January  2016, 21(1): 313-335. doi: 10.3934/dcdsb.2016.21.313

Dynamics of harmful algae with seasonal temperature variations in the cove-main lake

1. 

Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333

2. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300

3. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715

Received  April 2015 Revised  July 2015 Published  November 2015

In this paper, we investigate two-vessel gradostat models describing the dynamics of harmful algae with seasonal temperature variations, in which one vessel represents a small cove connected to a larger lake. We first define the basic reproduction number for the model system, and then show that the trivial periodic state is globally asymptotically stable, and algae is washed out eventually if the basic reproduction number is less than unity, while there exists at least one positive periodic state and algal blooms occur when it is greater than unity. There are several types of productions for dissolved toxins, related to the algal growth rate, and nutrient limitation, respectively. For the system with a specific toxin production, the global attractivity of positive periodic steady-state solution can be established. Numerical simulations from the basic reproduction number show that the factor of seasonality plays an important role in the persistence of harmful algae.
Citation: Feng-Bin Wang, Sze-Bi Hsu, Wendi Wang. Dynamics of harmful algae with seasonal temperature variations in the cove-main lake. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 313-335. doi: 10.3934/dcdsb.2016.21.313
References:
[1]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis,, Math. Biosci., 38 (1978), 113.  doi: 10.1016/0025-5564(78)90021-4.  Google Scholar

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

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S. Chakraborty, S. Roy and J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model,, Ecol. Model., 213 (2008), 191.  doi: 10.1016/j.ecolmodel.2007.12.008.  Google Scholar

[4]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[5]

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

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E. I. R. Falconer and A. R. Humpage, Cyanobacterial (bluegreen algal) toxins in water supplies: cylindrospermopsins,, Environ. Toxicol., 21 (2006), 299.   Google Scholar

[7]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone,, Math. Biosci., 222 (2009), 42.  doi: 10.1016/j.mbs.2009.08.006.  Google Scholar

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E. Graneĺi and N. Johansson, Increase in the production of allelopathic substances by Prymnesium parvum cells grown under N- or P-deficient conditions,, Harmful Algae, 2 (2003), 135.   Google Scholar

[9]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration,, Journal of Plankton Research, 33 (2011), 211.  doi: 10.1093/plankt/fbq070.  Google Scholar

[10]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423.  doi: 10.1137/0516030.  Google Scholar

[11]

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

[12]

P. R. Hawkins, E. Putt and I. Falconer, et al, Phenotypical variation in a toxic strain of the phytoplankter, Cylindrospermopsis raciborskii (Nostocales, Cyanophyceae) during batch culture,, Environ. Toxicol., 16 (2001), 460.   Google Scholar

[13]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats,, J. Diff. Eqns., 255 (2013), 265.  doi: 10.1016/j.jde.2013.04.006.  Google Scholar

[14]

J. Jiang, On the global stability of cooperative systems,, Bull London Math. Soc., 26 (1994), 455.  doi: 10.1112/blms/26.5.455.  Google Scholar

[15]

N. Johansson and E. Graneĺi, Cell density, chemical composition and toxicity of Chrysochromulina polylepis (Haptophyta) in relation to different N:P supply ratios,, Mar. Biol., 135 (1999), 209.  doi: 10.1007/s002270050618.  Google Scholar

[16]

D. Lekan and C. R. Tomas, The brevetoxin and brevenal composition of three Karenia brevis clones at different salinities and nutrient conditions,, Harmful Algae, 9 (2010), 39.  doi: 10.1016/j.hal.2009.07.004.  Google Scholar

[17]

C. G. R. Maier, M. D. Burch and M. Bormans, Flow management strategies to control blooms of the cyanobacterium, Anabaena circinalis, in the river Murray at Morgan, South Australia,, Regul. Rivers Res. Mgmt., 17 (2001), 637.  doi: 10.1002/rrr.623.  Google Scholar

[18]

S. M. Mitrovic, L. Hardwick and R. Oliver, et. al., Use of flow management to control saxitoxin producing cyanobacterial blooms in the Lower Darling River, Australia,, J. Plankton Res., 33 (2011), 229.   Google Scholar

[19]

C. S. Reynolds, Potamoplankton: Paradigms, Paradoxes and Prognoses, in Algae and the Aquatic Environment,, F. E. Round, (1990).   Google Scholar

[20]

D. L. Roelke, G. M. Gable and T. W. Valenti, Hydraulic flushing as a Prymnesium parvum bloom terminating mechanism in a subtropical lake,, Harmful Algae, 9 (2010), 323.  doi: 10.1016/j.hal.2009.12.003.  Google Scholar

[21]

D. L. Roelke, J. P. Grover and B. W. Brooks et al, A decade of fishkilling Prymnesium parvum blooms in Texas: Roles of inflow and salinity,, J. Plankton Res., 33 (2011), 243.   Google Scholar

[22]

H. L. Smith, Microbial growth in periodic gradostats,, Rocky Mountain J. Math., 20 (1990), 1173.  doi: 10.1216/rmjm/1181073069.  Google Scholar

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr 41, (1995).   Google Scholar

[24]

G. M. Southard, L. T. Fries and A. Barkoh, Prymnesium parvum: the Texas experience,, J. Am. Water Resources Assoc., 46 (2010), 14.  doi: 10.1111/j.1752-1688.2009.00387.x.  Google Scholar

[25]

H. L. Smith and P. E. Waltman, The Theory of the Chemostat,, Cambridge Univ. Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[26]

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

[27]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Differ. Equ., 20 (2008), 699.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[28]

K.F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[29]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications,, Commun. Appl. Nonlinear Anal., 3 (1996), 43.   Google Scholar

[30]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis,, Math. Biosci., 38 (1978), 113.  doi: 10.1016/0025-5564(78)90021-4.  Google Scholar

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[3]

S. Chakraborty, S. Roy and J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model,, Ecol. Model., 213 (2008), 191.  doi: 10.1016/j.ecolmodel.2007.12.008.  Google Scholar

[4]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[5]

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

[6]

E. I. R. Falconer and A. R. Humpage, Cyanobacterial (bluegreen algal) toxins in water supplies: cylindrospermopsins,, Environ. Toxicol., 21 (2006), 299.   Google Scholar

[7]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone,, Math. Biosci., 222 (2009), 42.  doi: 10.1016/j.mbs.2009.08.006.  Google Scholar

[8]

E. Graneĺi and N. Johansson, Increase in the production of allelopathic substances by Prymnesium parvum cells grown under N- or P-deficient conditions,, Harmful Algae, 2 (2003), 135.   Google Scholar

[9]

J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration,, Journal of Plankton Research, 33 (2011), 211.  doi: 10.1093/plankt/fbq070.  Google Scholar

[10]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere,, SIAM J. Math. Anal., 16 (1985), 423.  doi: 10.1137/0516030.  Google Scholar

[11]

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

[12]

P. R. Hawkins, E. Putt and I. Falconer, et al, Phenotypical variation in a toxic strain of the phytoplankter, Cylindrospermopsis raciborskii (Nostocales, Cyanophyceae) during batch culture,, Environ. Toxicol., 16 (2001), 460.   Google Scholar

[13]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats,, J. Diff. Eqns., 255 (2013), 265.  doi: 10.1016/j.jde.2013.04.006.  Google Scholar

[14]

J. Jiang, On the global stability of cooperative systems,, Bull London Math. Soc., 26 (1994), 455.  doi: 10.1112/blms/26.5.455.  Google Scholar

[15]

N. Johansson and E. Graneĺi, Cell density, chemical composition and toxicity of Chrysochromulina polylepis (Haptophyta) in relation to different N:P supply ratios,, Mar. Biol., 135 (1999), 209.  doi: 10.1007/s002270050618.  Google Scholar

[16]

D. Lekan and C. R. Tomas, The brevetoxin and brevenal composition of three Karenia brevis clones at different salinities and nutrient conditions,, Harmful Algae, 9 (2010), 39.  doi: 10.1016/j.hal.2009.07.004.  Google Scholar

[17]

C. G. R. Maier, M. D. Burch and M. Bormans, Flow management strategies to control blooms of the cyanobacterium, Anabaena circinalis, in the river Murray at Morgan, South Australia,, Regul. Rivers Res. Mgmt., 17 (2001), 637.  doi: 10.1002/rrr.623.  Google Scholar

[18]

S. M. Mitrovic, L. Hardwick and R. Oliver, et. al., Use of flow management to control saxitoxin producing cyanobacterial blooms in the Lower Darling River, Australia,, J. Plankton Res., 33 (2011), 229.   Google Scholar

[19]

C. S. Reynolds, Potamoplankton: Paradigms, Paradoxes and Prognoses, in Algae and the Aquatic Environment,, F. E. Round, (1990).   Google Scholar

[20]

D. L. Roelke, G. M. Gable and T. W. Valenti, Hydraulic flushing as a Prymnesium parvum bloom terminating mechanism in a subtropical lake,, Harmful Algae, 9 (2010), 323.  doi: 10.1016/j.hal.2009.12.003.  Google Scholar

[21]

D. L. Roelke, J. P. Grover and B. W. Brooks et al, A decade of fishkilling Prymnesium parvum blooms in Texas: Roles of inflow and salinity,, J. Plankton Res., 33 (2011), 243.   Google Scholar

[22]

H. L. Smith, Microbial growth in periodic gradostats,, Rocky Mountain J. Math., 20 (1990), 1173.  doi: 10.1216/rmjm/1181073069.  Google Scholar

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr 41, (1995).   Google Scholar

[24]

G. M. Southard, L. T. Fries and A. Barkoh, Prymnesium parvum: the Texas experience,, J. Am. Water Resources Assoc., 46 (2010), 14.  doi: 10.1111/j.1752-1688.2009.00387.x.  Google Scholar

[25]

H. L. Smith and P. E. Waltman, The Theory of the Chemostat,, Cambridge Univ. Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[26]

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

[27]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Differ. Equ., 20 (2008), 699.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[28]

K.F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[29]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications,, Commun. Appl. Nonlinear Anal., 3 (1996), 43.   Google Scholar

[30]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar

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