Phosphorus Threshold for the Growth of Microcystis wesenbergii, Microcystis aeruginosa, and Chlorella vulgaris Based on the Monod Formula

Abstract

The outbreak of algae in freshwater bodies poses an important threat to aquatic ecosystems, making finding an effective method for controlling algal blooms imperative. Numerous key factors influence algal bloom outbreaks, with nutrient levels in the water body being the decisive factor. Current research regarding the effect of nutrient levels on algal growth shows that phosphorus is a nutrient that influences algal blooms. Herein, we propose the concept of a modified Monod model for the relationship between algal specific growth rate and phosphorus concentration. Through this improved Monod model, we inferred that the phosphorus concentration at a specific growth rate of zero is the lower threshold of phosphorus concentration that limits algal growth and can effectively control algal outbreaks. This lower threshold is denoted as S′. On the basis of this concept, we designed algal growth experiments. Our results provided an equation that effectively describes the relationship between algal growth and nutrient concentration. When three algal species grow under phosphorus-limited conditions, the corresponding phosphorus concentrations at which they maintain a growth rate of 0 are 0.0565, 0.0386, and 0.0205 mg/L as reflected by the following order of their S′ values: Microcystis wesenbergii S′ < Microcystis aeruginosa S′ < Chlorella vulgaris S′. Furthermore, with the increase in phosphorus concentration, the growth of M. aeruginosa becomes faster than that of M. wesenbergii and C. vulgaris. Consequently, M. aeruginosa becomes the dominant population in the water, leading to its predominance in algal blooms. This situation explains the common occurrence of cyanobacterial blooms. Our findings provide a theoretical basis for regulating the concentration of phosphorus to control algal outbreaks. Therefore, our study is of great importance for controlling the eutrophication of water bodies.

1. Modified Monod Model and Its Significance

The introduction of large quantities of nitrogen, phosphorus, and other nutrients into stagnant water bodies can lead to eutrophication and trigger algal blooms, particularly those involving harmful species, such as cyanobacteria, with far-reaching and detrimental effects on aquatic ecosystems. The harmful algal bloom toxins produced during these blooms pose considerable risks to aquatic life, causing fish kills and poisoning various organisms. Additionally, these toxins can jeopardize human health through the consumption of contaminated water or aquatic organisms [1]. Therefore, effectively controlling the occurrence of algal blooms is crucial. Several factors influence algal outbreaks [2,3], with nutrient levels in the water playing a decisive role [4] and affecting the biomass and net productivity of algae [5]. Nutrient supply not only influences the synthesis of algal ATP [6,7], it also affects protein composition [8]. Among nutrients, nitrogen and phosphorus are considered crucial factors affecting cyanobacterial bloom outbreaks [9,10,11,12]. A case study using the water quality data of Lake Taihu from the past decade found that the annual average concentrations of total nitrogen (TN) and total phosphorus (TP) in Lake Taihu fluctuated from 1.26 mg/L to 2.43 mg/L and from 0.060 mg/L to 0.103 mg/L, respectively. The nitrogen-to-phosphorus ratios in the aquatic ecosystems of major lakes in China often exceed 16. Research on Dianchi Lake [13] indicated the importance of controlling phosphorus concentration in the water. By drawing on the research od Redfield et al. [14] and Liebig’s law of the minimum, [15] this study proposes the exploration of a lower phosphorus threshold [16,17,18] that limits algal growth with the aim of regulating algal proliferation and preventing the occurrence of algal blooms.

Reports showed that the dominant algal species in algal blooms in major lakes in China [19,20,21,22] are primarily members of Cyanobacteria and Chlorophyta. Among these species, Microcystis aeruginosa and Chlorella are the most prevalent. In consideration of the representative algal species in Chinese lake ecosystems and laboratory conditions, this study selected M. aeruginosa, Microcystis wesenbergii, and Chlorella vulgaris as research subjects. These three algal species exhibit distinctive biological characteristics. M. aeruginosa, a cyanobacterium with a spherical to ellipsoidal shape, belongs to the genus Microcystis, possesses robust nitrogen-fixing capabilities and resilience to eutrophic conditions; however, its bloom formation may produce microcystins, thus posing potential threats to ecosystems and human health. M. wesenbergii, also a cyanobacterium and belonging to the Microcystis, typically forms blooms in nutrient-rich waters and may produce toxins. C. vulgaris, a green alga belongs to the genus Chlorella, is known for its rapid growth and adaptability, often serving as a model organism in environmental and ecological research. While these algae play crucial roles in aquatic environments, their excessive proliferation can disrupt ecological balance and contribute to water quality issues. Therefore, exploring a method for controlling the excessive proliferation of algae to prevent the occurrence of algal blooms is necessary.

Although numerous studies have been conducted on the threshold of algal growth, their final conclusions do not elucidate the relationship between the growth-to-growth rate ratio of algae and thresholds. Furthermore, their threshold results vary considerably, making determining the precise numerical value that restricts algal growth challenging. For example, the conclusion derived by Xu et al. [23] in an in situ cultivation experiment on summer cyanobacterial blooms in Lake Taihu suggested an upper threshold for algal growth that is not limited by nutrients, with TP ranging from 0.15 mg/mL to 0.20 mg/L. Wu et al. [24], by using the frequency distribution method, estimated that the lower TP threshold value limiting algal growth is 0.059 mg/L, which is comparable to the findings of our study. Additionally, several studies have discussed TP thresholds for maintaining water bodies in a sub-bloom state. For example, the in situ cultivation experiments [25] conducted by Xu et al. on Lake Taihu and Meiliang Bay [26] suggested that maintaining TP concentrations below 0.08 mg/L is effective for controlling the frequency and intensity of harmful algal blooms. The OECD [27], on the basis of expert opinions, considers the TP threshold for eutrophication in water bodies to be 0.084 mg/L. Gibson et al. [28] proposed that the occurrence of cyanobacterial blooms becomes possible when TP concentrations exceed 0.01 mg/L.

As can be observed from the above studies, numerous conclusions regarding the threshold of algae exists, but a consensus on which concentration can effectively restrict algal growth has not been reached. In response to this uncertainty, we formulated a scientific hypothesis based on the Monod equation [29]. We propose that a significant correlation exists between the growth-to-growth rate ratio of algae and the phosphorus threshold. We posit the existence of a specific phosphorus concentration, denoted as S′, at which the growth-to-growth rate ratio of algae reaches zero. When the growth-to-growth rate ratio is zero, algal outbreaks are effectively controlled. We introduce an improved Monod equation and design experiments for verification to validate our hypothesis. Our conclusive experimental results confirm the feasibility of our hypothesis.

1.1. Monod Equation

In 1949, Monod published a study that systematically examined results from static reactors, associating cell-specific growth rates with the limiting substrate concentration in a functional manner [29]. He suggested that the relationship between microbial specific growth rate and substrate concentration could be described using the classical Michaelis–Menten equation, leading to the development of the Monod equation:

$$\frac{1}{X}\frac{dX}{dt}=\mu ={\mu }_{max}\frac{S}{K_s+S}.$$

(1)

In this context, μ_max represents the maximum specific growth rate of microorganisms, and K_s stands for the saturation constant, which corresponds to the substrate concentration when μ equals half of μ_max. It is also known as the half-velocity constant. S represents the concentration of organic substrate, and X is the concentration of microorganisms. The curve of the Monod equation is depicted in Figure 1 [30]. The Monod equation, in its empirical form, is widely utilized in contemporary wastewater biological treatment as a powerful tool to guide indicators, such as organic load, in biological treatment.

1.2. Modified Monod Model

The Monod equation, as one of the most widely used mathematical models in microbial growth studies, holds considerable theoretical and practical importance. However, it still exhibits certain limitations. As observed in Figure 1, the Monod equation represents a curve starting from the point (0, 0), indicating that when the substrate concentration S is zero, the specific growth rate of microorganisms is also zero. As noted in related studies [31], the Monod equation provides the relationship between the specific growth rate of fast-growing bacteria and the electron donor concentration that limits their growth:

$${\mu }_{syn}={\left(\frac{1}{X_a}\frac{d{X}_a}{dt}\right)}_{syn}=\hat{\mu }\frac{S}{K+S}.$$

(2)

Here, μ_syn represents the synthesized specific growth rate, X_a is the concentration of active bacterial cells (MxL⁻³), t is time (T), S is the substrate concentration limiting the growth rate (MSL⁻³), $\hat{\mu }$ is the maximum specific growth rate (T⁻¹), and K is the substrate concentration at which the growth rate is half of the maximum specific growth rate (MSL⁻³). Figure 2 illustrates the variation in μ with S and the condition where μ = $\hat{\mu }$/2 when K = S.

Researchers focusing on slowly growing bacteria, such as environmental engineers, have observed that active bacterial cells require energy to sustain their life activities, including movement, repair and synthesis, osmoregulation, transport, and heat dissipation, among other cellular functions. Environmental engineers often use the term “endogenous decay” to describe the energy and electron flow required to sustain growth. In other words, cells oxidize themselves to meet the energy requirements necessary for maintenance.

The rate of endogenous decay is determined by the following equation:

$${\mu }_{dec}={(\frac{1}{X_a}\frac{d{X}_a}{dt})}_{decay}=-b.$$

(3)

In the above equation, b represents the endogenous decay coefficient (T⁻¹), and μ_dec denotes the specific growth rate considering decay (T⁻¹).

Equation (3) illustrates that the loss of active bacterial cells can be described by using a first-order function. However, in reality, not all lost active bacterial cells undergo oxidation to generate the energy required for microbial maintenance. Although the majority of decaying cells undergo oxidation, a small fraction accumulates in the form of inert cells. The oxidation rate (respiration, which is the process that is truly utilized to generate energy) is determined by

$${\left(\frac{1}{X_a}\frac{d{X}_a}{dt}\right)}_{resp}=-f_{d}b.$$

(4)

In the above equation, f_d represents the fraction of biodegradable compounds within active bacterial cells. The rate at which active bacterial cells transform into inert cells is determined by the difference between the overall decay rate and the oxidation rate:

$$-\frac{1}{X_a}\frac{d{X}_i}{dt}={(\frac{1}{X_a}\frac{d{X}_a}{dt})}_{inert}=-(1-f_d)b$$

(5)

In the above equation, X_i represents the concentration of inert bacterial cells (MxL⁻³).

Overall, the net specific growth rate (μ) of active bacterial cells is the sum of new growth (Equation (2)) and decay (Equation (3)):

$$\mu =\frac{1}{X_a}\frac{d{X}_a}{dt}={\mu }_{syn}+{\mu }_{dec}=\hat{\mu }\frac{S}{K+S}-b$$

(6)

Figure 2 also illustrates the variation in μ with S, indicating that under sufficiently low S conditions, μ may be negative.

The specific growth rate μ of algae can also be negative. In terms of the effect of phosphorus on algal cell growth, phosphorus is an essential component of algal cell phospholipids, nucleic acids, and other important substances. It directly influences the storage and transfer of energy and information within active cells, making it a crucial factor in algal cell growth. Research indicates the occurrence of luxury phosphorus uptake [32,33] in the absorption of phosphorus by algae. The phosphorus absorbed during this process is stored intracellularly in the form of polyphosphate bodies. These internal phosphorus [34,35,36] reserves emerge as critical factors influencing cell division [37,38]. Excessive reductions in phosphorus concentrations not only affect the absorption of other elements, such as nitrogen [39], by cells but also hinders the ability of cells to undergo photosynthesis [40], ultimately suppressing cell growth [41]. Therefore, in low-phosphorus concentration culture media, some cells may die due to lysis, and overall, the cell proliferation rate is lower than the decay rate, resulting in a negative net growth rate.

In this work, we attempt to apply the theory developed above to the study of algal growth and propose a modified Monod model that describes the relationship between the specific growth rate of algae and the limiting concentration of phosphorus nutrients:

$$\frac{1}{X}\frac{dX}{dt}=\mu ={\mu }_{max}\frac{S}{K_s+S}+C.$$

(7)

In the above equation, μ represents the specific growth rate of algae, expressed in day⁻¹; S denotes the concentration of the limiting substrate phosphorus, measured in mg/L; t stands for time, measured in days; X represents algal density; μ_max corresponds to the maximum specific growth rate, expressed in day⁻¹; K_s is the phosphorus concentration at which algal growth reaches half of the maximum specific growth rate, measured in mg/L; and C represents the specific death rate of algal growth, expressed in day⁻¹.

Setting Equation (7) to zero yields:

$$S^{\prime }=\frac{K_s{\mu }_{max}C}{1-{\mu }_{max}C}.$$

(8)

The term S′ is defined as the lower threshold concentration of phosphorus at which the algal specific growth rate becomes zero. Its importance lies in the fact that when phosphorus becomes the limiting factor for algal growth, thus maintaining the phosphorus concentration at the level of S′, the specific growth rate of algae equals their specific death rate. At this point, algal growth reaches a dynamic equilibrium state. This situation has practical implications for controlling algal blooms.

2. Materials and Methods

Cultivation experiments were designed for algae under various phosphorus concentrations to validate the proposed modified Monod model described above. The nonlinear curve fitting analysis of the specific growth rate μ of algae against phosphorus concentration was conducted to assess the correlation between algal growth patterns under phosphorus limitation and the modified model.

2.1. Experimental Materials

The original algal strains used in this experiment were obtained from the Freshwater Algae Culture Collection of the Chinese Academy of Sciences. They included M. aeruginosa (FACHB-315), C. vulgaris (FACHB-8), and M. wesenbergii (FACHB-908). Other experimental materials are detailed in the Appendix A.

2.2. Medium Design

Three types of media were used in the experiments: initial culture medium, phosphate-deficient medium, and phosphate concentration gradient medium. The initial culture medium was prepared by using BG11 medium. The phosphate-deficient medium was derived from BG11 medium by excluding a phosphate source (K₂HPO₄). The specific formulations for the three types of culture media are provided in the Appendix A.

2.3. Experimental Design

All experimental operations were conducted under aseptic conditions. First, algae were subjected to nutrient deprivation in a phosphorus-free culture medium. Subsequently, they were inoculated into a culture medium with a phosphorus concentration gradient and placed in an experimental incubator. Sampling was conducted at 24 h intervals to measure the algal density of each species. Refer to the Appendix A for detailed experimental procedures and conditions.

2.4. Data Analysis Methods

The raw data obtained after 7 days of experimentation included the daily algal density X at various concentrations of the culture medium along with the time interval t between each sampling. The formula used to calculate the algal specific growth rate is as follows:

$$\mu =\frac{1}{X}\frac{dX}{dt}.$$

(9)

By integrating both sides of the equation, we obtain

$$\mu t=\ln \left(X\right)+C.$$

(10)

By compiling results, we obtain

$$\ln X=\mu t-C.$$

(11)

By utilizing the Matlab(R2022a) processing tool, we performed linear regression on the natural logarithm of algal density values (ln[X]) against the time interval t, obtaining the slope of this equation as the specific growth rate μ of the algae at different concentrations.

Furthermore, we employed the AR toolbox within Matlab to conduct nonlinear regression on the relationship between the algal specific growth rate μ and phosphorus concentration, as per Equation (7), to investigate the accuracy of the modified Monod equation.

3. Results and Analysis

The experiment involved fitting data for each algal species twice to minimize random errors during data processing. The linear regression of algal specific growth rate was performed by using the logarithm of the average of 15 algal density values ($\overline{X}$) obtained from three parallel samples at each phosphorus concentration. Additionally, the logarithm of the average algal density values ($\overline{X_1}$, $\overline{X_2}$, $\overline{X_3}$) for each set of parallel samples at each phosphorus concentration was calculated, and linear regression analysis was conducted against the cultivation time t.

3.1. Specific Growth Rate of C. vulgaris

The fitting plot of ln($\overline{X}$) for C. vulgaris against time t is shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, and the fitting results are presented in

Table 1. Specific growth rate of C. vulgaris at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fitted R-Squared Value (R²)
0.0356	−0.0347	0.7798
0.0711	−0.0251	0.9716
0.1423	0.1696	0.7126
0.2134	0.1802	0.6222
0.3557	0.2363	0.7709
1.0671	0.3167	0.9039
2.1342	0.3301	0.9112
2.7745	0.3271	0.8985

.

The fitted curve of the ln($\overline{X_1},\overline{X_2},\overline{X_3}$) values of C. vulgaris with respect to time are shown in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, and the fitting results are presented in

Table 2. Specific growth rate of C. vulgaris at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fitted R-Squared Value (R²)
0.0356	−0.0286	0.3596
0.0711	−0.0244	0.8491
0.1423	0.178	0.6019
0.2134	0.1824	0.6214
0.3557	0.2375	0.7763
1.0671	0.3154	0.8986
2.1342	0.3144	0.8899
2.7745	0.3226	0.8954

.

As can be observed in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, a clear linear relationship exists between the ln(X) values of C. vulgaris and time t. In accordance with Equation (11), the slope of the fitted linear equation represents the specific growth rate of C. vulgaris. Combining the results in Table 1 and Table 2 show that the fitting results from both sets of data are close and that the R² values of the fitting are generally high, indicating a significant linear relationship between the two variables. This situation suggests the validity of the results.

C. vulgaris growth exhibits a certain degree of inhibition at low phosphorus concentrations (Table 1 and Table 2) because the specific growth rates are negative. However, no significant cell death occurs, and algal cells experience a slow rate of decline. As phosphorus concentration increases, the specific growth rate of C. vulgaris approaches zero, indicating a dynamic equilibrium between growth and decline, whereas the algal cell population remains relatively constant. When the phosphorus concentration exceeds 0.07 mg/L, the growth and reproduction of C. vulgaris improve with increasing initial phosphorus concentrations, and thus algal biomass increases. When the phosphorus concentration reaches 1.06–2.13 mg/L, the variation in the specific growth rate of C. vulgaris is not significant. This result indicates that the concentration has approached the saturation point for phosphorus absorption by C. vulgaris, representing the upper threshold limit at which phosphorus concentration affects the growth of C. vulgaris. This result is consistent with the findings of Sun et al. [42].

3.2. Growth Rate of M. aeruginosa

The fitting plots of ln($\overline{\mathrm{X}}$) for M. aeruginosa in relation to time (t) are presented in Appendix B, Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7 and Figure A8, and the fitting results are shown in

Table 3. Specific growth rate of M. aeruginosa at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fitted R-Squared Value (R2)
0.0356	−0.0397	0.9486
0.0711	0.2308	0.9238
0.1423	0.3048	0.9411
0.2134	0.3562	0.9434
0.3557	0.4339	0.9675
1.0671	0.4863	0.9846
2.1342	0.5108	0.9773
2.7745	0.4842	0.9873

.

The fitted curves of ln($\overline{X_1},\overline{X_2},\overline{X_3}$) values of M. aeruginosa in relation to time are shown in Appendix B, Figure A9, Figure A10, Figure A11, Figure A12, Figure A13, Figure A14, Figure A15 and Figure A16, and the fitting results are presented in

Table 4. Specific growth rate of M. aeruginosa at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fitted R-Squared Value (R2)
0.0356	−0.0369	0.7486
0.0711	0.2317	0.8806
0.1423	0.3047	0.9338
0.2134	0.3564	0.9386
0.3557	0.4339	0.9613
1.0671	0.4873	0.9793
2.1342	0.5065	0.9783
2.7745	0.4876	0.9815

.

The natural logarithm of M. aeruginosa (ln[X]) values show a clear linear relationship with time (t) (Appendix B, Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12, Figure A13, Figure A14, Figure A15 and Figure A16), indicating a good fit of the data. When the phosphorus concentration is 0.0356 mg/L, the growth of M. aeruginosa is somewhat inhibited (Table 3 and Table 4). However, when the phosphorus concentration reaches 0.0711 mg/L, M. aeruginosa exhibits rapid growth and reproduction. The lower limit threshold at which phosphorus limits the growth of M. aeruginosa falls between 0.0356 and 0.0711 mg/L. Furthermore, once phosphorus concentration exceeds this lower threshold, the proliferation rate of M. aeruginosa significantly increases. This rate continues to increase with phosphorus concentration, reaching a maximum value of 2.1342 mg/L. Beyond this point, the specific growth rate of M. aeruginosa slightly decreases, indicating that excessively high phosphorus levels may have a mild inhibitory effect on its growth.

3.3. Growth Rate of M. wesenbergii

The fitting plots of ln($\overline{X}$) for M. wesenbergii in relation to time (t) are presented in Appendix B, Figure A17, Figure A18, Figure A19, Figure A20, Figure A21, Figure A22, Figure A23 and Figure A24, and the fitting results are shown in

Table 5. Specific growth rate of M. wesenbergii at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fitted R-Squared Value (R2)
0.0356	0.0955	0.3495
0.0711	0.1891	0.8780
0.1423	0.2416	0.9411
0.2134	0.2907	0.9814
0.3557	0.3231	0.9864
1.0671	0.3099	0.9765
2.1342	0.3535	0.9946
2.7745	0.3123	0.9907

.

The fitted curves of the ln($\overline{X_1},\overline{X_2},\overline{X_3}$) values of M. wesenbergii with time are shown in Appendix B, Figure A25, Figure A26, Figure A27, Figure A28, Figure A29, Figure A30, Figure A31 and Figure A32, and the fitting results are presented in

Table 6. Specific growth rate of M. wesenbergii at different phosphorus concentrations.

Phosphorus Concentration (mg/L)	Specific Growth Rate (day−1)	Fit R-Squared Value (R2)
0.0356	0.1019	0.4399
0.0711	0.1926	0.8860
0.1423	0.2455	0.9493
0.2134	0.2868	0.9843
0.3557	0.3259	0.9890
1.0671	0.3136	0.9722
2.1342	0.3621	0.9929
2.7745	0.3177	0.9846

.

The natural logarithm values of M. wesenbergii exhibit a clear linear relationship with time (t) (Appendix B Figure A17, Figure A18, Figure A19, Figure A20, Figure A21, Figure A22, Figure A23, Figure A24, Figure A25, Figure A26, Figure A27, Figure A28, Figure A29, Figure A30, Figure A31 and Figure A32), indicating favorable fitting results. When the phosphorus concentration is relatively low (0.0356 mg/L), M. wesenbergii exhibits minimal growth rates (Table 5 and Table 6). However, the specific growth rate of M. wesenbergii gradually increases with phosphorus concentration (0.0711–0.3557 mg/L), suggesting that the limiting phosphorus nutrient threshold for the growth of M. wesenbergii is likely below 0.0356 mg/L. Mathematical modeling can be used to calculate the theoretical lower threshold that restricts its growth.

When the phosphorus concentration exceeds this lower threshold, the proliferation rate of M. wesenbergii considerably increases, continues to increase until a phosphorus concentration of 2.1342 mg/L, and then decreases. Thus, excessively high phosphorus nutrient levels may exert inhibitory effects on the growth of M. wesenbergii.

4. Comparison and Discussion Based on the Experiment of Specific Growth Rate of Three Kinds of Algae

4.1. Comparison of the Specific Growth Rates of Three Algal Species

The comparative line chart depicting the specific growth rates of C. vulgaris, M. aeruginosa, and M. wesenbergii is presented in Figure 19.

Variations in the specific growth rates of C. vulgaris, M. aeruginosa, and M. wesenbergii exhibit similar trends at the same phosphorus concentrations (Figure 19).This is characterized by inhibited or reduced growth at low phosphorus concentrations, followed by an increase in specific growth rates as phosphorus concentration increases. However, as phosphorus concentration exceeds a certain threshold, the rate of increase in specific growth rates decreases.

In the comparative graph, M. aeruginosa and C. vulgaris have similar specific growth rates at a low phosphorus concentration (A1 0.0356 mg/L), whereas M. wesenbergii shows positive growth but with an extremely low specific growth rate. When the phosphorus concentration reaches the A2 concentration (0.0711 mg/L), C. vulgaris growth is still inhibited, and the specific growth rates of M. aeruginosa and M. wesenbergii exhibit a geometrically increasing trend, which suggests that the phosphorus nutrient thresholds of M. aeruginosa and M. wesenbergii are lower than the phosphorus nutrient threshold of C. vulgaris. M. aeruginosa has a higher specific growth rate than C. vulgaris and M. wesenbergii at various phosphorus nutrient concentrations. Thus, M. aeruginosa is more likely to gain a competitive advantage and become the dominant species, possibly explaining the prevalence of cyanobacterial blooms in China. When the phosphorus concentration is at the A5 concentration (0.3557 mg/L), the specific growth rates of C. vulgaris and M. wesenbergii nearly reach maximum values. Subsequent increases in phosphorus concentration have minimal effects on these two algal species, indicating that the upper threshold for their growth lies at approximately 0.3557 mg/L. This results aligns with existing research conclusions. By contrast, the specific growth rate of M. aeruginosa continues to increase with phosphorus concentration. At high phosphorus concentrations (>A7), C. vulgaris is essentially in a stable growth state and is almost unaffected by phosphorus concentration. The specific growth rate of M. aeruginosa slightly decreases, whereas that of M. wesenbergii shows a considerable decrease. This result suggests that the inhibitory effect of high phosphorus concentrations may not apply to all algal species and phosphorus concentration considerably affects the growth of M. aeruginosa and M. wesenbergii.

4.2. Fitting of the Modified Monod Model for C. vulgaris

The nonlinear fitting of the specific growth rates of C. vulgaris against phosphorus nutrient concentration follows Equation (7) and is illustrated in Figure 20.

The conventional model used for fitting is the modified Monod equation (Equation (7)): $\mu ={\mu }_{max}\frac{S}{K_s+S}-C$. The coefficients of the equation (with a 95% confidence interval) obtained from the fitting results are as follows: μ_max (0.5711, 0.5658), K_s (0.09321, 0.08543), and C (0.2156, 0.2196). The goodness-of-fit indicators include sum of squared errors (SSE) (0.005515, 0.006661), coefficient of determination (R²) (0.964, 0.954), adjusted R² (0.9498, 0.9357), and root-mean-squared error (RMSE) (0.03321, 0.0365).

The R² values obtained from the fitting indicate that the proposed modified Monod equation effectively reflects the nonlinear relationship between C. vulgaris growth rate and phosphorus nutrient concentration, validating the feasibility of the research hypothesis. This equation can describe the growth of C. vulgaris subjected to limited phosphorus nutrient. Additionally, the threshold phosphorus nutrient concentrations limiting C. vulgaris growth are 0.057 and 0.054 mg/L according to Equation (8). Wu et al. [19] concluded that the threshold for algal growth is a total phosphorus content of 0.059 mg/L, which aligns well with the results of this experiment.

Therefore, the proposed model provides a specific method for calculating the phosphorus limitation threshold in controlling nutrient-deficient algae. By maintaining the phosphorus concentration in water at approximately 0.054 mg/L, the growth of C. vulgaris can be maintained in a dynamic equilibrium state, and the excessive proliferation and the outbreak of algal blooms in aquatic ecosystems can thereby be effectively prevented.

4.3. Fitting of the Modified Monod Model for M. aeruginosa

The growth rate of M. aeruginosa in relation to phosphorus nutrient concentration was fitted by using Equation (7). The fitting graph is presented in Figure 21.

The conventional model used for fitting is the modified Monod equation presented in Equation (7): $\mu ={\mu }_{max}\frac{S}{K_s+S}-C$. The coefficients obtained from the fitting results (with 95% confidence intervals) are as follows: μ_max (1.838, 1.793), K_s (0.0145, 0.0149), and C (1.336, 1.291). The goodness-of-fit indicators include SSE (0.003874, 0.003697), R² (0.9837, 0.9843), adjusted R² (0.977, 0.978), and RMSE (0.02784, 0.02719).

The obtained R² values from the fitting indicate that the modified Monod equation proposed in this study (Equation (7)) effectively represents the nonlinear relationship between M. aeruginosa growth rate and phosphorus nutrient concentration. This validates the feasibility of the research hypothesis. The modified Monod model can describe the growth of M. aeruginosa under phosphorus nutrient limitation. Furthermore, the calculated phosphorus nutrient lower threshold for limiting the growth of M. aeruginosa is 0.039 mg/L (or 0.038 mg/L). This result provides a specific reference for nutrient control under oligotrophic conditions, implying that controlling phosphorus concentration in water at approximately 0.038 mg/L can help maintain a dynamic equilibrium in M. aeruginosa growth and thereby prevent the excessive proliferation of algae and occurrence of algal blooms.

4.4. Fitting of the Modified Monod Model for M. wesenbergii

The growth rate of M. wesenbergii was fitted against phosphorus nutrient concentration by using Equation (7) through nonlinear fitting. The fitting results are illustrated in Figure 22.

The conventional model used for fitting is the modified Monod equation as previously introduced in Equation (7): $\mu ={\mu }_{max}\frac{S}{K_s+S}-C$. The fitting results yielded the coefficients (with 95% confidence intervals) of μ_max: (0.7112, 0.6112); K_s: (0.0186, 0.0234); and C: (0.3732, 0.2675). The goodness-of-fit metrics included SSE: (0.001879, 0.001902); R²: (0.9633, 0.9626); adjusted R²: (0.9486, 0.9477); and RMSE: (0.01939, 0.0195).

The R² values obtained from the fitting indicate that the modified Monod equation (Equation (7)) proposed in this study effectively reflects the nonlinear relationship between the growth rate of M. wesenbergii and concentration of phosphorus nutrients, thus validating the feasibility of our research hypothesis. This modified Monod model can describe the growth of M. wesenbergii under phosphorus limitation. Furthermore, Equation (8) allows the calculation of the phosphorus lower threshold that restricts the growth of M. wesenbergii. The thresholds were approximately 0.039 and 0.038 mg/L. This study provides a specific phosphorus limitation threshold as a reference for controlling phosphorus concentration in water. Maintaining the phosphorus concentration in the range of approximately 0.038 mg/L in water bodies potentially maintains the dynamic equilibrium of M. wesenbergii growth, thus preventing the excessive proliferation of algae and occurrence of algal blooms.

4.5. Comparison of Modified Monod Models for C. vulgaris, M. aeruginosa, and M. wesenbergii

A comparative chart of the modified Monod models for C. vulgaris, M. aeruginosa, and M. wesenbergii is shown in Figure 23.

The coordinates of the equilibrium points (S′) are (0.0565, 0), (0.0386, 0), and (0.0205, 0), and when phosphorus is limited, the corresponding phosphorus concentrations at which they maintain a growth rate of 0 are 0.0565, 0.0386, and 0.0205 mg/L. This concentration represents the phosphorus lower threshold for algal growth.

The phosphorus lower threshold for M. wesenbergii (S′ = 0.0205 mg/L) is smaller than that for M. aeruginosa (S′ = 0.0386 mg/L), and both thresholds are smaller than the threshold for C. vulgaris (S′ = 0.0565 mg/L). In other words, these thresholds follow the order of M. wesenbergii S′ < M. aeruginosa S′ < C. vulgaris S′. Furthermore, M. wesenbergii and M. aeruginosa show a greater increase in their growth rates than C. vulgaris at low phosphorus concentrations and increasing phosphorus nutrient levels. This result indicates that M. wesenbergii and M. aeruginosa require low phosphorus nutrient concentrations as a lower threshold for growth and exhibit fast growth rates at the same phosphorus concentration. Although M. wesenbergii has a lower phosphorus nutrient threshold than M. aeruginosa, the growth rate of M. aeruginosa is higher when phosphorus concentration further increases. The maximum growth rate of M. aeruginosa is considerably higher than that of M. wesenbergii. Thus, M. aeruginosa tends to become the dominant species. Therefore, algal blooms in Chinese lakes are primarily composed of cyanobacteria, and other algal species do not undergo blooms when cyanobacterial blooms occur.

5. Conclusions

In this study, phosphorus was considered as the controlled factor, and C. vulgaris, M. aeruginosa, and M. wesenbergii were selected as research subjects. While ensuring an excess supply of other nutrients, experiments were designed to investigate the relationship between algal growth and phosphorus concentration gradients. Additionally, a modified Monod model was established to determine the lower threshold of phosphorus nutrient concentration for the growth of different algal species. The following conclusions were drawn:

(a)

The conventional Monod equation has certain limitations. In this study, we proposed the modified Monod equation $\mu ={\mu }_{max}\frac{S}{K_s+S}+C$ and designed experiments to validate it. The resulting fit had R² values of 0.954, 0.964, 0.977, 0.978, 0.9633, and 0.9626, indicating that the modified Monod equation effectively describes algal growth when phosphorus is limited.

(b)

By using the modified Monod model, we calculated the lower threshold of phosphorus nutrients, which is the phosphorus nutrient concentration at which algal growth reaches a dynamic equilibrium. The calculated values were 0.0565 mg/L for C. vulgaris, 0.0386 mg/L for M. aeruginosa, and 0.0205 mg/L for M. wesenbergii. Controlling phosphorus concentrations at or near these S′ values can theoretically prevent excessive algal proliferation, providing guidance for algal growth control using nutrient limitation.

(c)

The S′ of the three algal species followed the order of M. wesenbergii S′ < M. aeruginosa S′ < C. vulgaris S′. M. wesenbergii requires the lowest theoretical phosphorus nutrient concentration for growth, followed by M. aeruginosa, and C. vulgaris requires the highest. The results suggest that cyanobacteria (Microcystis and similar species) have lower phosphorus nutrient thresholds, explaining why algal blooms in China are mainly composed of cyanobacteria and why other algal species do not bloom during cyanobacterial bloom events.

(d)

Among the three algal species, M. aeruginosa exhibited the highest maximum specific growth rate, whereas C. vulgaris had the lowest. This result suggests that in natural water bodies with fluctuating phosphorus concentrations, M. aeruginosa would dominate in terms of biomass, followed by M. wesenbergii and C. vulgaris. This observation implies that cyanobacterial biomass tends to be higher than that of green algae, making cyanobacteria the dominant species in freshwater ecosystems.

(e)

At high phosphorus concentrations (>2 mg/L), the growth of M. aeruginosa and M. wesenbergii is inhibited to some extent and that of C. vulgaris is unaffected. This result indicates that phosphorus inhibition does not occur in all algal species. The results of this experiment suggest that phosphorus inhibition is evident in cyanobacteria, and future research may explore this phenomenon further by investigating internal phosphorus forms and phosphorus absorption gene sequences in different algal species.

This study, combined with international and domestic research on algal growth under nitrogen and phosphorus thresholds, is in line with China’s water quality conditions. It proposes the construction of a modified Monod model to describe algal growth, allowing the determination of the lower phosphorus nutrient threshold. The growth distribution trends of different algal species in freshwater ecosystems can be predicted and analyzed by comparing their lower thresholds. This information is crucial for controlling algal blooms through nutrient limitation. Moreover, this work constitutes a theoretical study based on laboratory conditions involving three algal species. Subsequent research can be extended to in situ experiments in lake environments. The experimental methods employed in this study are equally applicable to lake ecosystems; however, results may vary because of some factors, such as temperature variations and the influence of coexisting phytoplankton in lake water.