April  2022, 21(4): 1361-1384. doi: 10.3934/cpaa.2022022

Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth

1. 

College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

2. 

State Key Laboratory of Desert and Oasis Ecology, Xinjiang Institute of Ecology and Geography, Chinese Academy of Sciences, Urumqi 830011, China

* Corresponding author

Received  September 2021 Revised  December 2021 Published  April 2022 Early access  January 2022

Fund Project: This work was supported by the Natural Science Foundation of China (Grant No. 11771373, 11861065)

In this paper, the periodic solution and extinction in a periodic chemostat model with delay in microorganism growth are investigated. The positivity and ultimate boundedness of solutions are firstly obtained. Next, the necessary and sufficient conditions on the existence of positive $ \omega $-periodic solutions are established by constructing Poincaré map and using the Whyburn Lemma and Leray-Schauder degree theory. Furthermore, according to the implicit function theorem, the uniqueness of the positive periodic solution is obtained when delay $ \tau $ is small enough. Finally, the necessary and sufficient conditions for the extinction of microorganism species are established.

Citation: Ningning Ye, Zengyun Hu, Zhidong Teng. Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1361-1384. doi: 10.3934/cpaa.2022022
References:
[1]

P. AmsterG. Robledo and D. Sepúlveda, Existence of $\omega$-periodic solutions for a delayed chemostat with periodic inputs, Nonlinear Anal., Real World Appl., 55 (2020), 103134.  doi: 10.1016/j.nonrwa.2020.103134.

[2]

P. AmsterG. Robledo and D. Sepúlveda, Dynamics of a chemostat with periodic nutrient supply and delay in the growth, Nonlinearity, 33 (2020), 5839-5860.  doi: 10.1088/1361-6544/ab9bab.

[3]

P. Amster, Topological Methods in the Study of Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-8893-4.

[4]

E. Beretta and Y. Kuang, Global stability in a well known delayed chemostat model, Comm. Appl. Anal., 4 (2000), 147-155. 

[5]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Birkh$\ddot{a}$user, Boston, 2014. doi: 10.1007/978-3-319-11794-2.

[6]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[7] P. M. Doran, Bioprocess Engineering Principles, Academic Press, Kidlington, 2013. 
[8]

X. He and S. Ruan, Global stsblity in chemostat-type plankon models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.  doi: 10.1007/s002850050128.

[9]

S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.  doi: 10.3934/cpaa.2011.10.1479.

[10]

Y. Li, Periodic solutions of non-autonomous cellular neural networks with impulses and delays on time scales, IMA J. Math. Control Inform., 31 (2014), 273-293.  doi: 10.1093/imamci/dnt012.

[11]

S. Liu, Bioprocess Engineering: Kinetics, Sustainability, and Reactor Design, Elsevier, Kidlington, 2017.

[12]

F. MazencM. Malisoff and P. D. Leenheer, On the stability of periodic solutions in the perturbed chemostat, Math. Biosci. Eng., 4 (2007), 319-338.  doi: 10.3934/mbe.2007.4.319.

[13]

F. MazencS. Niculescu and G. Robledo, Stability analysis of mathematical model of competition in a chain of chemostats in series with delay, Appl. Math. Model., 76 (2019), 311-329.  doi: 10.1016/j.apm.2019.06.006.

[14]

H. Nie and F.-B. Wang, Competition for one nutrient with recycling and allelopathy in an unstirred chemostat, Discrete Contin. Dyn. Syst - B, 20 (2015), 2129-2155.  doi: 10.3934/dcdsb.2015.20.2129.

[15]

H. NieW. Xie and J. Wu, Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor, Commun. Pure Appl. Anal., 12 (2013), 1279-1297.  doi: 10.3934/cpaa.2013.12.1279.

[16]

Q. Peng and H. I. Freedman, Global attractivity in a periodic chemostat with general uptake functions, J. Math. Anal. Appl., 249 (2000), 300-323.  doi: 10.1006/jmaa.2000.6757.

[17]

W. PuD. JiangY. Wang and Z. Bai, Spatial dynamics of a nonlocal delayed unstirred chemostat model with periodic input, Int. J. Biomath., 12 (2019), 1950065.  doi: 10.1142/S1793524519500657.

[18]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete Contin. Dyn. Syst. B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

[19]

M. Rehim and Z. Teng, Permanence, average persistence and exctinction in nonautonomous single-species growth chemostat models, Adv. Complex Syst., 9 (2006), 41-58.  doi: 10.1142/S0219525906000616.

[20]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different rates and variable yields, Math. Biosci. Eng., 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.

[21] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[22]

Z. TengZ. Li and H. Jiang, Permanence criteria in non-autonomous predator-prey Kolmogorov systems and its applications, Dyn. Syst., 19 (2004), 171-194.  doi: 10.1080/14689360410001698851.

[23]

Z. Teng and Z. Li, Permanence and asymptotic behavior of the N-Species nonautonomous Lotka-Volterra competitive systems, Comput. Math. Appl., 39 (2000), 107-116.  doi: 10.1016/S0898-1221(00)00069-9.

[24]

P. Waltman, Competition Models in Population Biology, SIAM, Philadelphia, 1983. doi: 10.1137/1.9781611970258.

[25]

L. WangD. Jiang and D. $O^{'}$Regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Commun. Nonlinear Sci. Numer. Simulat., 37 (2016), 1-13.  doi: 10.1016/j.cnsns.2016.01.002.

[26]

G. S. K. Wolkowicz and X.-Q. Zhao, N-Species competition in a periodic chemostat, Diff. Integral Equat., 11 (1998), 465-491.

[27]

S. Yuan and M. Han, Bifurcation analysis of a chemostat model with two distributed delays, Chaos Solitons Fractals, 20 (2004), 995–1004. doi: 10.1016/j.chaos.2003.09.048.

[28]

S. Yuan and T. Zhang, Dynamics of a plasmid chemostat model with periodic nutrient input and delayed nutrient recycling, Nonlinear Anal., Real World Appl., 13 (2012), 2104-2119.  doi: 10.1016/j.nonrwa.2012.01.006.

show all references

References:
[1]

P. AmsterG. Robledo and D. Sepúlveda, Existence of $\omega$-periodic solutions for a delayed chemostat with periodic inputs, Nonlinear Anal., Real World Appl., 55 (2020), 103134.  doi: 10.1016/j.nonrwa.2020.103134.

[2]

P. AmsterG. Robledo and D. Sepúlveda, Dynamics of a chemostat with periodic nutrient supply and delay in the growth, Nonlinearity, 33 (2020), 5839-5860.  doi: 10.1088/1361-6544/ab9bab.

[3]

P. Amster, Topological Methods in the Study of Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-8893-4.

[4]

E. Beretta and Y. Kuang, Global stability in a well known delayed chemostat model, Comm. Appl. Anal., 4 (2000), 147-155. 

[5]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Birkh$\ddot{a}$user, Boston, 2014. doi: 10.1007/978-3-319-11794-2.

[6]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[7] P. M. Doran, Bioprocess Engineering Principles, Academic Press, Kidlington, 2013. 
[8]

X. He and S. Ruan, Global stsblity in chemostat-type plankon models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.  doi: 10.1007/s002850050128.

[9]

S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.  doi: 10.3934/cpaa.2011.10.1479.

[10]

Y. Li, Periodic solutions of non-autonomous cellular neural networks with impulses and delays on time scales, IMA J. Math. Control Inform., 31 (2014), 273-293.  doi: 10.1093/imamci/dnt012.

[11]

S. Liu, Bioprocess Engineering: Kinetics, Sustainability, and Reactor Design, Elsevier, Kidlington, 2017.

[12]

F. MazencM. Malisoff and P. D. Leenheer, On the stability of periodic solutions in the perturbed chemostat, Math. Biosci. Eng., 4 (2007), 319-338.  doi: 10.3934/mbe.2007.4.319.

[13]

F. MazencS. Niculescu and G. Robledo, Stability analysis of mathematical model of competition in a chain of chemostats in series with delay, Appl. Math. Model., 76 (2019), 311-329.  doi: 10.1016/j.apm.2019.06.006.

[14]

H. Nie and F.-B. Wang, Competition for one nutrient with recycling and allelopathy in an unstirred chemostat, Discrete Contin. Dyn. Syst - B, 20 (2015), 2129-2155.  doi: 10.3934/dcdsb.2015.20.2129.

[15]

H. NieW. Xie and J. Wu, Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor, Commun. Pure Appl. Anal., 12 (2013), 1279-1297.  doi: 10.3934/cpaa.2013.12.1279.

[16]

Q. Peng and H. I. Freedman, Global attractivity in a periodic chemostat with general uptake functions, J. Math. Anal. Appl., 249 (2000), 300-323.  doi: 10.1006/jmaa.2000.6757.

[17]

W. PuD. JiangY. Wang and Z. Bai, Spatial dynamics of a nonlocal delayed unstirred chemostat model with periodic input, Int. J. Biomath., 12 (2019), 1950065.  doi: 10.1142/S1793524519500657.

[18]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete Contin. Dyn. Syst. B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

[19]

M. Rehim and Z. Teng, Permanence, average persistence and exctinction in nonautonomous single-species growth chemostat models, Adv. Complex Syst., 9 (2006), 41-58.  doi: 10.1142/S0219525906000616.

[20]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different rates and variable yields, Math. Biosci. Eng., 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.

[21] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[22]

Z. TengZ. Li and H. Jiang, Permanence criteria in non-autonomous predator-prey Kolmogorov systems and its applications, Dyn. Syst., 19 (2004), 171-194.  doi: 10.1080/14689360410001698851.

[23]

Z. Teng and Z. Li, Permanence and asymptotic behavior of the N-Species nonautonomous Lotka-Volterra competitive systems, Comput. Math. Appl., 39 (2000), 107-116.  doi: 10.1016/S0898-1221(00)00069-9.

[24]

P. Waltman, Competition Models in Population Biology, SIAM, Philadelphia, 1983. doi: 10.1137/1.9781611970258.

[25]

L. WangD. Jiang and D. $O^{'}$Regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Commun. Nonlinear Sci. Numer. Simulat., 37 (2016), 1-13.  doi: 10.1016/j.cnsns.2016.01.002.

[26]

G. S. K. Wolkowicz and X.-Q. Zhao, N-Species competition in a periodic chemostat, Diff. Integral Equat., 11 (1998), 465-491.

[27]

S. Yuan and M. Han, Bifurcation analysis of a chemostat model with two distributed delays, Chaos Solitons Fractals, 20 (2004), 995–1004. doi: 10.1016/j.chaos.2003.09.048.

[28]

S. Yuan and T. Zhang, Dynamics of a plasmid chemostat model with periodic nutrient input and delayed nutrient recycling, Nonlinear Anal., Real World Appl., 13 (2012), 2104-2119.  doi: 10.1016/j.nonrwa.2012.01.006.

Figure 1.  (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solution of model (1.2) with initial function $ (u(\theta),v(\theta)) = (6,7) $ for all $ \theta\in[-0.15,0] $ converges to a positive periodic solution as $ t\rightarrow \infty $
Figure 2.  (a)-(b): the solutions of model (1.2) with the delay $ \tau = 0 $ and initial points $ (3.5,4.3) $, $ (1.5,1.7) $ and $ (0.5,0.7) $ converge to the unique positive periodic solution $ (u_0^*(t),v_0^*(t)) $ as $ t\rightarrow \infty $; (c): the numerical simulation of $ w^*(t) $; (d)-(e): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((2.1,2.5), (1.5,1.7),(1.1,1.3)) $ for all $ \theta\in[-0.2,0] $ converge to the unique positive periodic solution $ (u^*(t),v^*(t)) $ as $ t\rightarrow \infty $
Figure 3.  (a)-(b): the solutions of model (1.2) with the delay $ \tau = 0 $ and initial points $ (1.1,1.3) $, $ (1.5,1.7) $ and $ (0.5,0.8) $ converge to the unique positive periodic solution $ (u_0^*(t),v_0^*(t)) $ as $ t\rightarrow \infty $; (c): the numerical simulation of $ w^*(t) $; (d)-(e): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((1,2),(1.1,1.5),(1.4,1.9)) $ for all $ \theta\in[-0.1,0] $ converge to the unique positive periodic solution $ (u^*(t),v^*(t)) $ as $ t\rightarrow \infty $
Figure 4.  (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((2.8,1.7),(1.4,2.5)) $ for all $ \theta\in[-0.2,0] $ converge to $ (w^*(t),0) $ as $ t\to\infty $
Figure 5.  (a): the numerical simulation of $ w^*(t) $; (b)-(c): the solutions of model (1.2) with initial functions $ (u(\theta),v(\theta)) = ((0.8,1),(0.7,2),(1,0.6)) $ for all $ \theta\in[-0.3,0] $ converge to $ (w^*(t),0) $ as $ t\to\infty $
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