doi: 10.3934/dcdsb.2021208
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An analysis approach to permanence of a delay differential equations model of microorganism flocculation

1. 

School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China

2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

* Corresponding author

Received  February 2021 Revised  July 2021 Early access August 2021

Fund Project: This work is supported in part by the National Natural Science Foundation of China (Nos. 11901027, 11871093 and 11971055), the Scientific Research Project of Beijing Municipal Education Commission (No. KM201910016001), the Pyramid Talent Training Project of BUCEA (JDYC20200327) and the Bill & Melinda Gates Foundation (INV-005834)

In this paper, we develop a delay differential equations model of microorganism flocculation with general monotonic functional responses, and then study the permanence of this model, which can ensure the sustainability of the collection of microorganisms. For a general differential system, the existence of a positive equilibrium can be obtained with the help of the persistence theory, whereas we give the existence conditions of a positive equilibrium by using the implicit function theorem. Then to obtain an explicit formula for the ultimate lower bound of microorganism concentration, we propose a general analysis method, which is different from the traditional approaches in persistence theory and also extends the analysis techniques of existing related works.

Citation: Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021208
References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326.  doi: 10.1016/S0022-5193(89)80211-5.

[2]

J. R. Beddington, Mutual interference between parasites or predator and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[3]

A. W. Bush and A. E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol., 63 (1976), 385-395.  doi: 10.1016/0022-5193(76)90041-2.

[4]

J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845.

[5]

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

[6]

T. Chatsungnoen and Y. Chisti, Harvesting microalgae by flocculation-sedimentation, Algal Res., 13 (2016), 271-283.  doi: 10.1016/j.algal.2015.12.009.

[7]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.  doi: 10.2307/1467324.

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. 

[9]

D. M. Di Toro, D. J. O'Connor and R. V. Thomann, A dynamic model of the phytoplankton population in the Sacramento–San Joaquin Delta, in Nonequilibrium Systems in Natural Water Chemistry (ed. J. D. Hem), Adv. Chem. Series, No. 106, American Chemical Society, Washington, (1971), 131–180.

[10]

O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, et al., Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[11]

Q. Dong and W. Ma, Qualitative analysis of the chemostat model with variable yield and a time delay, J. Math. Chem., 51 (2013), 1274-1292.  doi: 10.1007/s10910-013-0144-9.

[12]

S. F. Ellermeyer, Competition in the chemostat: Global asymptotic behavior of a model with delayed response in growth, SIAM J. Appl. Math., 54 (1994), 456-465.  doi: 10.1137/S003613999222522X.

[13]

S. Guo and W. Ma, Global behavior of delay differential equations model of HIV infection with apoptosis, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 103-119.  doi: 10.3934/dcdsb.2016.21.103.

[14]

S. Guo and W. Ma, Global dynamics of a microorganism flocculation model with time delay, Commun. Pure Appl. Anal., 16 (2017), 1883-1891.  doi: 10.3934/cpaa.2017091.

[15]

S. GuoW. Ma and X.-Q. Zhao, Global dynamics of a time-delayed microorganism flocculation model with saturated functional responses, J. Dynam. Differential Equations, 30 (2018), 1247-1271.  doi: 10.1007/s10884-017-9605-3.

[16]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[17]

M. P. Hassell and G. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.  doi: 10.1038/2231133a0.

[18]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[19]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[20]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.

[21] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961. 
[22]

R. E. Kooij and A. Zegeling, A predator-prey model with Ivlev's functional response, J. Math. Anal. Appl., 198 (1996), 473-489.  doi: 10.1006/jmaa.1996.0093.

[23]

Y. Kuang, Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.  doi: 10.1137/0149107.

[24] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. 
[25]

B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.

[26]

B. LiY. Kuang and H. L. Smith, Competition between plasmid-bearing and plasmid-free microorganisms in a chemostat with distinct removal rates, Canad. Appl. Math. Quart., 7 (1999), 251-281. 

[27]

Z. Li and R. Xu, Stability analysis of a ratio-dependent chemostat model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243-253.  doi: 10.1142/S1793524510000921.

[28]

C. Liu, Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation, Appl. Math. Model., 37 (2013), 6899-6908.  doi: 10.1016/j.apm.2013.02.021.

[29]

A. J. Lotka, Elements of Mathematical Biology, Dover Publications, New York, 1956.

[30]

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.

[31] H. L. Smith and P. Waltman, Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[32]

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.

[33]

K. Song, W. Ma and S. Guo, et al., A class of dynamic model describing microbial flocculant with nutrient competition and metabolic products in wastewater treatment, Adv. Differ. Equ., 2018 (2018), Paper No. 33, 14 pp. doi: 10.1186/s13662-018-1473-6.

[34] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. 
[35]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[37]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Accad. Naz. dei Lincei, 2 (1926), 31-113. 

[38]

W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.  doi: 10.1016/S0893-9659(01)00153-7.

[39]

W. Wang, W. Ma and H. Yan, Global dynamics of modeling flocculation of microorganism, Appl. Sci., 6 (2016), 221. doi: 10.3390/app6080221.

[40]

G. S. K. WolkowiczH. Xia and S. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Math. Anal., 57 (1997), 1281-1310.  doi: 10.1137/S0036139995289842.

[41]

H. XiaG. 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.

[42]

T. Zhang, N. Gao and T. Wang, et al., Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants, Math. Biosci. Eng., 17 (2020), 179-201. doi: 10.3934/mbe.2020010.

[43]

T. Zhang, W. Ma and X. Meng, Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input, Adv. Difference Equ., 2017 (2017), Paper No. 115, 17 pp. doi: 10.1186/s13662-017-1163-9.

[44]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ ed., Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

show all references

References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326.  doi: 10.1016/S0022-5193(89)80211-5.

[2]

J. R. Beddington, Mutual interference between parasites or predator and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[3]

A. W. Bush and A. E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol., 63 (1976), 385-395.  doi: 10.1016/0022-5193(76)90041-2.

[4]

J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845.

[5]

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

[6]

T. Chatsungnoen and Y. Chisti, Harvesting microalgae by flocculation-sedimentation, Algal Res., 13 (2016), 271-283.  doi: 10.1016/j.algal.2015.12.009.

[7]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.  doi: 10.2307/1467324.

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. 

[9]

D. M. Di Toro, D. J. O'Connor and R. V. Thomann, A dynamic model of the phytoplankton population in the Sacramento–San Joaquin Delta, in Nonequilibrium Systems in Natural Water Chemistry (ed. J. D. Hem), Adv. Chem. Series, No. 106, American Chemical Society, Washington, (1971), 131–180.

[10]

O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, et al., Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[11]

Q. Dong and W. Ma, Qualitative analysis of the chemostat model with variable yield and a time delay, J. Math. Chem., 51 (2013), 1274-1292.  doi: 10.1007/s10910-013-0144-9.

[12]

S. F. Ellermeyer, Competition in the chemostat: Global asymptotic behavior of a model with delayed response in growth, SIAM J. Appl. Math., 54 (1994), 456-465.  doi: 10.1137/S003613999222522X.

[13]

S. Guo and W. Ma, Global behavior of delay differential equations model of HIV infection with apoptosis, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 103-119.  doi: 10.3934/dcdsb.2016.21.103.

[14]

S. Guo and W. Ma, Global dynamics of a microorganism flocculation model with time delay, Commun. Pure Appl. Anal., 16 (2017), 1883-1891.  doi: 10.3934/cpaa.2017091.

[15]

S. GuoW. Ma and X.-Q. Zhao, Global dynamics of a time-delayed microorganism flocculation model with saturated functional responses, J. Dynam. Differential Equations, 30 (2018), 1247-1271.  doi: 10.1007/s10884-017-9605-3.

[16]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[17]

M. P. Hassell and G. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.  doi: 10.1038/2231133a0.

[18]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[19]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.

[20]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.

[21] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961. 
[22]

R. E. Kooij and A. Zegeling, A predator-prey model with Ivlev's functional response, J. Math. Anal. Appl., 198 (1996), 473-489.  doi: 10.1006/jmaa.1996.0093.

[23]

Y. Kuang, Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.  doi: 10.1137/0149107.

[24] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. 
[25]

B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.

[26]

B. LiY. Kuang and H. L. Smith, Competition between plasmid-bearing and plasmid-free microorganisms in a chemostat with distinct removal rates, Canad. Appl. Math. Quart., 7 (1999), 251-281. 

[27]

Z. Li and R. Xu, Stability analysis of a ratio-dependent chemostat model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243-253.  doi: 10.1142/S1793524510000921.

[28]

C. Liu, Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation, Appl. Math. Model., 37 (2013), 6899-6908.  doi: 10.1016/j.apm.2013.02.021.

[29]

A. J. Lotka, Elements of Mathematical Biology, Dover Publications, New York, 1956.

[30]

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.

[31] H. L. Smith and P. Waltman, Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[32]

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.

[33]

K. Song, W. Ma and S. Guo, et al., A class of dynamic model describing microbial flocculant with nutrient competition and metabolic products in wastewater treatment, Adv. Differ. Equ., 2018 (2018), Paper No. 33, 14 pp. doi: 10.1186/s13662-018-1473-6.

[34] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. 
[35]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[37]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Accad. Naz. dei Lincei, 2 (1926), 31-113. 

[38]

W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.  doi: 10.1016/S0893-9659(01)00153-7.

[39]

W. Wang, W. Ma and H. Yan, Global dynamics of modeling flocculation of microorganism, Appl. Sci., 6 (2016), 221. doi: 10.3390/app6080221.

[40]

G. S. K. WolkowiczH. Xia and S. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Math. Anal., 57 (1997), 1281-1310.  doi: 10.1137/S0036139995289842.

[41]

H. XiaG. 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.

[42]

T. Zhang, N. Gao and T. Wang, et al., Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants, Math. Biosci. Eng., 17 (2020), 179-201. doi: 10.3934/mbe.2020010.

[43]

T. Zhang, W. Ma and X. Meng, Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input, Adv. Difference Equ., 2017 (2017), Paper No. 115, 17 pp. doi: 10.1186/s13662-017-1163-9.

[44]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ ed., Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

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