# American Institute of Mathematical Sciences

## Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics

 a. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China b. School of Computer Science and Network Security, Dongguan University of Technology, Dongguan, 523808, Guangdong, China c. College of Finance and Statistics, Hunan University, Changsha, 410079, Hunan, China d. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada

* Corresponding author: Lin Wang

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: The work of DZ was partially supported by the National Natural Science Foundation of China (No. 11501193) and the China Post Doctorial Fund (No. 2015M582335). LW was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC)

An inhibitory uptake function is incorporated into the discrete, size-structured nonlinear chemostat model developed by Arino et al. (Journal of Mathematical Biology, 45(2002)). Different from the model with a monotonically increasing uptake function, we show that the inhibitory kinetics can induce very complex dynamics including stable equilibria, cycles and chaos (via the period-doubling cascade). In particular, when the nutrient concentration in the input feed to the chemostat $S^0$ is larger than the upper break-even concentration value $\mu$, the model exhibits three types of bistability allowing a stable equilibrium to coexist with another stable equilibrium, or a stable cycle or a chaotic attractor.

Citation: Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018327
##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng., 10 (1968), 707-723. [2] J. Arino, J.-L. Gouze and A. Sciandra, A discrete, size-structured model of phytoplankton growth in the chemostat, J. Math. Biol., 45 (2002), 313-336. doi: 10.1007/s002850200160. [3] R. A. Armstrong and R. McGehee, Competitive exclusion, Am. Nat., 115 (1980), 151-170. doi: 10.1086/283553. [4] L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229. [5] B. Boon and H. Laudeuout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447. [6] A. W. Bush and A. E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theor. Biol., 63 (1976), 385-395. [7] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient outake, SIAM J. Appl. Math., 45 (1985), 138-151. doi: 10.1137/0145006. [8] E. P. Cohen and H. Eagle, A simplified chemostat for the growth of mammalian cells: characteristics of cell growth in continuous culture, J. Exp. Med., 113 (1961), 467-474. [9] J. M. Cushing, A competition model for size-structured species, SIAM J. Appl. Math., 49 (1989), 838-858. doi: 10.1137/0149049. [10] J. M. Cushing, An Introduction to Structured Population Dynamics, Reginal Conference Series in Applied Mathematics 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [11] D. E. Dykhuizen and A. M. Dean, Evolution of specialists in an experimental microcosm, Genetics, 167 (2005), 2015-2026. [12] T. B. K. Gage, F. M. Williams and J. B. Horton, Division synchrony and the dynamics of microbial populations: A size-specific model, Theor. Pop. Bio., 26 (1984), 296-314. doi: 10.1016/0040-5809(84)90035-2. [13] M. Golubitsky, E. B. Keeler and M. Rothschild, Convergence of the age-structure: Applications of the projective metric, Theor. Pop. Bio., 7 (1975), 84-93. doi: 10.1016/0040-5809(75)90007-6. [14] S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493. [15] S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in countinuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030. [16] L. Jones and S. P. Ellner, Effects of rapid prey evolution on predator-prey cycles, J. Math. Biol., 55 (2007), 541-573. doi: 10.1007/s00285-007-0094-6. [17] J. L. Jost, J. F. Drake, A. G. Fredrickson and H. M. Tsuchiya, Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandu and glucose in a minimal medium, J. Bacteriol., 113 (1973), 834-841. [18] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer-Verlag, New York, 1986. doi: 10.1007/978-3-662-13159-6. [19] S. Pavlou and I. G. Kevrekidis, Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies, Math. Biosci., 108 (1992), 1-55. doi: 10.1016/0025-5564(92)90002-E. [20] E. Senior, A. T. Bull and J. H. Slater, Enzyme evolution in a microbial community growing on the herbicide Dalapon, Nature, 263 (1976), 476-479. [21] H. L. Smith, A discrete, size-structured model of microbial growth and competition in the chemostat, J. Math. Biol., 34 (1996), 734-754. doi: 10.1007/BF00161517. [22] [23] H. L. Smith and X.-Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 183-191. doi: 10.3934/dcdsb.2001.1.183. [24] L. Wang and G. S. K. Wolkowicz, A delayed chemostat model with general delayed response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452-468. doi: 10.1016/j.jmaa.2005.08.014. [25] H. A. Wichman, J. Millstein and J. J. Bull, Adaptive molecular evolution for 13,000 phage generations: A possible arms race, Genetics, 170 (2005), 19-31. [26] L. M. Wick, H. Weilenmann and T. Egli, The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics, Microbiology (Reading, Engl.), 148 (2002), 2889-2902. [27] G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233. doi: 10.1137/0152012. [28] J. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885. doi: 10.1137/050627514. [29] H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillations induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530. doi: 10.1007/s00285-004-0311-5.

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##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng., 10 (1968), 707-723. [2] J. Arino, J.-L. Gouze and A. Sciandra, A discrete, size-structured model of phytoplankton growth in the chemostat, J. Math. Biol., 45 (2002), 313-336. doi: 10.1007/s002850200160. [3] R. A. Armstrong and R. McGehee, Competitive exclusion, Am. Nat., 115 (1980), 151-170. doi: 10.1086/283553. [4] L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt, Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229. [5] B. Boon and H. Laudeuout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447. [6] A. W. Bush and A. E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theor. Biol., 63 (1976), 385-395. [7] G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient outake, SIAM J. Appl. Math., 45 (1985), 138-151. doi: 10.1137/0145006. [8] E. P. Cohen and H. Eagle, A simplified chemostat for the growth of mammalian cells: characteristics of cell growth in continuous culture, J. Exp. Med., 113 (1961), 467-474. [9] J. M. Cushing, A competition model for size-structured species, SIAM J. Appl. Math., 49 (1989), 838-858. doi: 10.1137/0149049. [10] J. M. Cushing, An Introduction to Structured Population Dynamics, Reginal Conference Series in Applied Mathematics 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [11] D. E. Dykhuizen and A. M. Dean, Evolution of specialists in an experimental microcosm, Genetics, 167 (2005), 2015-2026. [12] T. B. K. Gage, F. M. Williams and J. B. Horton, Division synchrony and the dynamics of microbial populations: A size-specific model, Theor. Pop. Bio., 26 (1984), 296-314. doi: 10.1016/0040-5809(84)90035-2. [13] M. Golubitsky, E. B. Keeler and M. Rothschild, Convergence of the age-structure: Applications of the projective metric, Theor. Pop. Bio., 7 (1975), 84-93. doi: 10.1016/0040-5809(75)90007-6. [14] S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493. [15] S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in countinuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030. [16] L. Jones and S. P. Ellner, Effects of rapid prey evolution on predator-prey cycles, J. Math. Biol., 55 (2007), 541-573. doi: 10.1007/s00285-007-0094-6. [17] J. L. Jost, J. F. Drake, A. G. Fredrickson and H. M. Tsuchiya, Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandu and glucose in a minimal medium, J. Bacteriol., 113 (1973), 834-841. [18] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer-Verlag, New York, 1986. doi: 10.1007/978-3-662-13159-6. [19] S. Pavlou and I. G. Kevrekidis, Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies, Math. Biosci., 108 (1992), 1-55. doi: 10.1016/0025-5564(92)90002-E. [20] E. Senior, A. T. Bull and J. H. Slater, Enzyme evolution in a microbial community growing on the herbicide Dalapon, Nature, 263 (1976), 476-479. [21] H. L. Smith, A discrete, size-structured model of microbial growth and competition in the chemostat, J. Math. Biol., 34 (1996), 734-754. doi: 10.1007/BF00161517. [22] [23] H. L. Smith and X.-Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 183-191. doi: 10.3934/dcdsb.2001.1.183. [24] L. Wang and G. S. K. Wolkowicz, A delayed chemostat model with general delayed response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452-468. doi: 10.1016/j.jmaa.2005.08.014. [25] H. A. Wichman, J. Millstein and J. J. Bull, Adaptive molecular evolution for 13,000 phage generations: A possible arms race, Genetics, 170 (2005), 19-31. [26] L. M. Wick, H. Weilenmann and T. Egli, The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics, Microbiology (Reading, Engl.), 148 (2002), 2889-2902. [27] G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233. doi: 10.1137/0152012. [28] J. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885. doi: 10.1137/050627514. [29] H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillations induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530. doi: 10.1007/s00285-004-0311-5.
Bifurcation diagram of the limiting system (7). Here $f(s) = \frac{as}{1+0.1s+0.01s^2}, E = 0.1, S^{0} = 100, U_{0} = 70$ with $a\in[0.049, 0.059]$ Thus $aS^{0}>1$ and $S^{0}> \mu$. A cascade of period-doublings to chaos occurs as $a$ increases
Left: a numerical solution of the limiting system (7) with $U_0 = 60$; Right: a numerical solution of system (11) with $(U_0, S_0) = (60,6)$. Here $f(s) = \frac{0.05s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$
Numerical solutions of system (11) with $f(s) = \frac{0.05s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to E_0 = (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t) \to E_1 = (S^0-\lambda, \lambda)$ as $t\to\infty$, initial condition was $(U_0, S_0) = (80,6)$
Numerical solutions of system (11) with $f(s) = \frac{0.054s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t)$ approaches a stable $2-cycle$, initial condition was $(U_0, S_0) = (80,6)$
Numerical solutions of system (11) with $f(s) = \frac{0.059s}{1+0.1s+0.01s^2}$, $E = 0.1$, $S^{0} = 100$. Left: $(U_t,S_t)\to (0, S^0)$ as $t\to\infty$, initial condition $(U_0, S_0) = (10,6)$ was used; Right: $(U_t, S_t)$ approaches a chaotic attractor, initial condition was $(U_0, S_0) = (80,6)$
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