March  2020, 25(3): 877-901. doi: 10.3934/dcdsb.2019194

Global dynamics of a reaction-diffusion system with intraguild predation and internal storage

1. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

2. 

Department of Mathematics and National Center of Theoretical Science, National Tsing-Hua University, Hsinchu 300, Taiwan

3. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan

4. 

Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

* Corresponding author: fbwang@mail.cgu.edu.tw, fbwang0229@gmail.com

Received  November 2018 Revised  March 2019 Published  September 2019

Fund Project: The first author is supported by NSF of China (11671243, 11572180), the Fundamental Research Funds for the Central Universities (GK201701001).

This paper presents a reaction-diffusion system modeling interactions of the intraguild predator and prey in an unstirred chemostat, in which the predator can also compete with its prey for one single nutrient resource that can be stored within individuals. Under suitable conditions, we first show that there are at least three steady-state solutions for the full system, a trivial steady-state solution with neither species present, and two semitrivial steady-state solutions with just one of the species. Then we establish that coexistence of the intraguild predator and prey can occur if both of the semitrivial steady-state solutions are invasible by the missing species. Comparing with the system without predation, our numerical simulations show that the introduction of predation in an ecosystem can enhance the coexistence of species. Our mathematical arguments also work for the linear food chain model (top-down predation), in which the top-down predator only feeds on the prey but does not compete for nutrient resource with the prey. In our numerical studies, we also do a comparison of intraguild predation and top-down predation.

Citation: Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194
References:
[1]

C. J. Bampfylde and M. A. Lewis, Biological control through intraguild predation: Case studies in pest control, invasive species and range expansion, Bull. Math. Biol., 69 (2007), 1031-1066.  doi: 10.1007/s11538-006-9158-9.  Google Scholar

[2]

S. Diehl, Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships, Oikos, 68 (1993), 151-157.  doi: 10.2307/3545321.  Google Scholar

[3]

S. Diehl, Direct and indirect effects of omnivory in a littoral lake community, Ecology, 76 (1995), 1727-1740.  doi: 10.2307/1940706.  Google Scholar

[4]

S. Diehl and M. Feissel, Effects of enrichment on threelevel food chains with omnivory, Am. Nat., 155 (2000), 200-218.  doi: 10.1086/303319.  Google Scholar

[5]

S. Diehl and M. Feissel, Intraguild prey suffer from enrichment of their resources: a microcosm experiment with ciliates, Ecology, 82 (2001), 2977-2983.  doi: 10.2307/2679828.  Google Scholar

[6]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.  doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

[7]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

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J. P. Grover, Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model, Am. Nat., 138 (1991), 811-835.   Google Scholar

[9]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, Am. Nat., 178 (2011), 124-148.  doi: 10.1086/662163.  Google Scholar

[10]

J. P. GroverS. B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.  doi: 10.1007/s00285-011-0426-4.  Google Scholar

[11]

J. P. Grover and F.-B. Wang, Competition for one nutrient with internal storage and toxin mortality, Math. Biosci., 244 (2013), 82-90.  doi: 10.1016/j.mbs.2013.04.009.  Google Scholar

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.  Google Scholar

[13]

W. M. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

[14]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764.  doi: 10.1086/286018.  Google Scholar

[15]

S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[16]

S. B. HsuJ. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.  doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[17]

S. B. HsuK.-Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775-1825.  doi: 10.1007/s00285-017-1134-5.  Google Scholar

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S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.  Google Scholar

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[20]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[21]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theor. Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[23]

L. MeiS. B. Hsu and F.-B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 607-620.  doi: 10.3934/dcdsb.2016.21.607.  Google Scholar

[24]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137-150.  doi: 10.1111/j.1529-8817.1987.tb04436.x.  Google Scholar

[25]

H. NieS.-B. Hsu and F.-B. Wang, Steady-state solutions of a reaction-diffusion system arising from intraguild predation and internal storage, J. Differential Equations, 266 (2019), 8459-8491.  doi: 10.1016/j.jde.2018.12.035.  Google Scholar

[26] H. NieJ. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017.   Google Scholar
[27]

G. A. Polis and et al, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annu. Rew. Ecol. Syst., 20 (1989), 297-330.  doi: 10.1146/annurev.es.20.110189.001501.  Google Scholar

[28]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154.  doi: 10.1146/annurev.es.20.110189.001501.  Google Scholar

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[30]

J. A. RosenheimH. K. KayaL. E. EhleretJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.  Google Scholar

[31]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.  Google Scholar

[32]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[34]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[35]

F.-B. WangS.-B. Hsu and Y.-H. Ho, Mathematical analysis on a Droop model with intraguild predation, Taiwanese J. Math., 23 (2019), 351-373.  doi: 10.11650/tjm/181011.  Google Scholar

[36]

S. WilkenJ. M. H. VerspagenS. Naus-WiezerE. V. Donk and J. Huisman, Comparison of predator-prey interactions with and without intraguild predation by manipulation of the nitrogen source, Oikos, 123 (2014), 423-432.  doi: 10.1111/j.1600-0706.2013.00736.x.  Google Scholar

[37]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[38]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.  Google Scholar

[39]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

References:
[1]

C. J. Bampfylde and M. A. Lewis, Biological control through intraguild predation: Case studies in pest control, invasive species and range expansion, Bull. Math. Biol., 69 (2007), 1031-1066.  doi: 10.1007/s11538-006-9158-9.  Google Scholar

[2]

S. Diehl, Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships, Oikos, 68 (1993), 151-157.  doi: 10.2307/3545321.  Google Scholar

[3]

S. Diehl, Direct and indirect effects of omnivory in a littoral lake community, Ecology, 76 (1995), 1727-1740.  doi: 10.2307/1940706.  Google Scholar

[4]

S. Diehl and M. Feissel, Effects of enrichment on threelevel food chains with omnivory, Am. Nat., 155 (2000), 200-218.  doi: 10.1086/303319.  Google Scholar

[5]

S. Diehl and M. Feissel, Intraguild prey suffer from enrichment of their resources: a microcosm experiment with ciliates, Ecology, 82 (2001), 2977-2983.  doi: 10.2307/2679828.  Google Scholar

[6]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.  doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

[7]

J. P. Grover, Resource Competition, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4615-6397-6.  Google Scholar

[8]

J. P. Grover, Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model, Am. Nat., 138 (1991), 811-835.   Google Scholar

[9]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, Am. Nat., 178 (2011), 124-148.  doi: 10.1086/662163.  Google Scholar

[10]

J. P. GroverS. B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.  doi: 10.1007/s00285-011-0426-4.  Google Scholar

[11]

J. P. Grover and F.-B. Wang, Competition for one nutrient with internal storage and toxin mortality, Math. Biosci., 244 (2013), 82-90.  doi: 10.1016/j.mbs.2013.04.009.  Google Scholar

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.  Google Scholar

[13]

W. M. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.  Google Scholar

[14]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764.  doi: 10.1086/286018.  Google Scholar

[15]

S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[16]

S. B. HsuJ. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.  doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[17]

S. B. HsuK.-Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775-1825.  doi: 10.1007/s00285-017-1134-5.  Google Scholar

[18]

S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3169-3189.  doi: 10.3934/dcdsb.2014.19.3169.  Google Scholar

[19]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.  doi: 10.1137/0153051.  Google Scholar

[20]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[21]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theor. Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[23]

L. MeiS. B. Hsu and F.-B. Wang, Growth of single phytoplankton species with internal storage in a water column, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 607-620.  doi: 10.3934/dcdsb.2016.21.607.  Google Scholar

[24]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137-150.  doi: 10.1111/j.1529-8817.1987.tb04436.x.  Google Scholar

[25]

H. NieS.-B. Hsu and F.-B. Wang, Steady-state solutions of a reaction-diffusion system arising from intraguild predation and internal storage, J. Differential Equations, 266 (2019), 8459-8491.  doi: 10.1016/j.jde.2018.12.035.  Google Scholar

[26] H. NieJ. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017.   Google Scholar
[27]

G. A. Polis and et al, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annu. Rew. Ecol. Syst., 20 (1989), 297-330.  doi: 10.1146/annurev.es.20.110189.001501.  Google Scholar

[28]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154.  doi: 10.1146/annurev.es.20.110189.001501.  Google Scholar

[29]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[30]

J. A. RosenheimH. K. KayaL. E. EhleretJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.  Google Scholar

[31]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.  Google Scholar

[32]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.  Google Scholar

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[34]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[35]

F.-B. WangS.-B. Hsu and Y.-H. Ho, Mathematical analysis on a Droop model with intraguild predation, Taiwanese J. Math., 23 (2019), 351-373.  doi: 10.11650/tjm/181011.  Google Scholar

[36]

S. WilkenJ. M. H. VerspagenS. Naus-WiezerE. V. Donk and J. Huisman, Comparison of predator-prey interactions with and without intraguild predation by manipulation of the nitrogen source, Oikos, 123 (2014), 423-432.  doi: 10.1111/j.1600-0706.2013.00736.x.  Google Scholar

[37]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[38]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332.  doi: 10.1006/jdeq.2000.3870.  Google Scholar

[39]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-319-56433-3.  Google Scholar

Figure 1.  The effects of the nutrient supply concentration $R^{(0)} $: (A, C) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B, D) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. $R^{(0)} = 1.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (A, B), and $R^{(0)} = 2.5\times10^{-5}\, \mathrm{mol}\,\mathrm{l}^{-1}$ in (C, D)
Figure 2.  Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $R^{(0)}$ ranging from $0.5\times10^{-5}$ to $2.0\times10^{-4}\, \mathrm{mol}\,\mathrm{l}^{-1}.$ (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
Figure 3.  The effects of the diffusion rate $d $: (A, C, E) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B, D, F) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. $d = 0.08\, \mathrm{day}^{-1}$ in (A, B), $d = 0.12\, \mathrm{day}^{-1}$ in (C, D), and $d = 0.16\, \mathrm{day}^{-1}$ in (E, F)
Figure 4.  Bifurcation diagrams of positive steady state solutions to (5)-(7) with the bifurcation parameter $g_{\max}$ ranging from $0$ to $120\ \mathrm{cells}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $. (A) Intraguild predation with $a_{\max,2} = 24.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $, and (B) top-down predation with $a_{\max,2} = 0\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
Table 1.  Common parameters used in intraguild predation and top-down predation
Quantity Value Quantity Value
$\gamma $ $10\, \mathrm{day}^{-1} $ $a_{\max,1} $ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
$K_1 $ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $ $K_2 $ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $
$\mu_{\max,1} $ $0.7\,\mathrm{day}^{-1} $ $\mu_{\max,2} $ $2.2\,\mathrm{day}^{-1} $
$Q_{\min,1} $ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\min,2} $ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$Q_{\max,1} $ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\max,2} $ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$b $ 2.37 $K_0 $ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1} $
Quantity Value Quantity Value
$\gamma $ $10\, \mathrm{day}^{-1} $ $a_{\max,1} $ $12.0\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1}\,\mathrm{day}^{-1} $
$K_1 $ $9.0\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $ $K_2 $ $6.5\times10^{-7}\, \mathrm{mol}\,\mathrm{l}^{-1} $
$\mu_{\max,1} $ $0.7\,\mathrm{day}^{-1} $ $\mu_{\max,2} $ $2.2\,\mathrm{day}^{-1} $
$Q_{\min,1} $ $2.6\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\min,2} $ $1.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$Q_{\max,1} $ $9.5\times10^{-14}\, \mathrm{mol}\,\mathrm{cell}^{-1} $ $Q_{\max,2} $ $32.0\times10^{-13}\, \mathrm{mol}\,\mathrm{cell}^{-1} $
$b $ 2.37 $K_0 $ $4.0\times10^{8}\,\mathrm{cells}\,\mathrm{l}^{-1} $
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