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(2)
given that the limit exists, where ({boldsymbol{theta }}in {{mathbb{R}}}^{p}) is a p-dimensional vector of mortality parameters. The cumulative hazards rate is$$Uleft(x|{boldsymbol{theta }}right)=int_{0}^{x}mu (t|{boldsymbol{theta }}){dt},$$
(3)
which results in the survival function$$S(x|{boldsymbol{theta }})={{exp }}[-U(x|{boldsymbol{theta }})].$$
(4)
The Cumulative distribution function (CDF) of ages at death is F (x | θ) = 1 – S (x | θ), and the probability density function (PDF) of ages at death is f (x | θ) = μ (x | θ) S (x | θ), for x ≥ 0. The remaining life expectancy after age x is calculated as$$eleft(x{rm{|}}{boldsymbol{theta }}right) = frac{{int }_{x}^{{{infty }}}{tf}(t|{boldsymbol{theta }}){dt}}{Fleft({{infty }}right)-Fleft(xright)}\ =frac{{int }_{x}^{{{infty }}}S(t|{boldsymbol{theta }}){dt}}{Sleft(xright)},$$
(5)
which yields a life expectancy at birth given by$$eleft(0{rm{|}}{boldsymbol{theta }}right)={int }_{0}^{{{infty }}}S(x|{boldsymbol{theta }}){dx}.$$
(6)
The lifespan inequality at birth, as proposed by Demetrius16,36 and later by Keyfitz17, is given by$$H(0|{boldsymbol{theta }}) =-frac{{int }_{0}^{{{infty }}}S(x|{boldsymbol{theta }}){{log }}[S(x|{boldsymbol{theta }})]{dx}}{e(0|{boldsymbol{theta }})}\ = frac{{int }_{0}^{{{infty }}}S(x|{boldsymbol{theta }})U(x|{boldsymbol{theta }}){dx}}{e(0|{boldsymbol{theta }})}.$$
(7)
Following Colchero et al.13, we define the lifespan equality as$$varepsilon (x|{boldsymbol{theta }})=-{{log }}[H(x|{boldsymbol{theta }})].$$
(8)
For simplicity, henceforth we note the life expectancy, lifespan inequality and lifespan equality at birth as e(0 | θ) = e, H (0 | θ) = H, and ε (0 | θ) = ε, respectively.Survival analysisTo estimate age-specific survival for all the wild populations of non-human primates, we modified the Bayesian model developed by Colchero et al.13 and Barthold et al.37. This model is particularly appropriate for primate studies that follow individuals continuously within a study area and when individuals of one or both sexes can permanently leave the study area (out-migration), while other individuals can join the study population from other areas (in-migration). Thus, it allowed us to make inferences on age-specific survival (or mortality) and on the age at out-migration.Here we use the five parameter Siler mortality function25, as in Eq. (1) where θ = [a0, a0, c, b0, b1] is a vector of parameters to be estimated, and where a0, b0 ({mathbb{in }}{mathbb{R}}) and a1, c, b1 ≥ 0. For all species we studied, individuals of one or both sexes often leave their natal groups to join other neighbouring groups in a process commonly identified as natal dispersal. For some species, individuals who have undergone natal dispersal can then disperse additional times, described as secondary dispersal. Although dispersal within monitored groups (i.e. those belonging to the study area) does not affect the estimation of mortality, the fate of individuals that permanently leave the study area to join unmonitored groups can be mistaken for possible death. We identify this process as “out-migration”, which we classify as natal or immigrant out-migration, the first for natal and the second for secondary dispersals to unmonitored groups. This distinction is particularly relevant because not all out-migrations are identified as such, and therefore the fate of some individuals is unknown after their last detection. For these individuals we define a latent out-migration state at the time they were last detected, given by the random variable indicator O, with observations oij ∈ {0,1}, where oij = 1 if individual i out-migrated and oij = 0 otherwise, and where j = 1 denotes natal out-migration and j = 2 for immigrant out-migration. For known out-migrations, we automatically assign oij = 1. The model therefore estimates the Bernoulli probability of out-migration, πj, such that Oij ~ Bern(πj). Those individuals assigned as exhibiting out-migration, as well as known emigrants and immigrants, contribute to the estimation of the distribution of ages at out-migration. Here, we define a gamma-distributed random variable V for ages at out-migration, with realisations v ≥ 0, where Vj | Oj = 1 ~ Gam(γj1, γj2) and where γj1, γj2  > 0 are parameters to be estimated with j defined as above. The probability density function for the gamma distribution is gV(v | γj1, γj2) for v ≥ 0, with v = xl – αj, where xl is the age at last detection and αj is the minimum age at natal or immigrant out-migration.In addition, since not all individuals have known birth dates, the model samples the unknown births bi as xil = til – bi, where til is the time of last detection for individual i. The likelihood is then defined as$$p({x}_{{il}},{x}_{{if}},|,{boldsymbol{theta}},{boldsymbol{gamma}}_{1},{boldsymbol{gamma}}_{2},{pi }_{j},{o}_{ij})=left{begin{array}{cc}frac{fleft({x}_{il}right)}{Sleft({x}_{if}right)}({1-pi }_{j})hfill& {text{if}}; o_{{ij}}=0\ frac{Sleft({x}_{{il}}right)}{Sleft({x}_{{if}}right)}{pi }_{j}{g}_{V}({x}_{{il}}-{alpha }_{j})& {text{if}}; o_{{ij}}=1end{array}right.,$$
(9)
where xif is the age at first detection, given by xif = tif – bi, with tif as the corresponding time of first detection. The parameter vectors γ1 and γ2 are for natal and immigrant out-migration, respectively. In other words, individuals with oij = 0 are assumed to have died shortly after the last detection, while those with oij = 1 are censored and contribute to the estimation of the distribution of ages at out-migration. The full Bayesian posterior is then given by$$pleft({boldsymbol{theta }}{boldsymbol{,}}{{boldsymbol{gamma }}}_{1},{{boldsymbol{gamma }}}_{2},{boldsymbol{pi }},{{bf{b}}}_{u},{{bf{o}}}_{u},|,{{bf{b}}}_{k},{{bf{o}}}_{k},{{bf{t}}}_{f},{{bf{t}}}_{l}right) propto ; pleft({{bf{x}}}_{l},{{bf{x}}}_{f},|,{boldsymbol{theta }},{{boldsymbol{gamma }}}_{1},{{boldsymbol{gamma }}}_{2},{boldsymbol{pi }},{bf{d}}right)\ , times pleft({boldsymbol{theta }}right)pleft({{boldsymbol{gamma }}}_{1}right)pleft({{boldsymbol{gamma }}}_{2}right)pleft({boldsymbol{pi }}right),$$
(10)
where the first term on the right-hand-side of Eq. (10) is the likelihood in Eq. (9), and the following terms are the priors for the unknown parameters. The vector π = [π1, π2] is the vector of probabilities of out-migration while the subscripts u and k refer to unknown and known, respectively.Following Colchero et al.13, we used published data, expert information and an agent-based model to estimate the mortality and out-migration prior parameters for each population. We assumed a normal (or truncated normal distribution depending on the parameter’s support) for all the parameters. We used vague priors for the mortality and natal out-migration parameters (sd = 10), and informative priors for the immigrant out-migration parameters (sd = 0.5). We ran six MCMC parallel chains for 25 000 iterations each with a burn-in of 5000 iterations for each population, and assessed convergence using potential scale reduction factor38.For the zoo data we used a simplified version of the model described above, which omitted all parts that related to out-migration. In order to produce Supplementary Figs. 1 and 2, we used the same method as for the zoo data on the human life tables. To achieve this, we created an individual level dataset from the lx column of each population, and then fitted the Siler model to this simulated data. It is important to note that the Siler model provides a close fit to the nonhuman primate data and to high-mortality human populations, although it does not provide the best fit to low-mortality human populations, in part due to the late life mortality plateau common among human populations39 (Supplementary Fig. 6). It is therefore possible that the values of the mortality parameter b1 we report in Supplementary Data 2 for the human populations are under-estimated. Nonetheless, and for the purposes of our analyses, the Siler fits to the human populations we considered here are reasonable (Supplementary Fig. 6) and we can therefore confidently state that the limitations of the Siler model do not affect the generality of our results.Estimation of life expectancy and lifespan equalityBased on the results of the Bayesian inference models, we calculated life expectancy at birth as$$e= int_{0}^{{infty}}Sleft(t| {hat{boldsymbol{theta }}}right){dt},$$
(11)
where S (x) is the cumulative survival function as defined in Eq. (4) and where (hat{{boldsymbol{theta }}}) is the vector of mortality parameters calculated as the mean of the conditional posterior densities from the survival analysis described above. We calculated the lifespan inequality17,36, H, as$$H=-frac{1}{e}int_{0}^{{{infty }}}Sleft(x{rm{|}}hat{{boldsymbol{theta }}}right){{log }}left[Sleft(x|hat{{boldsymbol{theta }}}right)right]{dx},$$
(12)
from which we calculated lifespan equality, ε, as in Eq. (8). We calculated both measures for each of the study populations, and performed weighted least squares regressions for each genus, with weights given by the reciprocal of the standard error of the estimated life expectancies.Sensitivities of life expectancy and lifespan equality to mortality parametersAs we mentioned above, for simplicity of notation, we will express all demographic functions by their variable notation (e.g. e = e (0 | θ), S = S (x | θ), etc.), while we will alternatively note first partial derivatives, for instance the derivative of e with respect to a given mortality parameter θ ∈ θ, as eθ or ∂e / ∂θ.Proposition: If ({S:}{{mathbb{R}}}_{ge 0}to left[{mathrm{0,1}}right]) is a continuous non-increasing parametric survival function with parameter vector ({boldsymbol{theta }}{boldsymbol{in }}{{mathbb{R}}}^{{boldsymbol{p}}}), with continuous differentiable cumulative hazards function ({U:}{{mathbb{R}}}_{ge 0}to {{mathbb{R}}}_{ge 0}), and with life expetancy at birth, lifespan inequality and lifespan equality as in Eqs. (4)-(6), respectively, then the sensitivity of life expectancy, e, to a given parameter θ ∈ θ is$${e}_{theta }=frac{partial e}{partial theta }=int_{0}^{{{infty }}}{S}_{theta }{dx},$$
(13)
while the sensitivity of lifespan equality to θ is$${varepsilon }_{theta }=frac{partial varepsilon }{partial theta }=frac{{e}_{theta }left(1+{H}^{-1}right)-{H}^{-1}{int }_{0}^{{{infty }}}{S}_{theta }{Udx}}{e},$$
(14)
where$${S}_{theta }=frac{partial }{partial theta }S(x|{boldsymbol{theta }})$$
(15)
is the sensitivity of the survival function at age x to changes in parameter θ.Proof. The sensitivity of lifespan equality to changes in θ is derived from$${e}_{theta }=frac{partial }{partial theta }int_{0}^{{{infty }}}{Sdx},$$
(16)
which, by Leibnitz’s rule, Eq. (16) becomes$${e}_{theta }=int_{0}^{{{infty }}}frac{partial S}{partial theta }{dx}=int_{0}^{{{infty }}}{S}_{theta }{dx}.$$
(17)
The sensitivity of lifespan equality to changes in θ can be calculated as$${varepsilon }_{theta } =frac{partial }{partial theta }left(-{{log }}, Hright)\ =-frac{partial }{partial theta }{{log }}, H\ =-frac{1}{H}frac{partial H}{partial theta }\ =-frac{1}{H}frac{partial }{partial theta }left(frac{{int }_{0}^{{{infty }}}{SUdx}}{e}right).$$
(18)
By the quotient and Leibnitz’s rules, Eq. (18) can be modified as$${varepsilon }_{theta } =-frac{1}{H{e}^{2}}left[frac{partial }{partial theta }left(int _{0}^{{{infty }}}{SUdx}right)e-left(int _{0}^{{{infty }}}{SUdx}right)frac{partial e}{partial theta }right]\ =-frac{1}{{He}}int _{0}^{{{infty }}}frac{partial }{partial theta }left({SU}right){dx}+frac{1}{{He}}frac{int _{0}^{{{infty }}}{SUdx}}{e}frac{partial e}{partial theta }.$$
(19)
The first term in Eq. (19) can be further decomposed by the product rule, while the second term can be modified following the equality for H in Eq. (7), which yields$${varepsilon }_{theta } =-frac{1}{{He}}int _{0}^{{{infty }}}left(frac{partial S}{partial theta }U+Sfrac{partial U}{partial theta }right){dx}+frac{1}{e}{e}_{theta }\ =-frac{1}{{He}}left(int _{0}^{{{infty }}}{S}_{theta }{Udx}+int _{0}^{{{infty }}}Sfrac{partial U}{partial theta }dxright)+frac{1}{e}{e}_{theta }.$$
(20)
By the chain rule, we have that (frac{partial U}{partial theta }=-frac{partial }{partial theta }{{log }},S=-frac{1}{S}frac{partial S}{partial theta }), which modifies Eq. (20) as$${varepsilon }_{theta } = , -frac{1}{{He}}left(int _{0}^{infty }{S}_{theta }{Udx}-int _{0}^{infty }frac{partial S}{partial theta }{dx}right)+frac{1}{e}{e}_{theta }\ = , -frac{1}{{He}}left(int _{0}^{infty }{S}_{theta }{Udx}-{e}_{theta }right)+frac{1}{e}{e}_{theta }\ = , -frac{int _{0}^{infty }{S}_{theta }{Udx}}{{He}}+frac{{e}_{theta }}{e}left(1+frac{1}{H}right)\ =, frac{{e}_{theta }left(1+{H}^{-1}right)-{H}^{-1}int _{0}^{infty }{S}_{theta }{Udx}}{e},$$
(21)
hence completing the proof. ∎Changes in parameters along the genus linesFrom the results in Eqs. (13) and (14), we calculated the vectors of change (gradient vectors) at any point (leftlangle {e}_{j},{varepsilon }_{j}rightrangle) of the life expectancy-lifespan equality landscape, as a function of each of the Siler mortality parameters (See Fig. 2A, B).To quantify the amount of change of each parameter along the genus lines, we derived the sensitivities of a given mortality parameter θ to changes in life expectancy and lifespan equality, namely (frac{partial theta }{partial e}=frac{1}{{e}_{theta }}) for ({e}_{theta }, ne, 0,) and (frac{partial theta }{partial varepsilon }=frac{1}{{varepsilon }_{theta }}) for ({varepsilon }_{theta }, ne, 0). With these sensitivities we calculated the gradient vector$$nabla theta =leftlangle frac{partial theta }{partial e},frac{partial theta }{partial varepsilon }rightrangle$$
(22)
for any parameter at any point along the genus lines. Here we find a linear relationship between life expectancy and lifespan equality, given by$$mleft({e}_{{ik}}right)={hat{varepsilon }}_{{ik}}={beta }_{0k}+{beta }_{1k}{e}_{{ik}},$$
(23)
for i = 1, …, nk, where nk is the number of populations for genus k, and ({hat{varepsilon }}_{{ik}})is the fitted value of lifespan equality for population i in genus k, and β0k and β1k are linear regresssion parameters for genus k. To estimate the amount of change in parameter θ along the line for genus k, we can solve the path integral$${Theta }_{k}=int _{{C}_{k}}nabla theta d{bf{r}},$$
(24)
where path Ck is determined by the linear model for genus k and (d{bf{r}}=leftlangle {de},dhat{varepsilon }rightrangle =leftlangle {de},{d; m}left(eright)rightrangle) is the rate of change in the velocity vector ({bf{r}}=leftlangle e,hat{varepsilon }rightrangle =leftlangle e,mleft(eright)rightrangle).In order to compare results between the different mortality parameters in vector θ, we use the transformation g(θ) = log θ, which yields the following partial derivatives$$frac{partial }{partial e}gleft(theta right)=frac{1}{theta }frac{partial theta }{partial e}$$
(25)
and$$frac{partial }{partial varepsilon }gleft(theta right)=frac{1}{theta }frac{partial theta }{partial varepsilon }.$$
(26)
Thus the gradient vector becomes$$nabla theta =leftlangle frac{partial }{partial e}gleft(theta right),frac{partial }{partial varepsilon }gleft(theta right)rightrangle$$
(27)
while the path integral in Eq. (24) is modified accordingly. In short, the path integral ({Theta }_{j}) provides a measure of the relative change in parameter θ along the genus line (Fig. 3). To allow comparisons between all genera, we scaled the values of each path integral by the length of each line.Applications to the Siler mortality modelThe Cumulative hazards for the Siler mortality model in Eq. (7) is given by$$Uleft(xright)=frac{{e}^{{a}_{0}}}{{a}_{1}}left(1-{e}^{{-a}_{1}x}right)+{cx}+frac{{e}^{{b}_{0}}}{{b}_{1}}left({e}^{{b}_{1}x}-1right),$$
(28)
The sensitivities in Eqs. (13) and (14) require calculating Sθ for all θ ∈ θ. Treating S (x) as the function composition W (V), where W = exp(x) and V = – U, then Sθ is$${S}_{theta }=frac{{dW}}{{dV}}{V}_{theta }=-S{U}_{theta },$$
(29)
where Uθ is the first derivative of U(x | θ) with respect to θ. For each of the Siler mortality parameters, we then have$${S}_{{a}_{0}}=S(x|{boldsymbol{theta }})frac{{e}^{{a}_{0}}}{{a}_{1}}left({e}^{-{a}_{1}x}-1right)$$
(30)
$${S}_{{a}_{1}}=S(x|{boldsymbol{theta }})frac{{e}^{{a}_{0}}}{{a}_{1}}left[frac{1}{{a}_{1}}-{e}^{-{a}_{1}x}left(x+frac{1}{{a}_{1}}right)right]$$
(31)
$${S}_{c}=-S(x|{boldsymbol{theta }})x$$
(32)
$${S}_{{b}_{0}}=S(x|{boldsymbol{theta }})frac{{e}^{{b}_{0}}}{{b}_{1}}left(1-{e}^{{b}_{1}x}right)$$
(33)
$${S}_{{b}_{1}}=S(x|{boldsymbol{theta }})left[{e}^{{b}_{1}x}left(frac{1}{{b}_{1}}-xright)-frac{1}{{b}_{1}}right].$$
(34)
All analyses were performed in the free open source programme R40. The R functions we created for this project can be found in41.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

Spatial and temporal variations of the SRHII at daytime and nighttimeSignificant seasonal differences are observed in the SRHII in the YRDUA (Figure A1 and A2, Appendix A). In the daytime, RHI was concentrated in the Nanjing, “Su-Xi-Chang”, Ningbo, Shanghai, and Hangzhou metropolitan areas. Due to the high built-up areas and PD, the distribution of surface RHI is denser and stronger than that in the north and southwest of the YRDUA. The built-up area can absorb heat and store heat energy, which makes the surface temperatures rise rapidly. In spring and autumn, the spatial distribution of the RHI in spring or autumn was similar to that in summer except the spatial extent was tapered. However, the RHI gradually shrinks and transfers to the southern area of the YRDUA in winter, such as Linhai and Ningbo City, which is due to the relatively high solar radiation of the geographic location of the southern cities. The distance of the RHI is gradually shortened between cities and even into one piece from 2003 to 2017 due to long-term urban expansion and rapid growth of construction land (Figure A1, Appendix A). In the nighttime, the spatial pattern of the RHI is very different from that of the daytime. RHI mainly concentrates on Taihu Lake, Dianshan Lake, Ge Lake in the center part, Hongzhe Lake in the northwest, and Qiandao Lake in the southwest. Because water has a high specific heat capacity, it has the function of preserving heat at nighttime. Some cities like Shanghai, Hangzhou, and Nanjing have the strongest heat island in winter and the weakest heat island in summer. Urban areas usually have dense buildings, PD, and energy emissions, so there are more energy emissions at night. High surface albedo in urban areas at night leads to lower heat storage4,40 and ultimately resulting in smaller UHI at nighttime (Figure A2, Appendix A).From spring to summer and then summer to winter, RHI increases first and then decreases, and it reaches a peak in summer. For example, the proportion of the RHI was 12.65%, 31.03%, 21.12%, and 5.49% in spring, summer, autumn, and winter in 2017, respectively (Fig. 2d). An upward trend in the area of the RHI is observed from 2003 to 2017 in summer. In detail, the proportion of the heat island zone is 21.74%, 22.17%, and 31.03% in the summer of 2003, 2010, and 2017, respectively (Fig. 2d). It is because the urban areas of YRDUA have increased from 3571.01 km2 to 8760.26 km2 in 2003 and 2017, respectively (Figure B1, Appendix B). Moreover, the area of the medium heat island and strong heat island increased by 41.08% and 66.40% from 2003 to 2017 (Fig. 2b,c). A gradual decreasing trend is observed for the four grades of the SRHII (2–4 °C, 4–6 °C,  > 6 °C,  > 2 °C) in winter from 2003 to 2017 (Fig. 2a–d). The area of the RHI in winter was 18,481 km2, 8640 km2, and 6280 km2 in 2003, 2010, and 2017, respectively (Fig. 2d). Vegetation coverage is low in winter and bare soil is formed after harvest. It leads to the RHI decrease in winter. The above results indicated that the SRHII became increasingly hot in summer and increasingly cold in winter and that the trend became more obvious as the SRHII increased in the ranges of 2–4 °C, 4–6 °C,  > 6 °C. However, the seasonal variation of the RHI in the nighttime is opposite to that in the daytime. From spring to summer and then to winter, the area of the RHI decreases first and then increases, and it falls in the lowest value in summer (Fig. 2e–g). For example, the area of RHI is 19,209 km2, 5659 km2, 34,621 km2, and 38,596 km2 in spring, summer, autumn, and winter in 2017, respectively (Fig. 2h). The annual average of RHI regular increases, with values of 17,510 km2, 20,042 km2, and 20,097 km2 in 2003, 2010, and 2017, respectively (Fig. 2h).Figure 2Seasonal and inter-annual variations of the SRHII during the daytime (a–d) and nighttime (e–h) of the YRDUA.Full size imageRelationship between the SRHII and influencing factorsResults showed surface biophysical factors have a higher correlation with SRHII than socio-economic factors and climate factors in the day and night. NDBI and EVI have a stronger effect on SRHII than other biophysical factors in the day. NDBI showed a significant positive correlation with SRHII, while EVI showed a negative correlation with SRHII. In detail, NDBI (r = 0.567, p  autumn  > winter. The dominant influencing factor was the MNDWI in spring, autumn and winter, while EVI had the largest contribution in summer at night. More

Overview of the study areaThe experimental algae P. parvum was collected from the fishponds in Dawukou, Ningxia, China. Algal water samples were filtered by medium size filter paper and centrifuged, and cultured with F/2 culture medium in the following environmental conditions for 5 days; light intensity of 5000 lx, light/dark ratio of 12 h: 12 h, water temperature of 18.5 ± 0.5 ℃, pH of 8.5 ± 0.1 and salinity of 1.2 ± 0.1 mg/l20,21. The plate separation method was used to separate and purify the cultured algae22. After microscopy, the colony of pure algal cells were transferred to different volumes of triangular glass bottles which contains sterilized F/2 medium for expansion culture. Algae P. parvum propagates vegetatively by cell division, the cell density of algae increases exponentially during the process of propagation thus requires more space. To accommodate this increasing space requirement, different sizes of the triangular glass bottles were used as 50 ml, 250 ml and 10 l. The expansion cultures were maintained in the environmental conditions similar to the initial culture. The algal cells were used for the experiment when they reach the logarithmic growth stage (the logarithmic growth stage was reached in 10 days).Data collection and experimentationThe water sample from the 10 L expansion culture of P. parvum was collected. The initial nutrient concentrations and environmental factors were determined using appropriate methods and equipment in the laboratory. The initial nutrient concentrations and environmental conditions of the algae culture used in this experiment is presented in Table 1.Table 1 Initial nutrient concentrations and environmental conditions of algae culture used in the experiment.Full size tableExperimental factors and their levels for each nutrient concentrations and environmental factors were designed based on the above reference as shown in Table 2. We have designed eight levels for environmental factors (i.e., water temperature, pH and salinity) and ten levels for nutrient concentrations (i.e., nitrogen, phosphorous, silicon and iron).

1.

Evaluation of the effects of environmental factors on the growth of P. parvum

Table 2 Experimental factors and their levels designed for the experiment.Full size tableTo study the effects of environmental factors on the growth of P. parvum, water temperature, pH and salinity were used as the experimental factors by adopting the uniform design23,24,25 of three factors and eight levels as shown in Table 3.Table 3 Combination of environmental factors used for the different levels in the uniform design.Full size tableA 250 ml triangular glass bottle was used to implement each level of the above experiment with three replicates for each level (total of 24 bottles). The algae culture was allowed to grow in F/2 culture medium in the nutrient solution of 100 ml with an inoculation ratio of 1:10 (V/V). These bottles were kept in the light intensity of 5000 lx with light/dark ratio of 12 h: 12 h, while maintaining all other growth conditions to meet the experimental design requirements. The nutrient concentrations of N, P, Si and Fe were maintained at the level of initial concentrations (Table 1). Inoculated algae were cultured in a shaker for 10 days until it reaches its logarithmic growth stage and the growth rate was quantified.

2.

Evaluation of the effects of nutrient concentrations on the growth of P. parvum

To study the effects of nutrient concentrations on the growth of P. parvum, nitrogen, phosphorus, silicon and iron were used as experimental factors by adopting the uniform design5,26 of four factors and ten levels as shown in Table 4. The culture medium was prepared with sodium nitrate (NaNO3) as the nitrogen source, monosodium phosphate (NaH2PO4) as the phosphorus source, sodium metasilicate (Na2SiO3) as the silicon source, and ferric citrate (FeC6H5O7) as a source iron to obtain the appropriate concentrations of nitrogen, phosphorous, silicon and iron as designed for this experiment (Table 2).Table 4 Combination of nutrient concentration used for the different levels in the uniform design.Full size tableA 250 ml triangular glass bottle was used to implement each level of the above experiment with three replicates for each level (total of 30 bottles). The algae culture was allowed to grow in F/2 culture medium with a volume of 100 ml and an inoculation ratio of 1:10 (V/V). These bottles were kept in the light intensity of 5000 lx, light/dark ratio of 12 h: 12 h, water temperature of 18.5 ± 0.5 ℃, pH of 8.5 ± 0.1 and salinity of 1.2 ± 0.1 mg/l. Inoculated algae were cultured in a shaker for 10 days until it reaches its logarithmic growth stage and the biomass density was quantified.

3.

Determination of the growth rate of P. parvum

The algal cell density of the culture of each experimental level was measured using a 0.1 ml count plate under an optical microscope (Leica biological microscope DM1000, Leica Corporation, Oskar-Barnack-Straße, Germany) both at the beginning of the experiment and following 10 days of incubation period as the growth of the algae can reach its logarithmic growth stage at 10 days. Based on the algal cell density measurement, biomass density was calculated using the following formula (Eq. 1) described by Wei and Zhang;$$ Growth;rate;left( K right) = 3.322 times left( {log (N_{t} ) – log left( {N_{0} } right)} right)/left( {t – t_{0} } right) $$
(1)
where t is the duration of the experiment in days, N0 is the initial cell density (cell/ml) at the beginning of the experiment, and Nt is the cell density (cell/ml) at the end of day t of the experiment.Data analysis and results

1.

Establishment of the regression model between environmental factors and the growth rate

The growth rate of P. parvum under different levels of environmental factors are shown in Table 5, and the growth curve with time is shown in Fig. 1.Table 5 Growth rates of Prymnesium parvum under the different levels of environmental factors in the uniform design.Full size tableFigure 1The growth curve of P. parvum with time under different environmental factor levels.Full size imageIn multiple quadratic stepwise regression analysis, water temperature (X1), pH (X2) and salinity (X3) were taken as independent variables, and the growth rate (Y) was taken as the dependent variable. From this analysis a quadratic polynomial regression equation (Eq. 2) was developed as follows:$$ Y = – 11.0371 + 0.0682X_{1} + 2.5559X_{2} + 0.7953X_{3} – 0.0019X_{1} ^{2} – 0.1523{text{ }}X_{2} ^{2} – 0.3223{text{ }}X_{3} ^{2} $$
(2)
Correlation coefficient (R) of the above equation was 0.9994 and probability (P) of the regression equation was 0.025 (p  X3  > X1. Thus, the contribution of pH  > salinity  > water temperature on the growth rate of P. parvum.

2.

Evaluation of the effect of environmental factors on the growth rate of P. parvum

The environmental conditions that would result in the maximum growth rate of P. parvum were determined by optimizing the regression equation (Eq. 2). The following simple regression models (Eqs. 3–5) of multiple quadratic stepwise regression analyses reveal the relationships between individual environmental factors and the growth rate. These models were obtained by dimensionality reduction analysis in which the other factors were maintained at optimal levels.$$ X_{{1WT}} :;Yleft( {X_{1} } right) = 0.1768 + 0.0682X_{1} – 0.0019X_{1} ^{2} $$
(3)
$$ X_{{2pH}} :;Yleft( {X_{2} } right) = – 9.9345 + 2.5559X_{2} – 0.1523{text{ }}X_{2} ^{2} $$
(4)
$$ X_{{3salinity}} :;Yleft( {X_{3} } right) = 0.2982 + 0.7953X_{3} – 0.3223{text{ }}X_{3} ^{2} $$
(5)
The influence curves of each environmental factor on growth rate of P. parvum are shown in Fig. 2. The behavior of the curves is similar where the growth rate increases initially, then reaches a theoretical maximum and finally declines with increasing level of each environmental factor. Accordingly, P. parvum reaches its theoretical maximum growth rate (0.789) when the water temperature, pH and salinity is 18.11 ℃, 8.39 and 1.23‰, respectively. Therefore, Fig. 2 can be considered as the growth model of P. parvum as affected each of the respective environmental factors.

3.

Establishment of regression model between nutrient concentrations and the growth rate

Figure 2The growth rate of P. parvum as affected by the water temperature (a), pH (b) and salinity (c).Full size imageThe growth rates of P. parvum under the different levels of nutrient concentrations are shown in Table 7, and the growth curve with time is shown in Fig. 3.Table 7 Growth rate of Prymnesium parvum under the different levels of nutrient concentration in the uniform design.Full size tableFigure 3The growth curve of P. parvum with time under different nutrient concentrations factor levels.Full size imageA quadratic polynomial regression equation (Eq. 6) was generated using N (Xi), P (Xii), Si (Xiii) and Fe (Xiv) as independent variables and the growth rate (Y′) as the dependent variable by using multiple quadratic stepwise regression analysis as follows:$$ Y^{prime } = – 1.856686 + 1.371680X_{i} + 0.390361X_{{ii}} + 0.150656X_{{iii}} + 0.587990X_{{iv}} – {text{ }}0.2011178X_{i} ^{2} – 0.186640{text{ }}X_{{ii}} ^{2} – 0.108764{text{ }}X_{{iii}} ^{2} – 0.550523{text{ }}X_{{iv}} ^{2} $$
(6)
Correlation coefficient (R) of the above equation was 0.9994 and probability (P) of the regression equation was 0.035 ( Xii  > Xiv  > Xiii. Therefore, the contribution of nitrogen  > phosphorous  > iron  > silicon for the growth of P. parvum.

4.

Evaluation of the effect of nutrient concentrations on the growth rate of P. parvum

Multifactor square stepwise regression model was used to analyze the influence of individual nutrient concentration following the dimensionality reduction. To evaluate the influence of individual nutrient concentration on the growth rate, following sub-models (Eqs. 7–10) were developed by fixing other factors at the optimal level.$$ X_{i} nitrogen:Y^{prime } (X_{i} ) = – 1.4432 + 1.3717X_{i} – 0.2012X_{i} ^{2} $$
(7)
$$ X_{{ii}} phosphorus:Y^{prime } (X_{{ii}} ) = 0.6916 + 0.3904X_{{ii}} – 0.1866X_{{ii}} ^{2} $$
(8)
$$ X_{{iii}} silicon:Y^{prime } (X_{{iii}} ) = 0.8436 + 0.1507X_{{iii}} – 0.1088X_{{iii}} ^{2} $$
(9)
$$ X_{{iv}} iron:Y^{prime } (X_{{iv}} ) = 0.7388 + 0.5880X_{{iv}} – 0.5505X_{{iv}} ^{2} $$
(10)
The influence curves of each nutrients on growth rate of P. parvum are shown in Fig. 4. The behavior of the curves shows an initial increase of the growth rate, then the growth rate reaches a theoretical maximum and finally declines with increasing level of concentrations of each nutrient. Accordingly, P. parvum reaches its theoretical maximum growth rate (0.896) when the concentration of nitrogen, phosphorous, silicon and iron is 3.41, 1.05, 0.69, 0.53 mgl−1, respectively. Therefore, Fig. 4 may be considered as the growth model of P. parvum as affected each of the respective nutrients.Figure 4The growth rate of P. parvum as affected by nitrogen (a), phosphorus (b), silicon (c) and iron (d).Full size image More

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