Define D = {(, ) K : + R }, where R = 2 ma{, α, β } z, and let Ḋ denote the interior of D in K. Since the proof is long, we diide it into three lemmas. Lemma 3. For λ, the eqation T (, ) = λ(, ) has no soltion in K satisfying + = R. Proof. Sppose (, ) K satisfies T (, ) =λ(, ). Then we hae d + λ g (z β,z α ) =, d + λ g 2 (z β,z α ) =, with the bondary conditions as aboe. As in the proof of Lemma 2, it follows that +β < z,α+ <z.ths + <ma{, α, β }z. Hence there eists no fied point of T (, ) =λ(, ) ink satisfying + = R. Remark 4. It follows from Lemma 2. in [] that inde K (T,Ḋ) =. Let P σ (, ) = {(, ) K : + <σ} be the neighborhood of (, ) in K with radis σ. Lemma 4. For σ> small enogh, inde K (T,P σ (, ))=. Proof. Gien ɛ > sfficiently small, noting the definition of U, V, we can take <σ<σ sch that σ γ < min{u ɛ,v ɛ }. Denote S σ + = {(, ) K : + = σ γ }. Ths σz, σz wheneer (, ) S+ σ. Let ψ =(2+γ) γ 2. Then ψ>on[, ] and satisfies ψ <, <<, ψ ()=, ψ () + γψ()=. Take p =(ψ, ψ)( K). We show net (by contradiction) that for λ, (, ) T (, ) =λ(ψ, ψ) has no soltion on S σ + for small σ. Assme that this problem has a soltion (, ) ons σ +. Then (, ) satisfies d + g (z β,z α ) =dλψ, d + g 2 (z β,z α ) =dλψ, <<, <<. Hence by the definition of ψ, we hae d + g (( σβ)z, ( σ)z α), d + g 2 (( σ)z β,( σα)z ), <<, <<. Since λ i <, we can take sfficiently small σ, say, σ<σ, sch that λ (g (( σβ)z,( σ)z)) <, λ 2 (g 2 (( σ)z,( σα)z)) <, where λ (g (( σβ)z,( σ)z)), λ 2 (g 2 (( σ)z,( σα)z)) are the principal eigenales of (4) with g and g 2

8 26 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ replaced by g (( σβ)z,( σ)z) and g 2 (( σ)z,( σα)z), respectiely. As in the proof of Lemma 3.2 in [3] we can proe the eistence and niqeness of U,V of the following problem: du + U g (( σβ)z U, ( σ)z αu )=, dv + V g 2 (( σ)z βv, ( σα)z V )=, <<, <<, with the sal bondary conditions. By an L p estimate and the Sobole embedding theorem (see [27]), we proceed as in the proof of Theorem 2.5 in [32] to obtain lim U = U, lim V = V. σ σ Ths there eists σ 2 >, sch that for σ<σ 2,U >U ɛ,v >V ɛ. It follows from the monotone method and the niqeness of U, V that U, V. Now take σ< σ = min{σ,σ,σ 2 }. Then for σ< σ, we hae > σ γ, > σ γ, which contradicts (, ) S σ +. Lemma 2. of [] can be applied to complete the proof of this lemma. Let O + (U, ) be a small neighborhood of (U, ) in K. Then we hae the following lemma. Lemma 5. Sppose that T has no fied point in Ḋ. Then inde K(T,O + (U, ))= if ˆλ 2 >, λ <. Proof. Define T (θ)(, ) =(N(g (z θβ, z α θ)),n(g 2 (z θβ, z α θ))). It follows from (, ) =T (θ)(, ) that d + g (z θβ, z α θ) =, d + g 2 (z θβ, z α θ) =. If (, ) is a fied point of T (θ) on O + (U, ), the bondary of O + (U, ) in K, it is easy to see that >,. Frthermore, we hae >; otherwise we hae (, ) =(U, ), contradicting (, ) O + (U, ). We claim that for θ [, ], T(θ) has no fied point on O + (U, ). Otherwise, for θ =, by noting ˆλ 2 > and λ <, we find = U, =, a contradiction; for θ>, this implies that (, θ) > (, ) is a fied point of T in Ḋ, contradicting a hypothesis of this lemma. It follows from the homotopy inariance of topological degree that (7) inde K (T,O + (U, )) = inde K (T (),O + (U, )) = inde K (T (),O + (U, )), where T ()(, ) =(N(g (z, z α)),n(g 2 (z, z α))). The fied point (, ) oft () in O + (U, ) satisfies (8) d + g (z, z α) =, d + g 2 (z, z α) =, <<, <<, with the bondary conditions ()=, ()=, () + γ()=, () + γ()=. Since λ <, we hae = U. Noting ˆλ 2 >, we determine that the principal eigenale λ 2 of the following problem is negatie: dφ + φ g 2 (z U, z αu) =λ 2φ, φ ()=, φ () + γφ ()=.

9 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 27 Sbstitting = U into the second eqation of (8), we hae =. Hence (U, ) is the niqe fied point of T () in O + (U, ); ths (9) inde K (T (),O + (U, )) = inde K (T (), (U, )). Let I(θ) (θ [, ]) be defined by I(θ)(, ) =(N(g (z, z α)),n(g 2 (z (θu +( θ)),z α(θu +( θ))))). Then (, ) =I(θ)(, ) satisfies () d + g (z, z α) =, d + g 2 (z (θu +( θ)),z α(θu +( θ)))=, <<, <<, with the sal bondary conditions. We claim that I(θ) has no fied point on O + (U, ) in K. Otherwise, from the first eqation of (), we hae = U, and sbstitting this into the second eqation of (), we find =, so the only fied point of I(θ) on O + (U, ) is (U, ), a contradiction. By the definition of I(θ), we obtain () T () = I(), I() = T T 2, where T = N(g (z, z α)), T 2 = N(g 2 (z U, z αu)), (T T 2 )(, ) = (T, T 2 ). (, ) =I()(, ) satisfies d + g (z, z α) =, d + g 2 (z U, z αu) =, <<, <<. It follows from (7) () and the prodct theorem for fied points (see [33]) that (2) inde K (T (), (U, )) = inde K (I(), (U, )) = inde K (I(), (U, )) = inde CB (T,U) inde C + (T 2, ). B Since T 2 is a linear compact operator and ˆλ 2 >, then T 2 has no eigenale > with positie eigenfnction in C + B. It follows from Lemma 3. of [] that inde C + (T 2, ) = B. We show net that inde CB (T,U)=. Let τ = 2 min{, α } z, P τ = { C + B : τ}, P τ = { C + B : = τ}. For λ, if T = λ, dλ + g (z, z α) =. Arging as in the proof of Lemma, we hae min{, α }z<τ. Hence for λ, T = λ has no soltion on P τ. It follows from Lemma 2. of [] that inde C + B (T,P τ )=. Let <τ 2 min [,]{U()}. Sppose that for λ, p= ψ(), sch that T = λp has a soltion on P τ, where ψ() is defined as in the proof of Lemma 4. Then, d + g (z, z α) =dλψ. Ths is a sp-soltion of (5). From the monotone method and the niqeness of U it follows that U, a contradiction to = τ. Hence, inde C + (T,P τ )=. B Since = U is the niqe fied point of T in P τ \ P τ, we obtain inde CB (T,U)= inde C + B (T,P τ \ P τ )=inde C + (T,P τ ) inde B C + (T,P τ )=. B Combining the aboe reslt with eqations (7), (9), and (2), it follows that inde K (T,O + (U, ))=. Remark 5. Sppose that T has no fied point in Ḋ. We can proceed as aboe to obtain inde K (T,O + (,V))=ifˆλ >, λ 2 <. Now we trn to the proof of Theorem 3. Sppose that T has no fied point in Ḋ. Then the following eqation holds: inde K (T,Ḋ) =inde K(T,O + (, )) + inde K (T,O + (U, )) + inde K (T,O + (,V)),

10 28 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ contradicting Lemmas 3 5. This completes the proof of Theorem 3. Noting Lemma, and sing the same process as in the proof of Theorem 3.6 in [5], we hae the following theorem. Theorem 4. If λ > and λ 2 >, then the soltion of system () satisfies lim (S, R, U, V )=(z,z,, ). t If λ > and λ 2 <, then the soltion of system () satisfies lim (S, R, U, V )=(z βv,z V,,V). t If λ < and λ 2 >, then the soltion of system () satisfies lim (S, R, U, V )=(z U, z αu, U, ). t Theorem 5. Sppose ˆλ i <, i =, 2. Then the soltion of system () is niformly persistent ([]). Proof. It follows from system ( ) that (, t) V (, t), where V (, t) isthe soltion of the problem V t = dv + Vg 2 (ϕ βv,ϕ 2 V ), <<, t >, V (,t)=, V (,t)+γv (,t)=, V (, ) = (). Since ˆλ 2 <, then λ 2 <. We can proceed as in Theorem 2.5 in [32] to show that if λ 2 <, then lim t V (, t) =V (), where V () < min{, β }z is the niqe positie soltion of dv + Vg 2 (z βv,z V )=, V ()=, V () + γv ()=. <<, Since ˆλ <, we can take <ɛ, sch that for the following principal eigenale λ <, d φ + λ φ g (( ɛ( + β)/2)z βv (), ( ɛ)z V ())=, φ ()=, φ () + φ ()=. <<, There eists τ > sch that for [, ], t τ, the following ineqalities hold: ϕ z (ɛ/2)z, ϕ 2 z (ɛ/2)z, (, t) V ()+(ɛ/2)z. Using the comparison theorem, it follows that for t τ,(, t) (, t), where (, t) is the soltion of t = d + g (( (ɛ( + β)/2))z βv (), ( ɛ)z α V ()), <<,t>τ, (,t)=, (,t)+γ(,t)=, (, τ ) = min{( ɛ( + β)/2)z βv (), (( ɛ)z V ())/α, (, τ )}.

11 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 29 Noting λ <, we hae lim t (, t) = ɛ (), where ɛ () is the niqe positie soltion of d ɛ + ɛ g (( ɛ( + β)/2)z ɛ βv (), ( ɛ)z α ɛ V ())= with the sal bondary conditions. It follows from an L p estimate and the embedding theorem (see [27]) that lim ɛ ɛ () =û(), where û() is the niqe positie soltion of the following problem on [, ]: (3) dû +ûg (z û βv (),z αû V ())= with the sal bondary conditions. A similar reslt holds for if ˆλ 2 <. Hence, there eist constants η >, τ τ sch that (, t) η, (, t) η for [, ], t τ. By the eqation of S in system () and the definition of g i,i=, 2, we hae S t = ds g (S, { R) βg 2 } (S, R) ms ds ma K s, ms 2 K s2 S( + β). Then there eists τ >, and for t τ, the following ineqality holds: { ms S t ds ma, m } s 2 S(z + ε S). K s K s2 A similar reslt holds for R. Ths we can proceed as in Lemma 3.8 in [5] to show that there eist η 2 >, τ 2 > sch that S(, t) η 2,R(, t) η 2 for [, ], t τ 2. Denote τ = ma{τ,τ 2 },η= ma{η,η 2 }. Then we hae S η, R η, η, η for [, ], t τ. This completes the proof. 3. Nmerical simlations. The goal of this section is to present the reslts of nmerical simlations that complement the analytic reslts of the preios section. The simlations reported below represent a small fraction of those made. We wish to make a few general comments based on or obserations. First, in most simlations performed, conergence to eqilibrim was obsered. Second, competitie eclsion, the elimination of one poplation by another, was obsered. Finally, nonniqeness of the positie eqilibrim and bistability of the semitriial eqilibrim were obsered. Or simlations are consistent with the analytic reslts of the preios sections. Frthermore, the simlations reeal that mch more complicated dynamics are also possible in the region D defined below. Or nmerical simlations also seemed to indicate that coeistence is more likely in the case of competition for two limiting complementary resorces in the nstirred chemostat, than in the case of competition for a single limiting resorce in the nstirred chemostat (see [26]). Define A = {(ˆλ, ˆλ 2 ):< ˆλ <, < ˆλ 2 < }, B= {(ˆλ, ˆλ 2 ):< ˆλ <, ˆλ 2 > }, C= {(ˆλ, ˆλ 2 ):ˆλ >, < ˆλ 2 < }, and D = {(ˆλ, ˆλ 2 ):ˆλ >, ˆλ 2 > }. Or nmerical simlations seem to indicate the following: () Coeistence in the form of a positie eqilibrim can be obsered when (ˆλ, ˆλ 2 ) A B C (see Figres 2 and Tables 2), and apparently a globally stable positie eqilibrim can always be obsered when (ˆλ, ˆλ 2 ) A; (2) Competitie eclsion in the form of an apparently globally stable semitriial positie eqilibrim can occr when (ˆλ, ˆλ 2 ) B C (see Figres 2 and Tables 2);

12 22 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ (3) Both stable and nstable positie eqilibria can eist, and there can be bistability with two stable semitriial eqilibria and an nstable positie eqilibrim, reslting in initial condition dependent otcomes when (ˆλ, ˆλ 2 ) D (see Figres 3 4); (4) Eistence of mltiple stable and/or nstable positie eqilibria can be obsered when (ˆλ, ˆλ 2 ) D (see Figres 4 5); (5) The parameters hae an apparent effect on the density of both organisms, i.e., the density can be nondecreasing and the density can be nonincreasing as α increases (see Figres 6(a) 6(c)). Similar reslts for and hold as β increases. Bt the density of both organisms can decrease as γ increases (see Figres 6(b) and 6(d)). Now we describe an indirect method sed for determining either ˆλ i > orˆλ i < from nmerical simlations. The method will be described for determining the sign of ˆλ only, since the other case is similar. Consider the following system: (4) t = d + g (z β,z α ), t = d + g 2 (z β,z ), (,t)=, (,t)+γ(,t)=, (,t)=, (,t)+γ(,t)=, (, ) = (), (, ) = (),, <<, t>, <<, t>, where + β z, α + z. Taking initial conditions characterized by a ery small density of, we can proe and obsere nmerically that rapidly approaches the eqilibrim V (). Hence for large times, t, we take (, t) asv () in the first eqation of (4). Then we hae (5) t = d + g (z βv,z α V ), <<, t> with the sal bondary and initial conditions. We can se the comparison theorem and the Liapno fnction method to proe that the soltion (, t) of (5) satisfies lim t (, t) =ifˆλ and lim t (, t) =û if ˆλ <, where û is the niqe positie soltion of (3). Therefore, what happens to depends essentially on the sign of ˆλ. If ˆλ, we obsered the decay of the soltion of (5) to ery small ales; if ˆλ <, we obsered the growth of the soltion of (5) to the ale of the soltion of (3). Therefore, we can determine the sign of ˆλ nmerically by obsering whether there is decay to ery small ales or growth to the ale of the soltion of (3). We net simlate the corresponding time-dependent system of (3), which determines the limiting system of (): t = d + g (z β,z α ), t = d + g 2 (z β,z α ), (,t)=, (,t)+γ(,t)=, (,t)=, (,t)+γ(,t)=, (, ) = (), (, ) = (). <<, t>, <<, t>, We hae chosen to discretize the spatial ariables in the aboe system sing a second-order finite-difference scheme. The deriatie terms in the bondary conditions are approimated sing second-order centered differencing. The temporal ariable is approimated sing the Crank Nicholson method. In all of the simlations the domain is diided niformly into 4 cells.

13 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT 22 Table The eqilibria corresponding to the parameters gien in Tables and 2 are shown in Figres and 2. m s ˆλ ˆλ2 Area Eqilibrim (2.6, 2.4) < > B (U, ) (2.45, 2.55) < > B Coeistence (2.4, 2.6) < < A Coeistence (.8, 2.3) < < A Coeistence (.6, 2.5) < < A Coeistence (.55, 2.55) > < C (,V) Table 2 m s ˆλ ˆλ2 Area Eqilibrim (4.55, 4.45) < > B (U, ) (4.48, 4.52) < > B Coeistence (4.45, 4.55) < < A Coeistence (2.2, 6.8) < < A Coeistence (2., 6.9) > < C Coeistence (.95, 7.5) > < C (,V) In the following, we denote m s =(m s,m s2 ),m r =(m r,m r2 ),K s =(K s,k s2 ), K r =(K r,k r2 ). Coeistence and competitie eclsion. In Figres and 2 a seqence of simlations is reported where different growth rates were sed, bt all of the other parameter ales remain fied. The parameter ales sed were α = β =.5,γ =,K s = (, ),K r =(,.2). In Figre and Table, m r =(3, 3), and in Figre 2 and Table 2, m r =(6, 6). In Figre, m s took the ales indicated in Table, and in Figre 2, m s took the ales indicated in Table 2. In each simlation in Figres and 2, the densities were plotted at the final time, t =. This appeared to be long enogh to allow the soltions to be ery close to steady state. A similar procedre was sed in the other figres. We obsered from Figres and 2, as well as from many other simlations, that at the highest growth rate of and the lowest growth rate of, is dominant with barely present for any initial conditions. In this case, we checked that (ˆλ, ˆλ 2 ) B. As the growth rate of is decreased or the growth rate of is increased, the amont of increases at the epense of the amont of. Both organisms coeist at a positie eqilibrim. In this case, we also checked that (ˆλ, ˆλ 2 ) A or B or C. All the simlations show that the coeistence is niqe and an apparently globally stable positie eqilibrim eists when (ˆλ, ˆλ 2 ) A. As the growth rate of is frther decreased or the growth rate of is frther increased, is dominant with barely present for any initial condition. In addition, we checked (ˆλ, ˆλ 2 ) C in this case. Coeistence in the form of a positie eqilibrim can occr when (ˆλ, ˆλ 2 ) A or B or C. The noneistence of a positie eqilibrim can also occr when (ˆλ, ˆλ 2 ) B or C. Bistability and the eistence of positie eqilibria. (i) In Figre 3, we proide nmerical eidence of bistability; i.e., each of the two semitriial eqilibria is stable to inasion by its rial and attracts soltions corresponding to nearby initial data. As well, an nstable positie eqilibrim is obsered. We took m s =(3, 2), m r = (2.4, 3.6), and the other parameter ales as in Figre. In this case we checked that (ˆλ, ˆλ 2 ) D. The simlations in Figres 3(a) and 3(b) show a plot of the L norms of and of erss time t. In Figre 3(a) the initial conditions sed were =.5 and

14 222 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ (a) (b) (c) (d) (e) (f) Fig.. Eqilibria for m r =(3, 3) and the different ales of m s from Table (in the order gien in that table). The other parameters sed are K s =(, ), K r =(,.2), α = β =.5, and γ =.

15 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT (a) (b) (c) (d) (e) (f) Fig. 2. Eqilibria for m r =(6, 6) and the different ales of m s from Table 2 (in the order gien in that table). All the other parameters are the same as those in Figre.

16 224 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ L Norm L Norm Time (a) Time (b) L Norm (c) Time (d) L Norm L Norm Time (e) Time (f) Fig. 3. Conergence to eqilibria. All parameters ecept m s and m r are the same as those in Figre. In (a) (c) m s =(3, 2) and m r =(2.4, 3.6). In (a) (b) the L norms of and of erss time are shown for two semitriial eqilibria. In (c) a plot of the positie eqilibrim for each [, ] is shown. In (d) (f) m s =(2, 2) and m r =(2, 4). The L norms of and of erss time are shown for seeral different eqilibria.

17 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT (a) (b) (c) (d) Fig. 4. Seeral positie eqilibria. All parameters are the same as those in Figre, ecept m s =(2, 2) and m r =(2, 4). Note that and are indistingishable in (d). =.. In Figre 3(b) the initial conditions sed were =. and =.5. The positie eqilibrim is plotted in Figre 3(c). After many simlations, in this case we beliee that this is the only positie eqilibrim and that it is nstable. (ii) In Figres 3(d) (f), we took m s =(2, 2) and m r =(2, 4). All other parameters are the same as in Figre. Both semitriial eqilibria are stable. Only one of them is shown (see Figre 3(d)). As well, nonniqeness and stability of more than one positie eqilibrim are obsered. In this case, we checked that (ˆλ, ˆλ 2 ) D. Eistence of mltiple positie eqilibria. Based on etensie simlations, we beliee that mch more complicated dynamical behaior can occr when (ˆλ, ˆλ 2 ) D. (i) In Figre 4 we took the same parameters as in Figres 3(d) (f) and sed contination (nmerical analysis) to find the eqilibria. Simlations (not shown) seem to indicate that there are at least for positie eqilibria in this case, and strongly sggest that one of them is nstable (see Figre 4(d), where and are indistingishable), and that the other three are stable (see Figres 4(a) (c)). Note that the eqilibria depicted in Figres 4(b) (c) correspond to those in Figres 3(e) (f), respectiely. (ii) In Figres 5(a) (c) we took m s =(.5,.8) and m r =(.75,.42). In Figres 5(d) (f) we took m s =(2., 2.75) and m r =(2.8, 2.3). The other parameters

18 226 JIANHUA WU, HUA NIE, AND GAIL S. K. WOLKOWICZ (a) (b) (c) (d) (e) (f) Fig. 5. Positie eqilibria. All parameters are the same as those in Figre, ecept that in (a) (c) m s =(.5,.8) and m r =(.75,.42), and in (d) (f) m s =(2., 2.75) and m r =(2.8, 2.3).

19 COMPETING FOR RESOURCES IN THE UNSTIRRED CHEMOSTAT (a) (b) (c) (d) Fig. 6. Parameters α and γ hae an effect on the densities of the eqilibria. In (a) (d) m s =(2, 2.), m r =(3, 3), K s =(, ), K r =(,.2), and β =.5. In (a) (c) γ =, and in (d) γ =. In(a) α =.3, in (b) α =.5, in(c) α =.7, and in (d) α =.5. were taken as in Figre. In both cases, we checked that (ˆλ, ˆλ 2 ) D. For each case, we sed nmerical analysis to find the three positie eqilibria depicted in Figre 5. Sbseqent simlations strongly indicated that all these positie eqilibria are nstable. Effects of the parameters. In Figre 6 a seqence of simlations shows that the parameters α, β, γ hae an apparent effect on the density of both poplations. Parameter ales taken are m s =(2, 2.), m r =(3, 3), K s =(, ), K r =(,.2). Vales for α, β, and γ are gien in the caption of Figre 6. The initial data are = and =. We obsered that the density of can be nondecreasing and the density of can be nonincreasing as α increases (see Figres 6(a) 6(c)). A similar reslt holds for and as β increases. We also obsered that the density of both and can decrease as γ increases (see Figres 6(b) and 6(d)). Acknowledgment. The athors wold like to epress their sincere thanks to the anonymos referees of this paper for their carefl reading and alable sggestions leading to an improement of the paper.

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