Bifurcation Analysis of a Chemostat Model with a Distributed Delay

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 4, ARTICLE NO. 468 Bifurcation Analysis of a Chemostat Model with a Distributed Delay Shigui Ruan Department of Mathematics, Statistics and Computing Science, Dalhousie Uniersity, Halifax, Noa Scotia, Canada B3H 3J5 and Gail S. K. Wolkowicz Department of Mathematics and Statistics, McMaster Uniersity, Hamilton, Ontario, Canada L8S 4K1 Received August 7, 1995 A chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed. The model incorporates a general nutrient uptake function and a distributed delay. The delay indicates that the growth of the species depends on the past concentration of nutrient. Using the average time delay as a bifurcation parameter, it is proven that the model undergoes a sequence of Hopf bifurcations. Stability criteria for the bifurcating periodic solutions are derived. It is also found that the periodic solutions become unstable if the dilution rate is increased. Computer simulations illustrate the results Academic Press, Inc. 1. INTRODUCTION The chemostat is an important laboratory apparatus used to culture microorganisms. It is of both ecological and mathematical interest since it is one of the few places the mathematics is tractable, the parameters are measurable, and the experiments are reasonable. For recent development and mathematical analysis on chemostat models, we refer to the book by Smith and Waltman 35. * Research supported by the Natural Sciences and Engineering Research Council of Canada. -47X96 $18. Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 786

2 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 787 It is well known that time delays in ecological systems can have a considerable influence on the qualitative behavior of these systems. Some non-stationary phenomena, such as instabilities and periodic fluctuations, can be explained by incorporating time delays in model systems Žsee Cooke and Grossman 6, Cushing 11, Hale and Lunel, MacDonald 9, May 31, and the references cited there.. Discrete delay was first included in a chemostat model by Finn and Wilson 14 in order to model sustained oscillations that they observed in a yeast population in a chemostat. Caperon 4, who observed oscillatory transients in experimental populations, incorporated both discrete delay and distributed delay in chemostat models. Thingstad and Langeland 37 discussed in detail the stability analysis and numerical solutions of one of Caperon s models. Bush and Cook 3 also studied a model of growth of one organism in the chemostat with a discrete delay in the intrinsic growth rate of the microorganism. For other early work, see for example, Cunningham and Maas 8, Cunningham and Nisbet 9, and MacDonald 8. We also refer to a survey paper by MacDonald 3 regarding time delays in chemostat models. Recently, Freedman et al. 15 extended the model of Bush and Cook to a competition model with two microorganisms competing for the substrate. They provided a careful analysis of the bifurcation of a planar periodic orbit involving only the substrate and one population to a three-dimensional periodic orbit involving the substrate and both competitor populations. Freedman et al. 16 also considered another delayed chemostat model. For this model, recently Ellermeyer 1, Hsu et al., and Zhao 41 investigated the global asymptotic behavior of solutions, such as convergence and persistence of one of the competitors. Global asymptotic behavior of this model and a more general chemostat model with discrete delays was studied by Wolkowicz and Xia 4. For other related models, we refer to Beretta et al., Freedman and Xu 17 and Ruan 34. It has been found that continuously distributed delay models are more realistic Žsee Caswell. 5 and continuously distributed delays are more accurate than instantaneous time lags Žsee Caperon. 4. In this paper, we incorporate a distributed delay in a chemostat-type model of a single species feeding on a limiting nutrient supplied at a constant rate. The distributed delay is included since it is assumed that the growth of the species depends on the past concentration of nutrient. A general function is also used to describe the nutrient uptake. For both strong and weak kernels, using the average time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter passes through a critical value. In general, as pointed out by Cushing 1, it is quite difficult to determine the stability of bifurcating periodic solutions for delay systems

3 788 RUAN AND WOLKOWICZ Ž and even for ordinary differential equations.. However, there are a number of papers dealing with the stability of bifurcating periodic solutions for delay systems. We refer to Kazarinoff et al. 4, Huang et al. 3, and the reference cited therein for discrete delay models and Beretta et al. 1, Farkas 13, Gopalsamy and Aggarwala 19, and Stepan 36 for models involving distributed delays. In the case of our model with the weak kernel, by using the algorithm in Hassard et al. 1, we derive criteria to determine the stability of the bifurcating periodic solutions. Computer simulations illustrate the results. Morita 33 investigated the destabilization of periodic solutions arising in delaydiffusion equations. He showed that the stability regions of the bifurcation parameter depends on other factors such as the diffusion constant and the shape of the domain. One of his results indicates that the periodic solution becomes unstable near the bifurcation point if the diffusion coefficient is varied. See also Lin and Kahn 7 and Memory 3 for related results. Following the idea and the procedure of Morita 33, we find that for this chemostat model with a distributed delay, when one of the parameters Ž washout rate. is varied the bifurcating periodic solution loses its stability. The numerical simulation verifies our analysis. The paper is organized as follows. The model is described in Section. In Section 3, we study the existence of bifurcating periodic solutions. The stability analysis is carried out in Section 4. In Section 5, we study the destabilization of the bifurcating periodic solutions. A numerical example is given in Section 6. Finally, a discussion is presented in Section 7.. THE MODEL Ž. Ž. Let St and xt denote the concentration of the nutrient and the populations of microorganisms at time t. Our model is described by the integrodifferential equations ds Ž S SŽ t.. DaxŽ t. pž SŽ t.., dt dx t xž t. D1H FŽ t. pž S. d, dt 1. all of the parameters are nonnegative. S denotes the input concentration of nutrient. D is referred to as the dilution rate and D1 denotes the sum of the dilution rate and the death rate of the population of microorganisms. The function ps describes the species specific growth rate. The nutrient uptake rate is assumed to be proportional to the growth

4 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 789 rate with constant of proportionality a, and so 1a is referred to as the growth yield constant. We assume throughout that ps satisfies the following assumptions: p, p Ž S., and lim pž S. m.. S For example, these hypotheses are satisfied by the MichaelisMenten Ž Holling type II. function ms pž S. Ž m, k., 3. ks m denotes the maximal growth rate of the species and k is called the half-saturation constant. The distributed delay term in system 1. models the situation that the past history of consumption is important, but nutrient consumed either very recently or a long time ago has an insignificant effect on the growth of the organism. The weight function FŽ s. is a nonnegative bounded function defined on,. which describes the influence of nutrient experience on future growth of the organism. It is assumed in this model that the presence of the distributed time delay does not affect the equilibrium values, so we normalize the kernel so that H FŽ s. ds1. 4. As in MacDonald 9, we define the average time lag as H T sf Ž s. ds. 5. In particular, the weak kernel FŽ s. e s, 6. and the strong kernel FŽ s. se s, 7. are often used Žsee Cushing 11.. The average time lags for the weak and strong kernels are 1 T 8. and T, 9. respectively.

5 79 RUAN AND WOLKOWICZ We consider system.1 under the initial value conditions SŽ s. Ž s., s, x x, 1. Ž s. is a continuous function on Ž,. Ž Note that E S,. is always an equilibrium for system 1.. There is an interior equilibrium E ŽS, x. with DŽ S S. 1 S p Ž D 1., x 11. ad 1 provided S S and lim ps D. S 1 3. EXISTENCE OF HOPF BIFURCATIONS In this section we study the existence of bifurcating periodic solutions. Let u S S, u x x 1. Ž 3.1. System.1 can be written or du dt du dt ax pž S. Du až u x. pž u S., H t Ž. 1 1 u x D F t p u S d, du LuŽ t. H K Ž. už t. dhž u., Ž 3.. dt ž x p Ž S. F / D ax p Ž S. ap Ž S L., Ž 3.3. K, Ž 3.4. ap Ž S. uu 1 HŽ u. H.O.T. Ž 3.5. p Ž S. u H F u Ž t. d 1

6 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 791 The characteristic equation of the linearized system is D det I L e K d H 1 D ax p Ž S. ad x p Ž S. e F Ž. d. H Ž 3.6. as If F s is a weak kernel, i.e., F s e,, then 3.6 becomes a third-order algebraic equation: Define 3 Dax p Ž S. D ax p Ž S. ad x p S 1. b b D ax p S 1 1, b b Dax p S, b b ad x p S Then the characteristic equation takes the form 3 b1 b b3. Ž 3.7. Let :, R be a continuously differentiable function defined by 1 1 b1 b b3ž.. Ž 3.8. The RouthHurwitz criterion implies that the equilibrium ŽS, x. is locally asymptotically stable if. If 1 ad x p Ž S 1. Dax p Ž S., Ž 3.9. Dax p Ž S. then Ž. 1, and the characteristic equation has a pair of purely imaginary roots 1, i, ' bž. and a real root bž After some calculations, it follows that d 1 d 1 Re 1, Ž 3.1. d 4 b b d Ž 1. d 1 1 d ad x p S D ax p S.

7 79 RUAN AND WOLKOWICZ The above analysis can be summarized as follows: THEOREM 3.1. If, then the equilibrium ŽS, x. 1 of system Ž.1 is locally asymptotically stable. If ad. Ž. 1 x p S D ax p S, then as passes through the critical alue, there is a Hopf bifurcation at the equilibrium ŽS, x.. s If F s is a strong kernel, i.e., F s se,, then the characteristic equation is Define Dax p Ž S. ad1x p Ž S.. c c D ax p S 1 1, c c Dax p S, 3 3 c c Dax p S, c c ad x p S Then the characteristic equation can be rewritten Define 4 c1 3 c c3 c4. Ž c c d c c c. 3.1 The RouthHurwitz criterion implies that the equilibrium ŽS, x. of system 1. is locally asymptotically stable if. Let Ž i 1,, 3, 4. be the roots of the characteristic equation Ž i. Then we have c, c, c, c Ž If there exists R such that Ž., then by the RouthHurwitz criterion at least one root, say 1, has real part equal to zero. From the fourth equation of Ž it follows that Im 1, and hence there is another root, say, such that. Since 1 is a continuous function of its roots, 1 and are complex conjugate for in an open

8 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 793 interval including. Therefore, the equations in Ž have the following form at : 3 4c 1, c, 3 4 Ž. c, Ž c If and are complex conjugate, from the first equation of Ž it follows that Re 3c1. If 3 and 4 are real, from the first and fourth equations of Ž it follows that 3 and 4. Also, after some calculations it follows that d c1 d Re 1. d ccž cc c. d Thus, we have the following result. THEOREM 3.. If, then the equilibrium ŽS, x. of system 1. is locally asymptotically stable. If there exists R such that Ž. and Ž d d., then as passes through, a Hopf bifurcation occurs at ŽS, x.. 4. STABILITY OF BIFURCATING PERIODIC SOLUTIONS In this section, we study the stability of the bifurcating periodic solu- s tions. We suppose that the kernel is a weak kernel, i.e., F s e,. The case of the strong kernel can be discussed similarly. We first transform system Ž 3.. into an operator equation of the form du t Aut Fu t, Ž 4.1. dt u colž u, u., u ut, Ž, 1 t, and the operators A and F are defined as d, A d Ž 4.. L H K d,, ž/, F ap Ž S. Ž. Ž. Ž ,, p Ž S. H e d 1 L and K are defined as in 3.3 and 3.4.

9 794 RUAN AND WOLKOWICZ Note that the operator A depends on the bifurcation parameter. By Theorem 3.1, a Hopf bifurcation occurs when passes through. Set. Then the Hopf bifurcation occurs when. The adjoint operator A of A is defined as d, A d L T H K T d,, L T and K T are the transposes of the matrices L and K, respectively. Note that A and A can have complex eigenvectors. It is therefore. suitable to assume that, :,C. Define the bilinear form:, T T K Ž. d d. ² : H H To determine the Poincare normal form of the operator A, we need to calculate the eigenvector q of A belonging to the eigenvalue i and the eigenvector q of A belonging to the eigenvalue i. We find that and 1 i q e,, B x p Ž S. Ž i. B 1 i q E e,, C Dax p Ž S. i Dax p Ž S. C x p Ž S. 1 E. Cx p Ž S. 1BC Ž i., We compute that ² : ² : q, q 1, q, q.

10 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 795 Using the same notation as Hassard et al. 1, we first construct the coordinates to describe the centre manifold C at Ž.. Let zž t. ² q,u :, wž t,. u Re zž t. qž.. t t 4 Ž. On the centre manifold C, wt, w z t, zt,, Ž z z z wž z, z,. w w11 zz wž. w3. Ž z and z are local coordinates for the centre manifold C in the direction of q and q. Note that w is real if ut is real. We consider only real solutions. For solution u C of Ž 4.1., since. t ² Ž 4. : z t i z t q, F wre z t q We rewrite this as Ž 4. T i z t q F w z, z, Re z t q. z i zž t. gž z, z., Ž 4.6. Ž 4. T g z, z q F w z, z, Re z t q. 4.7 Using 4.1 and 4.5, we have w u zq t zq Aw Re² q Ž.,FŽ wre zž t. q 4. : q 4 Ž 4. F wre zž t. q Aw Re gž z, z. q 4 F wre zž t. qž. 4, which can also be rewritten as w Aw HŽ z, z,., Ž 4.8. HŽ z, z,. Re gž z, z. q 4 F wre zž t. qž. 4. Ž 4.9. We expand the function gž z, z. on the centre manifold C and z: z z z z in powers of z gž z, z. g g11 zzg g1. Ž 4.1.

11 796 RUAN AND WOLKOWICZ The coefficients of Ž 4.1. can be determined by comparing Ž 4.1. with Ž 4.7. w is replaced by its expansion Ž In order to determine the coefficients w Ž. of the expansion Ž 4.5., we expand the function HŽ z, z,. ij in powers of z and z on the manifold C : z z HŽ z, z,. H H11 zz HŽ.. Ž The argument of F is Thus Ž1. i iw w Ž. ze ze wzqž. zqž.. Ž. i i w zbe zbe ž/, Ž 4. 1 F wre z t q f f,, f ap Ž S. w z z w zb zb, 1 Ž1. Ž. Ž. Ž1. s Ž. ½H f x p S w zb zb w s e ds By 4.7, it follows that Therefore, Ž i. z Ž i. z. 5 1 g z, z Ef z, z ECf z, z. ½ 5 ž/, ž 1 f Ž z, z. /,. f Ž z, z. 1 H z, z, Re Ef z, z ECf z, z q

12 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 797 Since f Ž z, z. z zz ap Ž S. B, z zz f z, z x p S B i, if we define Cx Ž i. ž / GBp S a, Cx Ž i. G1Bp Ž S. a, by 4.9, we obtain H HŽ z, z,. z zz EGqŽ. EG1qŽ. ž/, ap Ž S. B Ž 4.1. x p Ž S. BŽ i.,. Similarly, since ap Ž S. Re B, zz zz 1 1 f z, z f z, z zz z z z z f z, z f z, z zz zz zz,

13 798 RUAN AND WOLKOWICZ we obtain HŽ z, z,. H 11 z z zz ap S Re B Eq Eq ž/, Ž ap Ž S. Re B,. ž / On the other hand, on the centre manifold C near the origin, w Ž z, z. wzwz. z z Ž Using the expansion Ž 4.5. to replace w and w and Eq. Ž 4.6. z z to replace ż and z, we obtain a second expression for w. By comparison of this result with Ž 4.8., we can derive equations for the coefficients w Ž. ij. These are Ž i IA. w H Ž., Ž and w w. Define w Ž1. Ž. w Aw11Ž. H11Ž. Ž Ž. w,. By substituting Ž 4.. and Ž 4.1. into Ž 4.15., when, it follows that d i ž Ž1. d w Ž. / d Ž. w i d When, we obtain i i EGe EG1e. Ž i i BEGe BEG e Ž1. i Dax p Ž S. apž S. w Ž. w i 1 w Ž1. Ž s. H Ž1. H s ds. Ž x p S e Ž. Ž. w Ž s. H

14 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 799 In order to obtain a continuous solution wž. on Ž,, we consider the above equations associated with the boundary condition The general solution of 4.17 is w Ž1. Ž. w Ž1. lim. Ž w Ž. w Ž. Ž. / w Ž1. Ž. k k1 k i i i e e e, Ž 4.. Ž. w ž l l1 l EG EG 1 k1 i, k ; 3 l k B, l k B, 1 1 and l and k are determined by 4.19, i.e., kw Ž1. Ž k1k., lw Ž. Ž l1l.. Ž1. Ž. Ž. To find w and w Ž., we substitute Ž 4.. into Eq. Ž to obtain i Dax p Ž S. apž S. ž Ž1. Ž1. w / ž c / x p Ž S. Ž i., Ž 4.1. i Ž. Ž. w c 4 c Ž1. H Ž1., Ž. Ž. c H i x p Ž S. 1 Ž BEG 3BEG1.Ž i. ½ 4 1 BEGŽ i. 3BEG1Ž i. 5 Define ad x p S 1 Ž i. i idax p Ž S.. 4.

15 8 RUAN AND WOLKOWICZ Then we find that i c Ž1. ap S c Ž. Ž1. w, x p Ž S. Ž i. i Dax p Ž S. c c 4 Ž. w Similarly, by Eq. 4.18, Ž. Ž1. w11 Ž1. p p1 i p i e e, Ž 4.. Ž. w q q1 q 11 ap S Re B E p1 i, q1pb, 1 p p, q p B, 1 pw11 Ž1. Ž p1p., qw11 Ž. Ž q1q.. Ž1. Ž. Ž. To find w and w, we substitute 4. into 4.16 to obtain Dax p Ž S. apž S. w11 Ž1. Ž. w Ž. x p S 11 H Ž1. 11 i i Ž. H11 x p Ž S. Ž p1p. p1 p H H11 Ž1. ad1 x p Ž S.Ž Re B. Ž. 11 c11 Ž1., Ž. c 11 i Ei i Ei 1 Ei Ei.

16 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 81 c11 Ž1. H11 Ž1. Ž., 4aD x p Ž S.Ž Re B. Im EŽ i. Im E c H. Ž. Ž Ž1. Ž. Ž. Ž. Solving for w and w, it follows that c11 Ž1. Ž1. 11 w, x p Ž S. w Ž. Ž1. Dax p S c11 x p S c Ž ad1x p Ž S. Now we consider Fwz,z, Ž Reztq Ž. Ž.4.. Since we find wž z, z,. Re zž t. q 4 z z w w11 zz w Re zž t. q w z w w z zz Ž. Ž. Ž. w w w Ž1. Ž1. Ž Re zž t. qž. 4, 4 z z 1 Ž1. Ž1. Ž1. f ap Ž S. w w11 zz w Ž z z. z z Ž. Ž. Ž. w w11 zz w Ž zb zb., z z Ž. Ž. Ž. f x p Ž S. w w11 zz w Ž zb zb. z z w w zz w Ž1. Ž1. Ž1. 11 Ž zž i. zž i..,

17 8 RUAN AND WOLKOWICZ H Ž1. s Ž1. w e w Ž s. ds s is is Ž1. is H 1 Ž Ž 1.. e k e k e w k k e ds k H i Ž1. s Ž i 1 k i Ž1. Ž w Ž k1 k.., 4 w e w Ž s. ds s i s i s Ž1. H e p e p e w Ž p p. ds Ž1. p1ž i. pž i. w11 Ž p1p., H Ž1. s Ž1. w e w Ž s. ds. Thus, Ž 4. T ž f 1 / Ž E, EC. g z, z q F w z, z, Re z t q 1 E f Cf. Ž 4.3. f Comparison of the coefficients of 4.1 and 4.3 yields g EG, g Eap S Re B, 11 Cx Ž i. gp Ž S. EB a,

18 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 83 ½ Ž1. Ž. Bw w Ž1. Ž. g1 Ep Ž S. a w11 Bw11 Ž1. Ž. Bw w Ž i. x C Ž i. Ž. Ž1. w11 Bw 11 Ž.. Therefore, we can compute the following parameters: i 1 g 1 c1 gg11 g11 g, 3 Re c1, Re Ž. 1 Im c Im 1 1, We obtain Rec1Ž., 5 4 T Ž 1 OŽ.., OŽ.. E p S c1 i až Re B. GapŽ S.Ž Re B. 1 Ž i. p Ž S. B aax C 3 ½ Ž1. Ž. Bw w Ž1. Ž. Ep Ž S. a w11 Bw11 Ž1. Ž. Bw w Ž i. x C Ž i. 5 Ž. Ž1. w11 Bw 11 Ž.. Now we can state the main result of this section.

19 84 RUAN AND WOLKOWICZ THEOREM 4.1. The direction of the Hopf bifurcation described in Theorem 3.1 is determined by the sign of : if Ž,. then the bifurcating periodic solutions exist for Ž.. The periodic solutions are stable Ž unstable. if Ž.The. period of the bifurcating periodic solutions of system 1. increases Ž decreases. if Ž.. 5. DESTABILIZATION OF THE PERIODIC SOLUTIONS By Theorem 3.1, under certain assumptions system 1. has a periodic solution bifurcating from the equilibrium E ŽS, x. when passes through a critical value. Let be a measure of the amplitude of the periodic solution pt;,max pt; t. According to Hassard et al. 1, there is an open interval Ž,. 1 such that for any in the interval 5 Ž 1. J1½, there is a unique Ž,. for which 1. For J 1, the period T and the Floquet characteristic exponent of the periodic solution pt; are 4 T 1T OŽ., Ž 5.1. Ž. OŽ 4., Ž 5.. OŽ 4., Ž 5.3. OŽ 4., Ž 5.4. OŽ.. Ž 5.5. By Theorem 4.1, if, then the periodic solution pt; is stable. In order to study the change of stability of pt; as D is varied, we write the linearized system of Ž 3.. around pt; as dz EzŽ t. f u Ž Ž., p t Ž ;.. z t, Ž 5.6. dt D E. Ž 5.7.

20 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 85 Define H d Ž LE. K d, L and K are defined as in 3.3 and 3.4, and E is defined by 5.7. Define two vectors and by and ž H / H i i e d Ž 5.8. ž H / i T i e d. Ž 5.9. After adopting new variables s s t, wž s. z zž t., system Ž 5.6. can be written as d wž s. EwŽ s. fuž Ž., ys, Ž. Ž ;.. w s,wž., Ž 5.1. ds s y s, p ;, Let w wž s.,. s, Ž. E E i.e., D D Ž for some matrix E Ž for some constant D.. Now Eq. Ž 5.1. has the same form as the one that appears in Morita 33. Applying the results in 33 it follows that for D D, the exponents of Ž 5.1. have the form Ž, D. Ž D. ˆŽ, D., ˆ Ž,., Ž 5.1. Ž D. satisfies and 14 Re Ž E,. B Ž E,. 4 1Ž. Re B E,, 5.13 d B1 i Re Ž., Ž d and are defined in Ž 5.3. and Ž and are defined by Ž 5.8. and Ž Thus we have the following result.

21 86 RUAN AND WOLKOWICZ THEOREM 5.1. Under the hypotheses of Theorems 3.1 and 4.1, if D D and Eq. Ž has a positie root, then there exists an such that for each Ž,., the periodic solution in system 1. is unstable. 6. AN EXAMPLE Consider system 1. ps takes the MichaelisMenten form and FŽ s. is the weak kernel. In particular, consider the following system: ds SŽ t..8ž 3.66 SŽ t.. 4.5xŽ t., dt 5.85 SŽ t. Ž 6.1. dx t SŽ. Žt. xž t H e d. dt 5.85 SŽ. By theorem 3.1, we can determine that.193,.98. Ž 6.. The positive equilibrium is E Ž ,.4.. Ž 6.3. By the results in Section 4, it follows that , , Ž 6.4. These calculations prove that the equilibrium E is stable when as is illustrated by the computer simulations Ž Figs. 1 and, When passes through the critical value.193, E loses its stability and a Hopf bifurcation occurs; i.e., a family of periodic solutions bifurcate from E. The individual periodic orbits are stable since. Choosing.1813, as predicted by the theory, Fig. 3 shows that there is an orbitally stable limit cycle. Since, the bifurcating periodic solutions exist at least for values of slightly less than the critical value. Recall that in 8., the average time delay is defined by T 1. Note that T increases when decreases. For a slightly smaller value of,.173, the orbitally stable periodic solution persists and is plotted in Fig. 4. The period is approximately 35 Ž Fig. 5.. Since, the period of the periodic solutions increases as decreases. From Theorem 5.1, the periodic solution loses its stability if the dilution rate D is varied. As we increase D from.8 to.14, the periodic solution in Fig. 4 becomes unstable and the equilibrium E regains its stability Ž Fig. 6..

22 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 87 FIG. 1. The equilibrium E ,.4 of system 5.1 is asymptotically stable when. Here FIG.. The trajectories of S and x with respect to time when.4645.

23 88 RUAN AND WOLKOWICZ FIG. 3. With.1813, there is an orbitally stable limit cycle attracting two trajectories. One trajectory has initial values inside the limit cycle and near the equilibrium; the other has initial values outside the limit cycle. FIG. 4. The bifurcating periodic orbit persists for values of slightly less than.a stable periodic orbit is plotted with.173.

24 CHEMOSTAT MODEL WITH A DISTRIBUTED DELAY 89 FIG. 5. The oscillations of S and x versus time when.173. The period is approximately 35. FIG. 6. The bifurcating periodic solution becomes unstable when D.14 and the equilibrium regains its stability.

25 81 RUAN AND WOLKOWICZ 7. DISCUSSION We have studied a chemostat model of a single species with a distributed delay. Using the average time delay as a bifurcation parameter, we have shown that a Hopf bifurcation occurs when this parameter passes through a critical value; i.e., a family of periodic orbits bifurcates from the positive equilibrium. The stability of the bifurcating periodic orbits is discussed. We have also studied the destabilization of the periodic solutions when the washout rate is varied. If the kernel is a delta function of the form FŽ s. Ž s.,, Ž 7.1. is a constant, then system 1. reduces to the following model with a discrete delay : ds Ž S SŽ t.. DaxŽ t. pž SŽ t.., dt Ž 7.. dx xž t. D 1 pž SŽ t... dt It has been shown by Freedman et al. 15 that there is a critical value of the discrete delay at which Hopf bifurcation occurs. Our Theorem 3.1 corresponds to one of their results. If the kernel is a delta function of the form FŽ s. Ž s., Ž 7.3. then system.1 becomes the system of ordinary differential equations ds Ž S SŽ t.. DaxŽ t. pž SŽ t.., dt dx xž t. D 1 pž SŽ t.., dt Ž 7.4. which has been investigated by many authors. By constructing a Liapunov function Žsee Wolkowicz et al. 39., it can be seen that if the positive equilibrium of Ž 7.4. exists, it is globally asymptotically stable. Thus our result shows that delays can destabilize an otherwise stable equilibrium. This phenomenon has been observed by many authors in other settings Žfor example, see Cushing 11, Kuang 5, MacDonald 9.. The stability of the bifurcating periodic orbits allows us to claim that for the delayed chemostat model, even if the positive equilibrium is not stable, all components may still coexist in an oscillatory mode.

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