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October  2017, 14(5&6): 1091-1117. doi: 10.3934/mbe.2017057

## Effect of seasonal changing temperature on the growth of phytoplankton

 1 School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, China 2 Department of Mathematics, Dalian Maritime University, 1 Linghai Road, Dalian, Liaoning, 116026, China 3 School of Urban and Environmental Sciences, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, China 4 Lamps and Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada

* Corresponding author: Meng Fan

Received  March 2016 Accepted  October 2016 Published  May 2017

Fund Project: Supported by NSFC-11671072 & 11271065, RFDP-20130043110001, RFCP-CMSP, and FRFCU-14ZZ1309

An non-autonomous nutrient-phytoplankton interacting model incorporating the effect of time-varying temperature is established. The impacts of temperature on metabolism of phytoplankton such as nutrient uptake, death rate, and nutrient releasing from particulate nutrient are investigated. The ecological reproductive index is formulated to present a threshold criteria and to characterize the dynamics of phytoplankton. The positive invariance, dissipativity, and the existence and stability of boundary and positive periodic solution are established. The analyses rely on the comparison principle, the coincidence degree theory and Lyapunov direct method. The effect of seasonal temperature and daily temperature on phytoplankton biomass are simulated numerically. Numerical simulation shows that the phytoplankton biomass is very robust to the variation of water temperature. The dynamics of the model and model predictions agree with the experimental data. Our model and analysis provide a possible explanation of triggering mechanism of phytoplankton blooms.

Citation: Ming Chen, Meng Fan, Xing Yuan, Huaiping Zhu. Effect of seasonal changing temperature on the growth of phytoplankton. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1091-1117. doi: 10.3934/mbe.2017057
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##### References:
Seasonal patterns of water temperature at monitor sites $T_i(i = 1,2,...,12)$ from March 2008 to April 2009
Seasonal patterns of Chlorophyll-a at monitor sites $T_i(i = 1,2,...,12)$ from June 2008 to April 2009
Location of Lake Tai and monitor stations. The longitude and latitude of each monitor station are listed in Table 1
Thermal growth rate curve and CTMI model calibrated for data of A. Formosa. (Reproduced from [4])
Time-series plot of phytoplankton (solid line in red) and limiting nutrient (dash line in black) in (6). It reveals that the conditions in Theorem 3.7 and Theorem 3.11 are sufficient for the cyclic dynamics and those two theorems can be further improved. (a) $R_0<1$ and the boundary $\omega$-periodic solution is GAS (Theorems 3.7 is valid), here $T(t)=5-5\sin(\pi/6t+8)$, $P_{in1}=0.05$. (b) $R_0>1$ and the internal $\omega$-periodic solution is GAS (Theorem 3.11 holds) here $T(t)=15-15\sin(8\pi t+8)$, $P_{in1}=5$, $\mu_{opt}=5$. (c) The conditions in Theorem 3.11 are not satisfied (i.e., $R_1<1<R_0$), but (6) still admits a positive periodic solution being GAS, here $T(t)=15-15\sin(\pi/6t+8)$, $P_{in1}=0.5$. All the other parameters take the default values listed in Table 2
Bifurcation diagram for (6) with $R_0$ being the bifurcation parameter. $R_0=1$ is the threshold that characterizes the global dynamics of (6), that is, the boundary $\omega$-periodic solution is GAS (phytoplankton tends to perish) if $R_0<1$ and the internal $\omega$-periodic solution is GAS (phytoplankton persists) if $R_0>1$
(a) Time-series plot of water temperature $T=T(t)$. (b) Time-series plot of phytoplankton ($A(t)$) (solid line in red) and limiting nutrient ($P(t)$) (dotted line in black) in the reference model (6). It reveals that the annual cycle of temperature is the primary driver of the biomass variability. Here $T=T(t)=15-15\sin(\pi/6t+8)$ and other parameters take the default values in Table 2
Solid lines show the bifurcation diagram of the minimum and maximum of periodic oscillation of phytoplankton in (6) with respect to the strength or intensity ($\alpha$) of external temperature forcing, which reveals a strong correlation between the water temperature and the cyclic dynamics of phytoplankton. Dashed line represents the bifurcation diagram for the amplitude (distance between two solid lines) of phytoplankton's periodic oscillation with respect to $\alpha$. Here $T(t)=15-\alpha\sin(\pi/6\cdot t+8)$
Cyclic fluctuation pattern of monthly mean temperature of Lake Taihu (from [35])
(a) Seasonal patterns of water temperature at monitor site $T_7$. In the figures, dot-solid curve represents field data while solid curve comes from fitting. (b) Comparison of model predictions (solid line) with real experimental data (stars) for the monitor station $T_7$, which shows the fundamental agreement of oscillatory population behavior and cycle phases between experiment and model prediction of Chlorophyll-a in Lake Tai
(a) Seasonal patterns of mean water temperature among 12 stations in Lake Tai. In the figures, dot-solid curve represents field data while solid curve comes from fitting. (b) Comparison of model predictions (solid line) with real experimental data (stars) for the average chlorophyll-a of 12 monitor stations, which shows the fundamental agreement of oscillatory population behavior and cycle phases between experiment and model prediction of Chlorophyll-a in Lake Tai
Time-series plot of phytoplankton ($A$) (solid line in black) and water temperature ($T$) (dashed line in green). The highest temperature occurs in the 5th day. The parameters are set as follows, $T_{opt}=30^oC$, (a) $T_{mean}=15$, $T_{high}=25$, $\delta=2$; (b) $T_{mean}=15$, $T_{high}=25$, $\delta=1$; (c) $T_{mean}=15$, $T_{high}=20$, $\delta=2$; (d) $T_{mean}=5$, $T_{high}=15$, $\delta=2$
Time-series plot of phytoplankton ($A$) (solid line in black) and water temperature ($T$) (dashed line in green). Phytoplankton with optimal temperature $T_{opt}=23^oC$. Other parameters are same with (a) in Fig. 12
Location of monitor stations (MS)
 MS Name Longitude Latitude $T_1$ Jia Xing Canal ${\rm E120°43'3''}$ ${\rm N30°48'5''}$ $T_2$ Wang Jiang Jing ${\rm E120°42'31''}$ ${\rm N30°53'6''}$ $T_3$ Ping Wang Bridge ${\rm E120°38'14''}$ ${\rm N30°59'49''}$ $T_4$ Hu Zhou Source ${\rm E120°4'23''}$ ${\rm N30°51'51''}$ $T_5$ Xiao Mei Kou ${\rm E120°6'1''}$ ${\rm N30°55'43''}$ $T_6$ New Port ${\rm E120°7'32''}$ ${\rm N30°56'18''}$ $T_7$ Yi Xing Industry ${\rm E119°47'48''}$ ${\rm N31°21'31''}$ $T_8$ Zhou Tie Sewage ${\rm E119°59'54''}$ ${\rm N31°27'20''}$ $T_9$ Yi Xing Rv/Lake ${\rm E120°0'30''}$ ${\rm N31°27'10''}$ $T_{10}$ Wu Xi Mei Liang ${\rm E120°7'25''}$ ${\rm N31°30'13''}$ $T_{11}$ Wu Xi Lake ${\rm E120°21'4''}$ ${\rm N31°27'57''}$ $T_{12}$ Su Zhou Lake Tai ${\rm E120°46'55''}$ ${\rm N31°13'17''}$
 MS Name Longitude Latitude $T_1$ Jia Xing Canal ${\rm E120°43'3''}$ ${\rm N30°48'5''}$ $T_2$ Wang Jiang Jing ${\rm E120°42'31''}$ ${\rm N30°53'6''}$ $T_3$ Ping Wang Bridge ${\rm E120°38'14''}$ ${\rm N30°59'49''}$ $T_4$ Hu Zhou Source ${\rm E120°4'23''}$ ${\rm N30°51'51''}$ $T_5$ Xiao Mei Kou ${\rm E120°6'1''}$ ${\rm N30°55'43''}$ $T_6$ New Port ${\rm E120°7'32''}$ ${\rm N30°56'18''}$ $T_7$ Yi Xing Industry ${\rm E119°47'48''}$ ${\rm N31°21'31''}$ $T_8$ Zhou Tie Sewage ${\rm E119°59'54''}$ ${\rm N31°27'20''}$ $T_9$ Yi Xing Rv/Lake ${\rm E120°0'30''}$ ${\rm N31°27'10''}$ $T_{10}$ Wu Xi Mei Liang ${\rm E120°7'25''}$ ${\rm N31°30'13''}$ $T_{11}$ Wu Xi Lake ${\rm E120°21'4''}$ ${\rm N31°27'57''}$ $T_{12}$ Su Zhou Lake Tai ${\rm E120°46'55''}$ ${\rm N31°13'17''}$
Values used for field application
 MS $T_{mean}$ $\alpha$ $\varphi$ $P_{in1}$ (mg$\cdot$L$^{-1}$) $T_7$ 19.18 -11.37 7.1527 0.364 $Mean$ 18.90 -11.68 0.8611 0.242
 MS $T_{mean}$ $\alpha$ $\varphi$ $P_{in1}$ (mg$\cdot$L$^{-1}$) $T_7$ 19.18 -11.37 7.1527 0.364 $Mean$ 18.90 -11.68 0.8611 0.242
Parameters of model (6) with default values used for numerical studies
 Par. Description Value Unit Ref. $U_p$ maximum nutrient uptake coefficient $0.05$ $\mathrm{day}^{-1}$ [40] $\mu_{opt}$ maximum growth rate of phytoplankton $0.3-1.6$ $\mathrm{day}^{-1}$ [2] $T_{opt}$ optimal water temperature $20-30$ $^{\circ}\mathrm{C}$ [2] $T_{min}$ minimum water temperature $-10-0$ $^{\circ}\mathrm{C}$ [2] $T_{max}$ maximum water temperature $30-40$ $^{\circ}\mathrm{C}$ [2] $T_{ref}$ reference water temperature $20$ $^{\circ}\mathrm{C}$ [8] $m_0$ natural mortality rate of phytoplankton $0.13$ $\mathrm{day}^{-1}$ [8] $K_1$ Respiration rate of phytoplankton $0.1-0.5$ $\mathrm{day}^{-1}$ [8] $\theta_1$ temperature dependence coefficient $1.11$ $-$ [8] $K_2$ temperature dependence coefficient $0.0004-0.1$ $\mathrm{day}^{-1}$ [26] $\theta_2$ temperature dependence coefficient $1.02-1.14$ $-$ [26] $H_1$ coefficient for nutrient uptake $0.6227$ $\mathrm{mg}^{2}\cdot \mathrm{L}^{-2}$ Defaulted $H_2$ coefficient for nutrient uptake $0.0320$ $\mathrm{mg}^{}\cdot \mathrm{L}^{-1}$ Defaulted $r_1$ coefficient for nutrient uptake $2$ $-$ Defaulted $r_2$ coefficient for nutrient uptake $1$ $-$ Defaulted $P_{in1}$ density of total input soluble phosphorus $0.364$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $P_{in2}$ density of total input solid phosphorus $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $P_{b}$ density of solid phosphorus in bottom sediments $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $a$ dilution rate $0.62$ $\mathrm{day}^{-1}$ Defaulted $b$ density limiting coefficient $0.023$ $\mathrm{L}\cdot(\mathrm{day}\cdot\mathrm{mg})^{-1}$ Defaulted
 Par. Description Value Unit Ref. $U_p$ maximum nutrient uptake coefficient $0.05$ $\mathrm{day}^{-1}$ [40] $\mu_{opt}$ maximum growth rate of phytoplankton $0.3-1.6$ $\mathrm{day}^{-1}$ [2] $T_{opt}$ optimal water temperature $20-30$ $^{\circ}\mathrm{C}$ [2] $T_{min}$ minimum water temperature $-10-0$ $^{\circ}\mathrm{C}$ [2] $T_{max}$ maximum water temperature $30-40$ $^{\circ}\mathrm{C}$ [2] $T_{ref}$ reference water temperature $20$ $^{\circ}\mathrm{C}$ [8] $m_0$ natural mortality rate of phytoplankton $0.13$ $\mathrm{day}^{-1}$ [8] $K_1$ Respiration rate of phytoplankton $0.1-0.5$ $\mathrm{day}^{-1}$ [8] $\theta_1$ temperature dependence coefficient $1.11$ $-$ [8] $K_2$ temperature dependence coefficient $0.0004-0.1$ $\mathrm{day}^{-1}$ [26] $\theta_2$ temperature dependence coefficient $1.02-1.14$ $-$ [26] $H_1$ coefficient for nutrient uptake $0.6227$ $\mathrm{mg}^{2}\cdot \mathrm{L}^{-2}$ Defaulted $H_2$ coefficient for nutrient uptake $0.0320$ $\mathrm{mg}^{}\cdot \mathrm{L}^{-1}$ Defaulted $r_1$ coefficient for nutrient uptake $2$ $-$ Defaulted $r_2$ coefficient for nutrient uptake $1$ $-$ Defaulted $P_{in1}$ density of total input soluble phosphorus $0.364$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $P_{in2}$ density of total input solid phosphorus $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $P_{b}$ density of solid phosphorus in bottom sediments $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$ Defaulted $a$ dilution rate $0.62$ $\mathrm{day}^{-1}$ Defaulted $b$ density limiting coefficient $0.023$ $\mathrm{L}\cdot(\mathrm{day}\cdot\mathrm{mg})^{-1}$ Defaulted
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