Simple Food Chain in a Chemostat with Distinct Removal Rates

Transcription

1 Journal of Mathematical Analyi and Application 242, 7592 Ž doi:0.006jmaa , available online at on Simple Food Chain in a Chemotat with Ditinct Removal Rate Bingtuan Li Department of Mathematic, Unierity of Utah, Salt Lake City, Utah 842 and Yang Kuang Department of Mathematic, Arizona State Unierity, Tempe, Arizona Submitted by V. Lakhmikantham Received March 23, 999 In thi paper, we conider a model decribing predatorprey interaction in a chemotat that incorporate both general repone function and ditinct removal rate. In thi cae, the conervation law fail. To overcome thi difficulty, we make ue of a novel way of contructing a Lyapunov function in the tudy of the global tability of a predator-free teady tate. Local and global tability of other teady tate, peritence analyi, a well a numerical imulation are alo preented. Our finding are largely in line with thoe of an identical removal rate cae Academic Pre Key Word: chemotat; predator; prey; food chain; peritence.. INTRODUCTION The chemotat i a laboratory apparatu ued for the continuou culture of microorganim. It can be ued to tudy competition between different population of microorganim, and ha the advantage that the parameter are readily meaurable. See the monograph of Smith and Waltman 0 for a detailed decription of a chemotat and for variou mathematical method for analyzing chemotat model. Reearch partially upported by NSF Grant DMS X00 $35.00 Copyright 2000 by Academic Pre All right of reproduction in any form reerved.

2 76 LI AND KUANG Conider a food chain in a chemotat with one predator and one prey. We aume that the predator feed ecluively on the prey, and the prey conume the nutrient in the chemotat. Thi i an intereting practical problem both mathematically and biologically. In a wate treatment proce, the bacteria live on the wate Ž or nutrient. while other organim uch a ciliate feed on the bacteria. The equation of interet are 0 SŽ t. S SŽ t. D F SŽ t. Ž t., Ž t. Ž t. Ž FŽ SŽ t.. D. F2Ž Ž t.. yž t., Ž.. 2 Ž 2 2. yž t. yž t. F Ž t. D, SŽ 0. 0, Ž 0. 0, yž 0. 0, where St Ž. denote the concentration of nutrient at time t; t Ž. denote the concentration of prey at time t; yt Ž. denote the concentration of predator at time t; S 0 denote the input concentration of the nutrient; and 2 denote the yield contant; F and F2 denote the pecific per-capita growth rate of prey and predator organim, repectively; D i the wahout rate of the chemotat; each Di D i, i, 2, where and 2 denote the pecific death rate of organim and y, repectively. The value D D and D2 D reult from auming that the death rate of and y are negligible o that the only lo of organim i due to wahout at the ame rate that the nutrient i lot. If an organim death rate i ignificant, the removal rate of thi organim hould be the um of D and the death rate. We make the following aumption on the repone function f i F i: R R, Ž.2. Fi i continuouly differentiable, Ž.3. Fi Ž 0. 0, Ž.4. F S 0 for all S 0 and F 2Ž. 0 for 0. Ž.5. By meauring concentration of nutrient in unit of S 0, time in unit of D, i unit of S 0, and y in unit of S 0, one reduce the number 2

3 DISTINCT REMOVAL RATES 77 of parameter and obtain the following differential equation S Ž t. SŽ t. fž SŽ t.. Ž t., Ž Ž.. Ž. t t f S t D f t y t, 2 Ž. yž t. yž t. f Ž t. D, 2 2 SŽ 0. 0, Ž 0. 0, yž 0. 0, Ž 0. i i 2 Ž.6. where D D D, i, 2; f S D F S S ; and f D F ŽS 0.. Clearly, f atify Ž.2. Ž i. If the functional repone function are of the MichaeliMenten form and D D, then ytem Ž.6. become 2 ms S S, a S ms m2 y, a S a2 Ž.7. m2 y y, a 2 SŽ 0. 0, Ž 0. 0, yž Sytem Ž.7. ha been tudied in, 4, 9, and 2 and related eperiment are decribed in 2, 4. In, Butler et al. took advantage of the conervation principle and gave a global analyi of thi model. The author howed that the interior critical point i globally aymptotically table if it i locally aymptotically table, and if it i untable then there eit a periodic orbit for Ž.7.. The analyi in critically depend on the aumption that removal rate for the prey and predator organim are both equal to the wahout rate of the chemotat. Conider the quantity T S y, where S,, and y are olution of Ž.7.. Then T atifie the differential equality T T. In thi pecial cae, the differential equality implie that all olution approach the plane S y at an eponential rate. Thi in turn implie that ytem Ž.7. can be reduced to a planar ytem by dropping the S equation and making the ubtitution S y in the remain-

4 78 LI AND KUANG ing two equation. Thi i often referred to a the conervation principle. However, a light change in the removal rate of or y detroy the form of the conervation principle and the reduction to a planar ytem i no longer poible. Thi paper i organized a follow: In the net ection, we preent reult on the poitivity and boundedne of olution. Section 3 deal with the eitence and local tability of teady tate. In Section 4, we hall provide global analyi, including global tability of the boundary teady tate and peritence analyi. Thi paper end with a dicuion ection which conit of comment and numerical imulation. 2. PRELIMINARIES In thi ection, we hall preent ome preliminary reult, including the poitivity and boundedne of olution. We conider firt the poitivity. LEMMA 2.. The olution SŽ. t, t, Ž. yt Ž. ofž.6. are poitie, and for large t, St Ž.. Proof. It i eay to ee that St Ž. tay poitive. Aume the lemma i fale. Let t mint : t 0, tyt Ž. Ž. 04. Aume firt that t 0. Then yt Ž. 0 for t 0, t. Let A min f ŽSŽ t.. 0 t t D Žf ŽŽ.. t Ž.. t yt Ž.4. Then, for t 0, t, Ž. t AŽ. 2 t, which implie that At t 0 e 0, a contradiction. A imilar argument how that yt 0 i aburd. Finally, S 0 for all S. Thi prove the lemma. Define Dma ma, D, D24 and Dmin min, D, D 24. Adding the three equation in.6 yield Thi lead to Ž S y. Ž S D D2 y.. D Ž S y. Ž S y. D Ž S y.. ma Solving thi inequality yield the following lemma. LEMMA 2.2. For 0, the olution SŽ. t, t, Ž. yt Ž. ofž.6. atify for large t. SŽ t. Ž t. yž t. Ž 2.. D D ma min min

5 DISTINCT REMOVAL RATES STEADY STATES AND THEIR STABILITY The wahout teady tate for ytem Ž.6. i denoted by E Ž, 0, 0.. There i only one poible teady tate involving prey organim but not predator organim, denoted by E Ž, Ž. D, 0. 2 where i defined a the unique olution of fž S. D 0 Ž if it eit.. The interior teady tate i denoted by E ŽS,, Žf ŽS. c D. D. where i defined a the unique olution of 2 f2ž. D2 0 if it eit, and S* i defined a the unique olution of S fž S. 0 Ž 3.. with S* Ž 0,.. One can ee that no teady tate can eit where there i a predator pecie but no prey pecie. We ay that E2 or Ec doe not eit if any one of it component i negative. We now dicu the eitence of teady tate. The wahout teady tate E Ž, 0, 0. alway eit. Since f i increaing with f Ž 0. 0, eit, atifying 0 and fž. D fž. D. Ž 3.2. In thi cae there i a predator-free teady tate E Ž, Ž. D,0. 2. Otherwie, no uch teady tate eit. In the cae where f Ž S. D for all S 0, we regard. Net conider the mied-culture Ž interior. teady tate E c. Since f2 i increaing with f Ž. 0 0, 2 eit, atifying f Ž. D lim f Ž. D. Ž For E to eit, f Ž S*. c D mut be poitive or S*. Note that FS S f Ž S. i decreaing in S with FŽ 0. 0, FS* 0, and FŽ. D. So, S* if and only if Ž. D. In the cae where f Ž S. 2 D2 for all S 0, we regard. Therefore E2 eit if and only if, and Ec eit if and only if and Ž. D. Net we invetigate the local tability of thee teady tate by finding the eigenvalue of the aociated Jacobian matrice.

6 80 LI AND KUANG The Jacobian matri of.6 take the form J fž S. fž S. 0 fž S. fž S. D yf2ž. f2ž. 0 yf Ž. f Ž. D Ž 3.4. At E, fž. 0 JŽ E. 0 fž. D 0. Ž D 2 The eigenvalue lie on the diagonal. They are all negative if and only if f Ž. D 0 or, equivalently,. When E eit, the Jacobian matri at E i 2 2 f Ž. f Ž. 0 D / ž D / JŽ E f Ž. 0 f 2. 2ž. Ž 3.6. D D 0 0 f D 2 2 The determinant of the upper left-hand 2 2 matri i poitive and it trace i negative, o it eigenvalue have negative real part. The third eigenvalue of JŽ E. i f ŽŽ. D. 2 2 D 2, the entry in the lower right-hand corner. Therefore E2 i aymptotically table if and only if f ŽŽ. D. D 0orŽ. 2 2 D. When Ec eit, the Jacobian matri at Ec take the form f Ž S*. f Ž S*. 0 2Ž. Ž. 2 f S* f f S* D D JŽ E. D c 2 Ž fž S*. D. 0 f Ž D 2 3.7

7 The eigenvalue of EE atify DISTINCT REMOVAL RATES 8 c 3 a 2 a2 a3 0, Ž 3.8. where and ž D 2 / 2Ž. D 2 a f S* f f S* D, 2 Ž. 2Ž. Ž. a f S* f f S* D f Ž S*. D f Ž. f Ž S*. f Ž S*., 2 c a f Ž. f Ž S*. f Ž S*. D. 3 2 Note that the contant term a i poitive, o the RouthHurwitz criterion 3 ay that E will be aymptotically table if and only if a 0 and c aa a. 2 3 We ummarize the above reult in the following theorem. THEOREM 3.. If, then only E eit and E i locally aymptoti- cally table. If and Ž. D, then only E and E2 eit, E i untable, and E2 i locally aymptotically table. If and Ž. D, then E, E 2, and Ec eit, and E and E2 are untable. Ec i locally aymptotically table if a 0 and aa2 a 3. If D D, then the limit plane of Ž.6. 2 i : S y. If we drop the S equation, then the limit ytem of Ž.6. take the form f Ž y. f Ž. y, 2 y y f2ž., Ž 3.9. Ž 0. 0, yž For thi reduced ytem, if Ec eit then the Jacobian matri at Ec i JŽ Ec. f Ž S*. f Ž. f Ž S*. f Ž S*. f Ž. 2 2 Ž fž S*.. f2ž. 0. Ž 3.0.

8 82 LI AND KUANG The determinant of thi matri i f Ž S*. f Ž. Žf Ž S*. 2 f Ž., which i poitive, and the trace i f Ž S*. Ž f Ž Žf Ž S*... Therefore if f Ž S*. Ž f Ž..Ž fž S* then E i locally aymptotically table; if f Ž S*. Ž f Ž.. c 2 Žf Ž S*.. 0 then Ec i a repeller in, and in thi cae there i a periodic olution in Ž by an application of the PoincareBendion teorem.. We ummarize thee in the following theorem. THEOREM 3.2. Aume D D and E eit Ž i.e.,.. 2 c If f Ž S*. Ž f Ž..Ž f Ž S* then Ec i locally aymp- totically table. If f Ž S*. Ž f Ž..Ž f Ž S* then Ec i untable, and there i a periodic olution in the plane : S y. 4. GLOBAL ANALYSIS In the previou ection, we howed that if only E eit then E i aymptotically table, if E and E2 eit then E i untable and E2 i Ž locally. aymptotically table, if E, E, and E eit then E and E are 2 c 2 untable, and in thi cae E may or may not be table. We hall how that c E i globally aymptotically table if only E eit. The proof i very traightforward. Mot importantly, we hall how that if only E and E 2 eit, under a reaonable additional aumption E2 i globally aymptotically table. The proof involve the contruction of a Lyapunov function and the application of the LyapunovLaSalle theorem. ŽWe hall ue Theorem 2. in Wolkowicz and Lu 3, which i a lightly modified verion of the tatement given in LaSalle 6 and Hale. 3. We hall alo how that ytem Ž.6. i uniformly peritent if Ec eit. The following theorem tate that E i a global attractor if it i the only teady tate Ž i.e.,.. THEOREM 4.. If, then all olution of.6 atify lim SŽ t., Ž t., yž t. Ž,0, 0.. t Proof. Since St Ž. for large t and f Ž. D 0 Ž i.e.,,. there i 0 uch that Ž. t t Ž. for t ufficiently large. Thi how lim t Ž. 0. It follow from the third equation of Ž.6. t that lim yt Ž. 0. Then the firt equation of Ž.6. yield lim St Ž. t t. The proof i complete. Note that if Ž. Dmin then D. That i, Dmin implie Ž. D.

9 DISTINCT REMOVAL RATES 83 THEOREM 4.2. If and D min, Ž 4.. then all olution of.6 atify lim Ž SŽ t., Ž t., yž t..,,0. D t Proof. We chooe d D and d D uch that ma 2 min and that, for large t, Ž 4.2. d 2 Let SŽ t. Ž t. yž t.. Ž 4.3. d d 2 ½ f Ž. Ž Ž. D. 5 ma 2 0Ž. Ž D f Ž D and ½ 5 f2ž. D2 ma. D D min 2 Let Cu be a continuouly differentiable function and CŽ u. i given by 0, if u, CŽ u. u, if u, d2 d2 d2, u. C u i imply linear on d,. In view of 4.3, 2 y if S. d 2 d 2

10 84 LI AND KUANG Therefore, if S, then CŽ y. 0. The graph of CŽ y. i hown in Fig.. Define the Lyapunov function VS, Ž, y. a follow S fž. D Ž. V H d DŽ. * *ln y CŽ y. Ž 4.4. * on the et Ž S,, y. : S Ž 0,.,, y Ž 0,., S y Ž d,d.4, where * Ž. 2 D. Then the time derivative of V along olution of the differential equation i Ž. fž S. V CŽ y. Ž fž S. D. DŽ S. f2ž. Ž f2ž. D2. DC 2 Ž y. y. D Firt, note that Ž. f Ž S. D Ž S.Žf Ž S. D. 2 i nonpoitive for 0 S and equal 0 for S 0,. if and only if S or 0. Since CŽ y. 0 for S and CŽ u. 0 for u 0, CŽ y. Žf Ž S. D. i nonpoitive for S 0,.. Therefore the firt term in V i alway nonpoitive and equal 0 for S 0,. if and only if S or 0.. FIG.. A graphical depiction of C y for S 0,.

11 DISTINCT REMOVAL RATES 85 Define f2ž. hž S,, y. Ž f2ž. D2. DC 2 Ž y.. D If 0 D, then f2ž. 0 and f 2 Ž. D 2 0 D and they are equal to 0 if and only if Ž. D. Note that CŽ y. i alway nonnegative. By the definition of, hs, Ž, y. 0 for 0 Ž. D and poibly hs, Ž, y. 0 if Ž. D. If Ž. D, all three term in hs, Ž, y. are nonpoitive and one can eaily ee hs, Ž, y. 0. If, then y and therefore CŽ y.. Note that, if, only the econd term in hs, Ž, y. i nonnegative. According to the definition of, hs, Ž, y. 0 if. Therefore hs, Ž, y. 0 for 0 and Ž. D, and it i poibly 0 if Ž. D. By Lemma 2. every bounded olution of Ž.6. i contained in, and hence by Theorem 2. in 3 every olution of Ž.6. approache the et, the larget invariant ubet M of Ž S,, y. : V 0. 4 i made up of point of the following form Ž S, 0, 0., where S 0,,,, y, where y 0,., ž D / Ž,, 0., where 0,.. Since V i bounded above, any point of the form Ž S, 0, 0. cannot be in the 3 -limit et of any olution initiating in the interior of R. Ž,,0. M implie that St Ž., which in turn lead to 0 SŽ. t f Ž. and hence Ž. D. Ž, Ž. D, y. M implie that St and t Ž. Ž. D. The econd equation of Ž.6. implie that 0, which yield y 0. Therefore M E 4. Thi complete the proof. THEOREM 4.3. If and Ž. D, then ytem Ž.6. i uniformly peritent; i.e., there eit a contant 0, independent of initial condition, uch that lim inf SŽ t., lim inf Ž t., lim inf yž t.. t t t 2

12 86 LI AND KUANG and Proof. Chooe 5 min min D min 5 D D 5 X ½Ž S,, y.;0s, 0, 0 y, D D Y S,,0 ;0 S, 0, ½ Y S,0, y ;0 S, 0 y, 2 ½ min X Y Y. 2 2 min FIG. 2. m 3.6, a 0.8, D.4, m 3, a 0.6, D.2. ŽSŽ 0,. Ž 0,y. Ž Ž 0.6, 0.2, The top curve depict St, the middle one depict t, and the bottom one depict yt. Ž. In thi cae,. Clearly, the olution approache the predator-free teady tate.

13 DISTINCT REMOVAL RATES 87 Then X and X2 are two dijoint ubet of R 3, X2 i compact, X X X i alo compact, and X and X are poitively invariant for Ž By Lemma 2.2, X2 and X are global attractor in the union of the S plane and S y plane and in R 3, repectively. We prove that X2 i a uniformly trong repeller for X Ž for the definition of a uniformly trong repeller a well a a weak repeller, ee Thieme.. E and E2 are the only teady tate in X 2. E i a addle in R 3 and it 4 3 table manifold i S,0, y ; y 0.E2 i alo a addle in R and it table manifold i Ž S,,0.; 04. Therefore E and E2 are weak repeller for X. The table manifold tructure of E and E Ž 2 E2 i a global attractor in the S plane. imply that they are not cyclically chained to each other on the boundary X. By Propoition.2 of Thieme 2, X2 i a uniform trong repeller for X ; that i, there are 0 and 2 0 uch that lim inf t Ž. and lim inf yt Ž. with and not dependt t 2 2 FIG. 3. m 3.6, a 0.8, D.4, m 3, a 0.6, D. ŽSŽ 0,. Ž 0,y. Ž Ž 0.6, 0.2, The top curve depict St, the middle one depict t, and the bottom one depict yt. Ž. Clearly, the olution approache a poitive teady tate.

14 88 LI AND KUANG ing on the initial value in X. Applying Propoition 2.2 of Thieme to the firt equation of Ž.6. yield that there i 3 0 uch that lim inf St Ž. t 3 with 3 not depending on the initial value of X. Thi complete the proof. 5. DISCUSSION In thi paper, we conidered a food chain with one prey and one predator in the chemotat. In thi model, the prey conume the nutrient and the predator conume the prey but the predator doe not conume the nutrient. We aumed that the functional repone function are general monotone repone function and the removal rate are different. The model we conidered i more general and realitic than the model in, 4, 9, 2. FIG. 4. m 8.5, a 0.6, D D, m 6, a 0.6. ŽSŽ 0,. Ž 0,y. Ž Ž 0., 0.7, The top curve depict St, the middle one depict yt, and the bottom one depict t. Ž. The olution appear to approach a periodic olution.

15 DISTINCT REMOVAL RATES 89 The main difficulty we faced wa the lack of a conervation principle which wa lot due to the different removal rate. In the cae of different removal rate, the ytem cannot be reduced to a two-dimenional ytem and we therefore mut look at the full ytem. We found that the wahout teady tate E i the global attractor if it i the only teady tate Ž thi happen when.. Thi confirm the intuition that both prey and predator cannot perit if the removal rate of the prey i relatively large. When E and the predator-free teady tate E2 are the only teady tate, we found that E i untable and E2 i locally aymptotically table. By contructing a Lyapunov function, we were able to how that if E and E2 are the only teady tate, under an additional aumption Žee Ž 4..., E i a global attractor. The contruction of the Lyapunov function i rather novel and nontrivial. Thi novel idea ha been ued in 7, 8. Thi condition doe not depend on the pecific propertie of the functional repone function, and it become neceary if D i cloe to both D and. The min 2 FIG. 5. m 8.5, a 0.6, D., m 6, a 0.6, D. ŽSŽ 0,. Ž 0,y. Ž Ž 0., 0.7, The top curve depict St, the middle one depict yt, and the bottom one depict t. Ž. The olution ocillate but eventually approache a poitive teady tate.

16 90 LI AND KUANG global tability of E2 implie that the predator will be wahed out in the chemotat regardle of the initial denity level of prey and predator. We alo howed that, when Ec eit, the prey and predator coeit in the ene that the ytem i uniformly peritent. In thi cae, a witch of the tability of the interior teady tate Ec may occur. E i a global attractor if it i locally aymptotically table and Ž hold. Baed on our etenive imulation work where the functional repone function take the MichaeliMenten form ms m2 fž S. and f2ž., a S a 2 we conjecture that thi teady tate remain globally aymptotically table a long a it i locally table Ž ee Fig. 2.. A E2 become untable, a locally aymptotically table interior teady tate Ec bifurcate from it. Our imulation work Ž Fig. 3. ugget that Ec i a global attractor if it i locally aymptotically table. A certain parameter increae or decreae further away, Ec loe it tability and ocillatory olution appear. Thee ocilla- tory olution Ž ee Fig. 4 and 6. appear to be the reult of Hopf bifurcation. Figure 4 how a cae in which D D and ytem Ž.6. 2 poee periodic olution. Figure 5 indicate that perturbing D Ž while keeping other parameter in Fig. 4 fied. lead to a bifurcation. For eample, changing D in Fig. 4 to D. in Fig. 5 eem to detroy the periodic olution and poibly lead to the global tability of E. c Therefore varying the value of D and D2 may affect the dynamic of Ž.6. in a very urpriing and ignificant way. Net, we ee how the parameter D and D affect the dynamic of Ž.6. 2 if S 0 i fied. D and D2 2, where and 2 denote the caled pecific death rate of the prey and predator, repectively. ŽNote that the analyi of the model require no aumption on the ign of, i a long a the D i all remain poitive. Thi leave the ID i open to other interpretation.. Aume that D and D are large enough o that, 2 then both prey and predator population will be wahed out Ž E i table. in the chemotat. A D i gradually decreaed, eventually there i a bifurcation when and Ž. D. In thi cae, E loe it tability and the new bifurcated teady tate E2 i aymptotically table. A D i gradually decreaed, the net bifurcation occur when and 2 Ž. D hold. In thi cae E2 loe it tability, and a new interior teady tate Ec appear. If D D2, the conervation principle hold; that i, the -limit et of olution of Ž.6. lie in the plane : S y. In thi cae, one can eaily how that E Ž if it eit. c i locally aymptotically table if and only if f Ž S*. Ž f Ž..Ž f Ž S*.. 0. When thi in- 2

17 DISTINCT REMOVAL RATES 9 FIG. 6. m 4, a 0.6, D., m 5, a 0.5, D.2. ŽSŽ 0,. Ž 0,y. Ž Ž 0., 0.7, In thi cae, it can be hown that Ec eit and i untable. The top curve depict St, Ž. the middle one depict t, Ž. and the bottom one depict yt. Ž. The olution ocillate and eem to approach a periodic olution. equality i revered, Ec will be a repeller in, and there will be at leat one periodic orbit Ž by an application of the PoincareBendion theorem.. Determining the number of periodic olution i a deep mathematical problem. Kuang 5 ha hown in the cae of MichaeliMenten-type repone function, if f Ž S*. Ž f Ž..Ž f Ž S*.. 2 i mall and poitive, then the limit cycle i unique and aymptotically table. We gue thi i true for general repone function. In, it wa hown that in the cae of MichaeliMenten-type repone function and D D2, Ec i globally aymptotically table if it i locally aymptotically table. It remain open if thi i true in the cae of general repone function and different removal rate.

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