July  2020, 40(7): 4427-4451. doi: 10.3934/dcds.2020185

A periodic-parabolic Droop model for two species competition in an unstirred chemostat

1. 

School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

2. 

Department of Mathematics, 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: Feng-Bin Wang

Received  August 2019 Revised  December 2019 Published  April 2020

We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diffusion rate is sufficiently small or large, the sign can be determined. We then show that for the competition model, when diffusion rates for both species are small, there exists a coexistence periodic solution.

Citation: Xiaoqing He, Sze-Bi Hsu, Feng-Bin Wang. A periodic-parabolic Droop model for two species competition in an unstirred chemostat. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185
References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203. 

[2]

A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture, in Mathematics in Microbiology, Academic Press, London, 1983.

[3]

M. R. Droop, Vitamin B12 and marine ecology. Ⅳ. The kinetics of uptake, growth and inhibition in monochrysis lutheri, Journal of the Marine Biological Association of the United Kingdom, 48 (1968), 689-733.  doi: 10.1017/S0025315400019238.

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. 

[5]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.

[6]

J. P. Grover, Dynamics of competition among microalgae in variable environments: Experimental tests of alternative models, Oikos, 62 (1991), 231-243. 

[7]

J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model, American Naturalist, 138 (1991), 811-835.  doi: 10.1086/285254.

[8]

J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theor. Biol., 158 (1992), 409-428. 

[9]

J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, American Naturalist, 173 (2009), E44–E61. doi: 10.1086/595751.

[10]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), E124–E148. doi: 10.1086/662163.

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Vol. 25, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Vol. 840, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.

[13]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Vol. 247, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.

[14]

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

[15]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.

[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.

[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.

[18]

S.-B. HsuJ. 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.

[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.

[20]

S.-B. HsuF.-B. Wang and X.-Q. Zhao, Competition for two essential resources with internal storage and periodic input, Differential Integral Equations, 29 (2016), 601-630. 

[21]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[22]

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.

[23]

J. Mallet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discrete Contin. Dyn. Syst., 8 (2002), 519-562.  doi: 10.3934/dcds.2002.8.519.

[24]

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

[25]

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.

[26]

J. Monod, La technique de culture continue théorie et applications, in Selected Papers in Molecular Biology by Jacques Monod, Academic Press, 1978. doi: 10.1016/B978-0-12-460482-7.50023-3.

[27]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, Journal of Phycology, 23 (1987), 137-150. 

[28]

R. M. Nisbet and W. S. C. Gurney, Modelling fluctuating populations, Acta Appl. Math., 4 (1985).

[29]

S. S. Pilyugin and P. Waltman, Competition in the unstirred chemostat with periodic input and washout, SIAM J. Appl. Math., 59 (1999), 1157-1177.  doi: 10.1137/S0036139997323954.

[30]

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

[31]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Vol. 41, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995.

[32]

H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat, J. Math. Biol., 35 (1997), 545-556.  doi: 10.1007/s002850050065.

[33]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 34 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[34]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Vol. 13, Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[35]

U. Sommer, Comparison between steady state and non-steady state competition: Experiments with natural phytoplankton, Limnology and Oceanography, 30 (1985), 335-346. 

[36]

H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, in Ordered Structures and Applications, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-27842-1_26.

[37]

M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat, Bull. Math. Biol., 71 (2009), 145-161.  doi: 10.1007/s11538-008-9357-7.

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Vol. 16, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, New York, 2013. doi: 10.1007/978-3-319-56433-3.

show all references

References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203. 

[2]

A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture, in Mathematics in Microbiology, Academic Press, London, 1983.

[3]

M. R. Droop, Vitamin B12 and marine ecology. Ⅳ. The kinetics of uptake, growth and inhibition in monochrysis lutheri, Journal of the Marine Biological Association of the United Kingdom, 48 (1968), 689-733.  doi: 10.1017/S0025315400019238.

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. 

[5]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.

[6]

J. P. Grover, Dynamics of competition among microalgae in variable environments: Experimental tests of alternative models, Oikos, 62 (1991), 231-243. 

[7]

J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model, American Naturalist, 138 (1991), 811-835.  doi: 10.1086/285254.

[8]

J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theor. Biol., 158 (1992), 409-428. 

[9]

J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, American Naturalist, 173 (2009), E44–E61. doi: 10.1086/595751.

[10]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), E124–E148. doi: 10.1086/662163.

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Vol. 25, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Vol. 840, Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.

[13]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Vol. 247, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.

[14]

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

[15]

S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.  doi: 10.1137/070700784.

[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.

[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.

[18]

S.-B. HsuJ. 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.

[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.

[20]

S.-B. HsuF.-B. Wang and X.-Q. Zhao, Competition for two essential resources with internal storage and periodic input, Differential Integral Equations, 29 (2016), 601-630. 

[21]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[22]

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.

[23]

J. Mallet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discrete Contin. Dyn. Syst., 8 (2002), 519-562.  doi: 10.3934/dcds.2002.8.519.

[24]

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

[25]

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.

[26]

J. Monod, La technique de culture continue théorie et applications, in Selected Papers in Molecular Biology by Jacques Monod, Academic Press, 1978. doi: 10.1016/B978-0-12-460482-7.50023-3.

[27]

F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, Journal of Phycology, 23 (1987), 137-150. 

[28]

R. M. Nisbet and W. S. C. Gurney, Modelling fluctuating populations, Acta Appl. Math., 4 (1985).

[29]

S. S. Pilyugin and P. Waltman, Competition in the unstirred chemostat with periodic input and washout, SIAM J. Appl. Math., 59 (1999), 1157-1177.  doi: 10.1137/S0036139997323954.

[30]

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

[31]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Vol. 41, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995.

[32]

H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat, J. Math. Biol., 35 (1997), 545-556.  doi: 10.1007/s002850050065.

[33]

H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 34 (1994), 1113-1131.  doi: 10.1137/S0036139993245344.

[34]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Vol. 13, Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[35]

U. Sommer, Comparison between steady state and non-steady state competition: Experiments with natural phytoplankton, Limnology and Oceanography, 30 (1985), 335-346. 

[36]

H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, in Ordered Structures and Applications, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-27842-1_26.

[37]

M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat, Bull. Math. Biol., 71 (2009), 145-161.  doi: 10.1007/s11538-008-9357-7.

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Vol. 16, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, New York, 2013. doi: 10.1007/978-3-319-56433-3.

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