CN109858132B - Blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm - Google Patents
Blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm Download PDFInfo
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Abstract
The invention provides a blue algae bloom outbreak early warning method based on a mutation theory and an improved cuckoo algorithm, and belongs to the technical field of water environment prediction early warning. Modeling nonlinear kinetics of blue algae growth; optimizing parameters in the blue algae growth nonlinear dynamics model by using an improved cuckoo algorithm and a Dragon-Gregory tower method; converting the nonlinear kinetic model of blue algae growth into a point mutation theoretical model; determining a critical point of water bloom outbreak according to the divergence and collection of the point mutation theoretical model; and finally, judging the outbreak of cyanobacteria bloom, and carrying out early warning. The invention adopts a method combining a numerical method and an intelligent algorithm, makes partial improvement on iterative optimization of the intelligent algorithm, accelerates the optimizing convergence speed, ensures that parameters are not simple isotropic random walk, ensures that a nonlinear dynamics model of blue algae growth has universality and practicability, can predict water bloom outbreak in time, and provides basis for decision departments to formulate water bloom prevention and control countermeasures.
Description
Technical Field
The invention relates to a forecasting and early-warning method for cyanobacteria bloom, and belongs to the technical field of water environment forecasting and early-warning. Specifically, according to the growth dynamics process of blue algae, the peak mutation theory is utilized to carry out mathematical modeling, and an improved cuckoo search algorithm is used to carry out parameter calibration, so that a new solution is provided for water bloom prediction and early warning.
Background
The water quality of the water body comprises water indexes such as physical factors, chemical elements, biological characteristics and the like of the water, and is an important reference factor for measuring the usability of the water body to society. Eutrophication is a phenomenon that primary producers such as blue algae and other phytoplankton propagate rapidly due to excessive content of plant nutrient substances such as nitrogen, phosphorus and the like, and further the water quality is spoiled. The main characteristics of the water-based nutrient solution are water bloom outbreak, water eutrophication and the like. Eutrophication of the water body can enrich nutrient salts beneficial to the growth of algae, and can further cause the water bloom outbreak phenomenon of the algae so as to destroy the ecological balance of the water environment. Therefore, the blue algae bloom outbreak is predicted and early-warned scientifically and efficiently through the research on the related aspects, and the method has very important ecological and social significance.
At present, a data driving model and a mechanism model of water quality of a water body can be used for predicting cyanobacterial bloom.
The data driving model is mainly established by intelligent algorithms such as a support vector machine, an artificial neural network, a decision tree and the like, such as a time sequence model, a neural network model, a genetic algorithm model, a support vector machine classification model and the like. Although the value of the future time characterization factor can be predicted by the characterization factor and the influence factor of the current time and the history time to a certain extent, the existing intelligent model cannot well extract the data characteristics, and the data driving model only simply performs sample analysis from the observed data, so that the mechanism and theoretical support are lacked, and the final prediction accuracy is limited due to poor generalization capability, local minimum problem and the like.
The mechanism mathematical model of the water body can be divided into a single nutrient salt model, a double nutrient salt model, a multi-factor coupling model and the like according to types. The mathematical expression is also expressed as differential equation or differential equation of mutual nested coupling. However, the mechanism model of the water quality of the water body is complex, more parameter variables are involved, necessary experimental support is lacked, and the problem of insufficient research on random influence of uncertainty is caused, so that a final simulation result is inaccurate.
Aiming at the defects of the water quality data driving model and the mechanism model, in order to establish a water bloom prediction early warning method with high prediction precision and high optimizing speed, the mechanism driving modeling and the data driving modeling are combined, a nonlinear dynamics model capable of reflecting the sudden characteristics of the cyanobacteria water bloom outbreak process is established, and the method is combined with an intelligent algorithm with higher convergence speed, so that the water bloom outbreak prediction early warning method which is applicable to various influencing factors and can reflect the actual sudden characteristics of the water bloom outbreak is established.
Disclosure of Invention
The invention aims to solve the problems that the existing water bloom predicting and early-warning method cannot fully reflect the sudden onset of water bloom, the model predicting precision is low, the intelligent algorithm late convergence speed is low and the like.
According to the invention, the blue algae growth nonlinear dynamics model with the nitrogen and phosphorus double nutrient salts is built by utilizing the measured data such as the nitrogen nutrient salt concentration, the phosphorus nutrient salt concentration, the chlorophyll concentration, the water temperature, the illumination and the like acquired in the actual blue algae growth process. And (3) carrying out parameter calibration on the constructed blue algae growth nonlinear dynamics model by adopting an improved cuckoo search algorithm and a Dragon lattice tower method. And establishing a mathematical correspondence relation between chlorophyll concentration and total phosphorus concentration, bringing the relation into a blue-green algae growth nonlinear dynamics model, and establishing a water bloom sudden-onset point mutation model based on a blue-green algae growth nonlinear dynamics mechanism by utilizing mathematical means such as Taylor expansion and the like and combining a mutation theory. According to the bifurcation set of the point mutation, the condition of the bloom is determined, and then the specific measured data is combined, so that a novel method for early warning the bloom of blue algae is finally constructed.
The blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo search algorithm provided by the invention comprises the following five steps:
step one, modeling nonlinear dynamics of blue algae growth;
the water bloom outbreak is a phenomenon that the oxygen content in water is consumed excessively by the mass propagation of blue algae in a short time and the water environment is rapidly deteriorated due to the mutual influence of various coupling factors. Among these various factors of mutual coupling, three factors, temperature, total nitrogen concentration, and total phosphorus concentration, play a major role. Therefore, a nonlinear kinetic model of blue algae growth is established around the three main factors. Wherein chlorophyll concentration is used as a characterizing factor reflecting cyanobacteria biomass in a body of water. The quantity of blue algae biomass is related to the influencing factors such as the water temperature, the nitrogen nutrient salt concentration, the phosphorus nutrient salt concentration and the like in the water, so that a nonlinear dynamics model of chlorophyll concentration and the temperature, the nitrogen nutrient salt concentration and the phosphorus nutrient salt concentration is established.
Step two, optimizing and calibrating parameters of a nonlinear dynamics model of blue algae growth;
and (3) carrying out parameter optimization calibration on the nonlinear dynamics model of blue algae growth established in the step one by combining with the water body data acquired in practice. The iterative optimization characteristic of the cuckoo algorithm in the intelligent algorithm is mainly utilized, and the Longguge tower method of the numerical algorithm is combined, so that the predicted value is compared with the actual observed value, and the error is continuously iteratively reduced to be within an acceptable threshold range. And finally determining the model parameters of the nonlinear dynamics of the blue algae growth.
Step three, converting the nonlinear kinetic model of blue algae growth into a point mutation theoretical model;
and (3) calculating a fitting curve of chlorophyll concentration-total phosphorus concentration by combining actual data, replacing the phosphorus nutrient salt concentration in the blue algae growth nonlinear dynamics model with a chlorophyll concentration value, performing Taylor expansion when the chlorophyll concentration is 0 through simplified calculation, and ignoring a higher term in the chlorophyll concentration value, thereby obtaining the point mutation theoretical model.
Step four, determining a critical point of water bloom outbreak according to the divergence and collection of the sharp point mutation theoretical model;
aiming at the established point mutation theoretical model, the mutation behavior of blue algae growing near the critical point of the system needs to be further researched, and according to a bifurcation set formula in the point mutation theory, actual data are combined and substituted into calculation; if the value of the bifurcation set is near 0, the critical point of the bloom of blue algae can be determined. The position near 0 means that the value of the divergence set is in the range (-0.499 to 0.499).
Fifthly, judging and early warning the outbreak of cyanobacteria bloom;
and judging that the current blue algae growth dynamic system is in an unstable state according to the critical point of water bloom outbreak obtained by calculation in the step four. And then, judging whether the blue algae grows and whether the water bloom outbreak occurs or not according to the predicted value of the chlorophyll concentration, and making early warning information.
The invention has the advantages that:
1. the invention establishes a nonlinear dynamics model of blue algae growth, has complete mathematical theoretical support, and each parameter variable has actual physical causal relationship, not only simple data correlation analysis. And the description of the time-varying system for blue algae growth is more practical, so that the description is more accurate by considering various influencing factors such as nitrogen and phosphorus nutritive salts in the actual water body, the water body temperature and the like.
2. According to the invention, when parameter optimization rate timing is carried out on the blue algae growth nonlinear dynamics model, a numerical method is combined with an intelligent algorithm, partial improvement is made on iterative optimization of the intelligent algorithm, the optimization convergence speed is accelerated, parameters are not simple isotropic random walk, and the blue algae growth nonlinear dynamics model has universality and practicability.
3. After the nonlinear kinetic model of blue algae growth is constructed, the blue algae growth nonlinear kinetic model is finally converted into a sharp point mutation model in a mutation theory by combining a fitting curve of measured data and utilizing mathematical means such as Taylor expansion and the like. The mutation theory is used for describing the advantages of a complex nonlinear system and reflecting the advantages to a blue algae growth nonlinear dynamics model, so that the prediction and early warning of the blue algae chlorophyll concentration are realized. The invention can utilize the curve fitted by the measured data of different water bodies to be substituted into the same sharp point mutation model, thereby having expansibility.
4. The invention uses the divergence and collection value of the sharp point mutation model as the observation value of the water bloom outbreak early warning, thereby leading the judgment condition to be visual and concise.
5. The invention provides a blue algae bloom outbreak early warning method based on nonlinear dynamics critical points of mutation theory and combined with actual water quality detection data. And researching the stability transition critical point of a nonlinear dynamics system of blue algae growth by mutation theory. Around the critical point, the value of the divergence set is around 0, indicating that the water bloom outbreak phenomenon occurs. The method can predict the sudden water quality disaster of the water bloom outbreak in time and provides basis for decision-making departments to formulate prevention and control countermeasures of the water bloom.
Drawings
FIG. 1 is a flow chart of a blue algae bloom early warning method based on mutation theory and improved cuckoo algorithm;
FIG. 2 is a flow chart for parameter optimization of the improved cuckoo algorithm in combination with the Dragon's lattice tower algorithm;
FIG. 3 is a graph showing the comparison of chlorophyll concentration measured values with predicted values;
FIG. 4 is a model diagram of the theory of sharp point mutations;
wherein M represents a balanced hypersurface in the space and is divided into a stable upper leaf, a stable lower leaf and a stable middle leaf; s represents an odd point set, namely a curve of which the curved surface is intersected with the vertical surface; b is a bifurcation set, which is a projection of the curved folded portion.
FIG. 5 is a numerical graph of a sharp point abrupt change bifurcation set;
wherein the values of the divergence set are partially negative and cannot be completely displayed in the logarithmic graph.
FIG. 6 is a graph of values of sharp point abrupt transitions around a critical point;
from the partial graph, we can see that the value of the bifurcation set is suddenly reduced to 0 at the 580 th measuring point, so we can indicate that the water environment near the date has the condition of blue algae outbreak, and the water bloom phenomenon can occur. Which also coincides with the actual observation.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
The invention provides a blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm, which is shown in a flow shown in figure 1, and comprises the following specific steps:
step one, modeling nonlinear dynamics of blue algae growth;
time is represented by t, chlorophyll concentration is A (t); the growth rate of blue algae is mu (t), and mu is the maximum growth rate; blue algae mortality is Ma (t), ma is the maximum mortality, and K represents the half-saturation constant of blue algae mortality; the ratio of Q to V is a hydrological parameter, where Q and V are expressed as water output and water volume, respectively; e, e H T (T) is the temperature, and T is the optimal temperature for blue algae growth; the real-time concentrations of total nitrogen and total phosphorus are TN (t) and TP (t), KN represents the half-saturation constant of algae absorption nitrogen salt, and KP represents the half-saturation constant of algae absorption phosphorus salt;
the nonlinear dynamics model expression of blue algae growth provided by the invention is as follows:
step two, optimizing and calibrating parameters of a nonlinear dynamics model of blue algae growth;
combining a cuckoo algorithm and a Longguge tower numerical analysis method, carrying out optimization rating on 7 parameters in a blue algae growth linear dynamics model, namely formulas (1) to (3): maximum growth rate mu of blue algae and temperature parameter e H The index H of (2), the half-saturation constant KN of algae absorbing nitrogen salt, the half-saturation constant KP of algae absorbing phosphorus salt, the maximum death rate Ma of blue algae, and the half-saturation constant of algae death rateK and hydrologic parameters Q/V. Hereinafter, C is used respectively 1 、C 2 ...C 7 To represent.
In order to reduce unnecessary redundant searching in the late iteration stage of the cuckoo algorithm, quicken the convergence speed in the late iteration stage of the algorithm and improve the optimizing efficiency, two fixed values in the cuckoo algorithm are calculated on the basis of the original standard cuckoo searching algorithm: the discovery probability pa and the step length theta of the random walk are dynamically improved, so that the two values can be changed along with continuous iteration penetration of the cuckoo search algorithm, and the parameter calibration optimizing process is quickened. Based on the improvement, relevant measured data are substituted, and parameters are optimized and calibrated.
In order to prevent the excessive difference between the data from influencing the final parameter optimizing result, the data of each sampling point are grouped according to the numerical value difference between the nitrogen-phosphorus ratios from the data statistics perspective. Then obtaining a plurality of groups of parameters C applicable to different nitrogen-phosphorus ratios through optimizing calibration 1 、C 2 ...C 7 。
Preprocessing the sampled data (the preprocessing refers to taking three times standard deviation of measured data as a limit, namely removing the data as abnormal data when the data exceeds the range), and carrying out parameter optimization on the preprocessed data by combining a cuckoo algorithm and a Dragon-Gregorian tower method, wherein a flow chart is shown in fig. 2, and the specific steps are as follows:
(1) Setting an objective function:
the left side of the equation is the chlorophyll concentration change rate, and the right side of the equation contains 7 unknown parameters C to be optimized 1 、C 2 ...C 7 The threshold Tol is set to 3%.
(2) Setting an initial population x (i) and population number n, substituting the initial population x (i) and population number n into actual measurement data, and operating a formula (4) to generate a group of current optimal solutions D1.
(3) Each population is subjected to a markov chain-based fly-by-fly operation, i.e., using random walk to shuffle the population, regenerating a new set of optimal solutions D2.
(4) And respectively calculating the predicted values fmin and fnew of the concentration of the patchouli corresponding to the two groups of optimal solutions D1 and D2 by using a Dragon library tower method.
Formulas (5) to (9) are related formulas of the fourth-order Dragon-George tower used in the invention, wherein the step h is set to 1 (h is used as a coefficient in the formulas, and no table is omitted).
Root mean square errors between the predicted values fmin, fnew and the measured value FR are calculated and denoted as w and v, respectively.
In the above formula, K1 represents an initial slope, K2 and K3 each represent a slope at the middle section, but the values are different, and K4 is an end slope.
(5) If the root mean square error v between fmin and the measured value is smaller than the root mean square error w of fmin, replacing the data value in fmin with fmin, namely letting fmin=fmew, and marking the current minimum root mean square error as v, and updating and eliminating the inferior population according to a certain discovery probability pa; if the root mean square error v between fnew and the measured value is not smaller than the root mean square error w of fmin, the inferior population is updated and eliminated directly according to the discovery probability pa.
(6) The counter n_iter+n is updated and a new cycle is entered, N being the population number. And (3) jumping out of the cycle until the current minimum root mean square error v meets the set threshold value Tol, outputting an optimal result, performing subsequent operations such as visualization processing and the like.
Step three, combining a Chla-TP curve, and converting a blue algae growth nonlinear dynamics model into a cusp mutation theoretical model;
(1) Firstly, according to the measured data, drawing a Chla-TP curve of a total phosphorus concentration and a chlorophyll concentration curve, and establishing a mathematical corresponding relation between the total phosphorus concentration and the chlorophyll concentration.
TP(t)＝a×A(t)+b (10)
Wherein TP (t) and A (t) represent the total phosphorus concentration and chlorophyll concentration of the measured data, respectively, and a and b represent the undetermined coefficients. Substituting the formula (10) into the formulas (2) and (3) to replace TP (t) in the formulas, substituting the replaced formulas (2) and (3) into the formula (1), and finally obtaining the formula of the chlorophyll concentration change rate as follows:
at this time, the total phosphorus concentration TP (t) is replaced by the chlorophyll concentration A (t). Therefore, the change rate of chlorophyll concentration in the formula is only related to the values of the water temperature T (T), the total nitrogen concentration TN (T), and the like.
The equation (11) is subjected to taylor expansion of 3 rd order at a (t) =0 (when a (t) →0 on the right of the equation, the left of the equation is also treated with approximately 0), the final result of taylor expansion is:
ignoring the higher terms of A (t), such as quadratic terms, quartic terms, sextuple terms, and the like, and only reserving the primary terms and the tertiary terms of A (t) in the formula (12), and finishing to obtain:
further finishing to obtain the following formula:
for convenience of representation, we set the coefficient of the first order term a (t) to U and the constant term to V. Namely:
therefore, equation (14) can be simplified as:
A 3 (t)+U·A(t)+V＝0 (17)
from the relevant mathematical information background of sharp point mutations in mutation theory, we can see that system jumps sometimes occur when a system is gradually changed along a smooth folding curved surface M. As shown in fig. 4: when the system moves from top to bottom along the track L1 from the smooth folding curved surface, a phenomenon that the system jumps directly to bottom along the track L2 may occur. The smooth folding curved surface M is called a balanced hypersurface, the intersection line of the curved surface folding part and the vertical plane is a singular point set S, and the equation is as follows:
M：4x 3 +2ux+v＝0 (18)
S：12x 2 +2u＝0 (19)
according to the point mutation theory, a bifurcation equation can be obtained by combining equations of the balanced hypersurface M and the odd point set S, and the bifurcation equation can be used for judging the stability of a system. Namely equation (18) and equation (19) to get the bifurcation set:
8u 3 +27v 2 ＝0 (20)
comparing the Taylor expansion (17) after simplifying the arrangement with the equilibrium hypersurface formula (18) of the point mutation theory, the expressions of the bifurcating and centralizing regularization factor u and the splitting factor v can be known:
so far, the invention has related specific variables such as chlorophyll concentration A (T), water temperature T (T), total nitrogen TN (T) and the like with a bifurcation equation in the point mutation theory through a regularization factor u and a division factor v. It is then necessary to calculate the value of the corresponding bifurcation set based on the substitution of the related data. Let B have the value:
B＝8u 3 +27v 2 (23)
step four, determining a critical point of water bloom outbreak according to the divergence and collection of the sharp point mutation theoretical model;
the outbreak of cyanobacterial bloom is an extremely complex high-dimensional dynamic system behavior. The method for determining nonlinear dynamics condition of cyanobacteria bloom outbreak comprises the following steps of optimizing and determining related parameters C according to the previous step two 1 ，C 2 ...C 7 And coefficients a and b determined by the fitting relation between the total phosphorus concentration TP (T) and the chlorophyll concentration A (T), and substituting the measured data of the chlorophyll concentration A (T), the water temperature T (T), the total nitrogen TN (T) and the like in the blue algae growth process into a bifurcation equation (20) together to calculate the value of the bifurcation, thereby determining the critical point of the onset of the blue algae bloom. The invention also provides the formula condition for meeting the critical point.
(1) Firstly, substituting actual measurement data of total phosphorus concentration TP (t) and chlorophyll concentration A (t) into a formula (11) to solve values of undetermined coefficients a and b;
(2) The measured data such as chlorophyll concentration A (T), water temperature T (T), total phosphorus concentration TP (T), total nitrogen concentration TN (T) and the like acquired by each sampling point are classified according to the difference of nitrogen-phosphorus ratio values, and the parameters C corresponding to the measured data are combined 1 ，C 2 ...C 7 Substituting the parameters u and v of the divergence set to obtain the following formula:
(3) The measured data are substituted into the above two formulas (24) and (25), and the numerical values of the respective groups u and v are calculated.
(4) Substituting u and v into a bifurcation equation formula (23) of the cusp mutation theory to calculate a numerical value B of the bifurcation equation. If the calculated numerical value B of the bifurcation equation is approximately equal to 0, the comprehensive effect influence of the water temperature T (T), the total phosphorus concentration TP (T) and the total nitrogen concentration TN (T) in the current blue algae water environment is indicated, and the critical point of the occurrence of bloom of the blue algae is reached. The value B being equal to about 0 means that the index value B is about 0, typically in the (-0.499 to 0.499) range.
Fifthly, judging and early warning the outbreak of cyanobacteria bloom;
if the numerical value B of the bifurcation equation of the nonlinear dynamics model of blue-green algae growth is approximately equal to 0 according to the point mutation theory, judging that the current water environment of blue-green algae growth reaches the critical point of water bloom outbreak, and the risk of water bloom outbreak exists. And combining with the predicted value of the chlorophyll concentration, when the predicted chlorophyll concentration is larger than the concentration threshold value of the normal water body, the behavior that water bloom outbreak occurs is indicated, and early warning information should be made immediately.
Example 1:
taking data of a monitoring site 2009-2013 of a natural lake in China as an example, the method provided by the invention is adopted to predict and early warn blue algae bloom. The data mainly comprise chlorophyll concentration A (T), water temperature T (T), total phosphorus concentration TP (T), total nitrogen concentration TN (T) and the like.
Step one, modeling nonlinear dynamics of blue algae growth;
see formulas (1) - (3) in the detailed description.
Step two, optimizing and calibrating parameters of a nonlinear dynamics model of blue algae growth;
in order to prevent the excessive difference between data from influencing the final parameter optimizing result, the actual measurement data between 2009 and 2012 are subjected to data grouping by taking 10 as a unit span according to the data difference of the nitrogen-phosphorus ratio. And (3) calibrating the parameters in the formula (4) by utilizing the parameter optimization calibration method combining the numerical algorithm and the intelligent optimization algorithm, wherein the calibration result is shown in the table 1.
TABLE 1 non-linear kinetic model parameter calibration results for blue algae growth
And classifying the actual measurement data in 2013 according to the same nitrogen-phosphorus ratio interval, and calculating to obtain prediction fitting data by using a Dragon-Gregory tower method by combining the corresponding rated parameter value of the interval. The result of comparing the fitting data with the actual data from fig. 3 is more desirable than the result of calibrating the visible parameters.
Step three, converting the nonlinear kinetic model of blue algae growth into a point mutation theoretical model;
see formulas (11) - (22) in the detailed description.
Step four, determining a critical point of water bloom outbreak according to the divergence and collection of the sharp point mutation theoretical model;
and substituting the measured data into a fitting relation between the total phosphorus concentration and the chlorophyll concentration, and calculating the numerical values of a and b to be-0.0373 and 0.0609 respectively by using a least square method.
The actual measurement data in 2013 is combined with the corresponding parameter values, the values of u and v corresponding to each point are calculated by formulas (24) and (25), and then the value of the bifurcation set corresponding to each point is calculated by a calculation formula (23) of the value B of the bifurcation set equation. Because the computing mode of the divergence set is the operation among indexes, the numerical range of the result is large (0 to 10 16 ) To facilitate viewing values at low orders of magnitude, a logarithmic coordinate system is employed, as shown in fig. 5.
Since the value of the divergence set corresponding to the partial data point is a negative number and cannot be completely displayed in the logarithmic coordinate system, the absolute value and the value of the divergence set at the local display critical point are utilized to obtain fig. 6.
The data points at 2013, 2 and 4 were calculated to have a value of 0.1907 approaching 0 for the bifurcation. From this, it is known that the cyanobacteria growth dynamics system is in an unstable state at this point, i.e., the critical point of onset of bloom is reached.
Fifthly, judging and early warning the outbreak of cyanobacteria bloom;
according to the obtained predictive fitting data, the chlorophyll concentration change range at 580-600 points of the corresponding data points around 2 months and 4 days is found to be 16.4-33.2, which is far higher than a constant value 6.097. The detection time period shows that the blue algae growth dynamics system is in an unstable state, reaches a critical point and generates the phenomenon of water bloom outbreak.
The nonlinear dynamics model of blue algae growth established by the invention not only can be used for predicting blue algae biomass in water environment, namely the concentration of chlorophyll, but also can be used for judging whether the phenomenon of water bloom outbreak occurs. The prediction and early warning of the water bloom outbreak are realized, and a new scheme is provided for the water bloom prevention work.
Claims (1)
1. A blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm is characterized by comprising the following steps:
modeling the nonlinear dynamics of blue algae growth;
establishing a blue algae growth nonlinear dynamics model of chlorophyll concentration, temperature, nitrogen nutrient salt concentration and phosphorus nutrient salt concentration;
the built nonlinear kinetic model of blue algae growth is as follows:
t represents time;
a (t) is chlorophyll concentration;
mu (t) is the growth rate of blue algae;
ma (t) is blue algae mortality;
the ratio of Q and eta is a hydrological parameter, namely Q/eta, wherein Q and eta are respectively expressed as water output and water capacity;
mu is the maximum growth rate of blue algae;
e H is a temperature parameter;
t is the optimal temperature for blue algae growth;
t (T) is the temperature;
TN (t) is the real-time concentration of total nitrogen;
KN is the half-saturation constant of the absorption of nitrogen salt by algae;
TP (t) is the real-time concentration of total phosphorus;
KP is the half-saturation constant of the algae to absorb phosphorus salt;
ma is the maximum mortality rate of blue algae;
k is the half-saturation constant of blue algae mortality;
step two, optimizing and calibrating the nonlinear dynamics model parameters of blue algae growth;
data of all sampling points are grouped according to the numerical value difference between the nitrogen and phosphorus ratios; optimizing parameters in a blue algae growth nonlinear dynamics model by using an improved cuckoo algorithm and a Longguge tower method to obtain a plurality of groups of parameters applicable to different nitrogen-phosphorus ratios, and respectively using C 1 、C 2 、C 3 、C 4 、C 5 、C 6 And C 7 A representation;
C 1 the maximum growth rate of the blue algae is replaced;
C 2 an index of temperature parameters that are alternatives;
C 3 the half-saturation constant of the absorption nitrogen salt for the substituted algae;
C 4 the half-saturation constant of the absorption of phosphorus salt for the substituted algae;
C 5 maximum mortality of blue algae for substitution;
C 6 a half-saturation constant that is a surrogate for blue algae mortality;
C 7 is a substituted hydrologic parameter;
the improved cuckoo algorithm and the Longguge tower method are utilized to carry out optimization calibration on parameters in a nonlinear dynamics model of blue algae growth, and the method comprises the following specific steps:
(1) Setting an objective function:
the left side of the equation in the formula (4) is the chlorophyll concentration change rate, and the right side comprises 7 unknown parameters C to be optimized 1 、C 2 、C 3 、C 4 、C 5 、C 6 And C 7 Setting the threshold Tol to 3%;
(2) Setting an initial population x (i) and a population number n, substituting the initial population x (i) and the population number n into actual measurement data, and operating a formula (4) to generate a group of current optimal solutions D1;
(3) Carrying out Lewy flight operation based on Markov chains on each population, namely using random walk to disturb the population, and regenerating a new set of optimal solutions D2;
(4) Calculating the predicted values fmin and fnew of the concentration of the patchouli corresponding to the two groups of optimal solutions D1 and D2 respectively by using a Dragon library tower method;
the related formulas of the Dragon-Gregory tower method are as follows:
the step length is set to be 1, root mean square errors between the predicted values fmin and fnew and the actual measured value FR are calculated respectively and are recorded as w and sigma;
K 1 representing an initial slope;
K 2 、K 3 representing the slope at the two intermediate segments;
K 4 is the end point slope;
(5) If the root mean square error sigma between fmin and the measured value is smaller than the root mean square error w of fmin, replacing the data value in fmin with fmin, namely letting fmin=fmew, and recording the current minimum root mean square error as sigma, and updating and eliminating the inferior population according to a certain discovery probability pa; if the root mean square error sigma between fnew and the measured value is not smaller than the root mean square error w of fmin, the inferior population is updated and eliminated directly according to the discovery probability pa;
(6) Updating the counter N_iter and entering a new loop; until the current minimum root mean square error sigma meets a set threshold value Tol, jumping out of the cycle, outputting an optimal result and performing visual processing operation;
step three, converting the nonlinear kinetic model of blue algae growth into a point mutation theoretical model by combining a chlorophyll concentration-total phosphorus concentration curve;
(1) Firstly, drawing a chlorophyll concentration and total phosphorus concentration curve A (t) -TP (t) curve according to measured data, and establishing a mathematical corresponding relation between the total phosphorus concentration and the chlorophyll concentration:
TP(t)＝a×A(t)+b (10)
wherein TP (t) and A (t) respectively represent the total phosphorus concentration and chlorophyll concentration of measured data, and a and b represent coefficients to be determined; substituting the formula (10) into the formulas (2) and (3) to replace TP (t) in the formulas, substituting the replaced formulas (2) and (3) into the formula (1), and finally obtaining the formula of the chlorophyll concentration change rate as follows:
at this time, since the total phosphorus concentration TP (t) is replaced by the chlorophyll concentration a (t); therefore, the change rate of chlorophyll concentration in the formula (11) is only related to the water temperature T (T), the total nitrogen concentration TN (T) value;
(2) The 3 rd order taylor expansion is performed at a (t) =0 with equation (11), and the final result of taylor expansion is:
ignoring the A (t) high-order items, only preserving one-time items and three-time items, and finishing to obtain the following steps:
further finishing to obtain the following formula:
for convenience of representation, the coefficient of the first order term a (t) is set to U, and the constant term is set to V; namely:
therefore, equation (14) is simplified as:
A 3 (t)+U·A(t)+V＝0 (17)
according to the point mutation theory, a bifurcation equation is obtained through the equation of the simultaneous equilibrium hypersurface M and the odd point set S, and is used for judging the stability of the system; namely equation (18) and equation (19) to get the bifurcation set:
M：4x 3 +2Ux+V＝0 (18)
S：12x 2 +2U＝0 (19)
8U 3 +27V 2 ＝0 (20)
comparing the simplified Taylor expansion (17) and the balanced hypersurface formula (18) of the point mutation theory, and branching and centralizing the expressions of the regularization factor u and the splitting factor v:
so far, specific variables of chlorophyll concentration A (T), water temperature T (T) and total nitrogen TN (T) are related with a bifurcation equation in a cusp mutation theory through a regularization factor u and a splitting factor v, so that:
B＝8u 3 +27v 2 (23)
step four, determining a critical point of water bloom outbreak according to the divergence and collection of the sharp point mutation theoretical model;
according to the optimizing rate of the second step, the obtained related parameter C 1 ，C 2 ，C 3 ，C 4 ，C 5 ，C 6 ，C 7 And the peak mutation theoretical model coefficients a and b determined by the fitting relation between the total phosphorus concentration TP (T) and the chlorophyll concentration A (T) are substituted into a bifurcation equation together by combining with the actual measurement data of the chlorophyll concentration A (T), the water temperature T (T) and the total nitrogen TN (T) in the blue algae growth process, and the numerical value of the bifurcation equation is calculated, and if the numerical value of the bifurcation equation is near 0, the critical point of blue algae bloom is determined;
the critical point satisfies the condition of the formula,
(1) Firstly, substituting actual measurement data of total phosphorus concentration TP (t) and chlorophyll concentration A (t) into a formula (11) to solve values of undetermined coefficients a and b;
(2) The measured data of chlorophyll concentration A (T), water temperature T (T), total phosphorus concentration TP (T) and total nitrogen concentration TN (T) acquired by each sampling point are classified according to the difference of nitrogen-phosphorus ratio values, and the parameters C corresponding to the measured data are combined 1 ，C 2 ，C 3 ，C 4 ，C 5 ，C 6 ，C 7 Substituting the parameters u and v of the bifurcation set B to obtain the following formula:
(3) Substituting each measured data into the formulas (24) and (25), and calculating the numerical values of each group u and v;
(4) Substituting u and v into a bifurcation equation formula (23) of the point mutation theory to calculate the value of B; if the calculated value B is near 0, the comprehensive effect influence of the water temperature T (T), the total phosphorus concentration TP (T) and the total nitrogen concentration TN (T) in the current blue algae water environment is demonstrated, and the critical point of the occurrence of water bloom outbreak of the blue algae is reached;
step five, judging the outbreak of cyanobacteria bloom and carrying out early warning;
and step four, if the critical point of water bloom outbreak is reached, then, the forecasting value of chlorophyll concentration is combined to make early warning on whether the water bloom outbreak occurs in the blue algae growth.
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