CN107292436B - Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model - Google Patents
Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model Download PDFInfo
- Publication number
- CN107292436B CN107292436B CN201710458468.9A CN201710458468A CN107292436B CN 107292436 B CN107292436 B CN 107292436B CN 201710458468 A CN201710458468 A CN 201710458468A CN 107292436 B CN107292436 B CN 107292436B
- Authority
- CN
- China
- Prior art keywords
- blue algae
- time
- growth
- blue
- algae
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q10/00—Administration; Management
- G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
Abstract
The invention discloses a cyanobacterial bloom prediction method based on a nonlinear dynamics time sequence model, and belongs to the technical field of water environment prediction. The method comprises the steps of establishing a blue algae growth nonlinear dynamics time sequence model with double nutrient salt circulation by taking the blue algae growth rate as a time-varying parameter, carrying out optimization calibration on a fixed parameter in the blue algae growth nonlinear dynamics time sequence model by combining a numerical algorithm and an intelligent evolution algorithm, realizing prediction on the time-varying parameter and the blue algae biomass by establishing a multivariate time sequence model, carrying out nonlinear dynamics analysis on a blue algae growth time-varying system by adopting a bifurcation theory and a central manifold theory, obtaining the condition of blue algae bloom outbreak, and further realizing early warning on the water bloom outbreak behavior. The invention not only determines the condition of the blue algae bloom outbreak, but also improves the accuracy of the bloom prediction, provides effective reference for environmental protection departments and provides treatment decision for water environment treatment.
Description
Technical Field
The invention relates to a blue algae bloom prediction method, belongs to the technical field of water environment prediction, and particularly relates to a blue algae bloom outbreak prediction early warning method based on a nonlinear dynamics time sequence model.
Background
With the development of economic society, water eutrophication has become a global major water environment problem. The eutrophication of water body means that under the influence of human activities, a great amount of nutrient substances such as nitrogen, phosphorus and the like required by organisms enter into slow-flow water bodies such as lakes, rivers, lakes, gulfs and the like, so that the phenomena of rapid propagation of algae and other plankton, reduction of dissolved oxygen of the water body and mass death of fishes and other organisms are caused. Algal bloom is a typical expression of water eutrophication and is one of the main problems of water environment pollution of rivers and lakes in China. The fulminant propagation of algae causes water quality deterioration, oxygen deficiency, generation of odorous substances such as fishy smell and the like, even generates algal toxins and causes poison to people, livestock and aquatic organisms through a food chain, thereby destroying the ecological stability of rivers, seriously affecting the safety of urban water supply and drinking water, harming human health, restricting the sustainable development of regional economy, bringing extremely bad influence to human life, and solving the problem of water bloom into the global water pollution. Therefore, the method deeply researches the algal bloom generation process, effectively simulates and predicts the unconventional sudden event of algal bloom outbreak, and has important significance for promoting water environment protection and technical progress.
The research of the current blue algae bloom prediction method mainly comprises a mechanism driving model and a data driving model. The mechanism driving model comprises a single nutritive salt model, a phytoplankton ecological model, an ecological dynamics model and the like, the models contain a plurality of ecological variables and undetermined parameters, and the modeling is carried out by describing the mechanism process of the occurrence of the bloom, so that the mechanism driving model plays an important role in recognizing the bloom and controlling the bloom development; the data driving model comprises a mathematical statistics model and an artificial intelligence model, relevant information hidden in the system is mined by the models from a large amount of measured data through technologies such as an intelligent algorithm and the like, a nonlinear process can be well simulated, the model is particularly suitable for a high-dimensional nonlinear system with an uncertain mechanism, and the model is applied to water bloom prediction. The two modeling methods have respective advantages, but have some problems, most mechanism-driven models are developed aiming at biological, chemical and physical factors and the like influencing the formation of the bloom, but the occurrence of the bloom is still a nonlinear problem with a unclear mechanism and needs to be deeply researched, and meanwhile, a large number of undetermined parameters in the models also limit the applicability of the models; however, the data-driven model lacks mechanism support, so that the causal relationship between various influencing factors and the generation of the bloom cannot be reasonably explained, and the accuracy of the model is difficult to ensure.
In the research on the water bloom prediction, most of the existing mechanism driving models regard a blue algae growth kinetic system as a time invariant system, and model parameters of the time invariant system are set to be constant values which do not change along with time, so that the influence of the change of influencing factors such as water temperature and illumination in an actual water body along with time on the model parameters such as blue algae growth rate and death rate is ignored, the kinetic characteristics and the dynamic process of the actual blue algae growth time variant system cannot be effectively described, and the accurate prediction of the blue algae water bloom is difficult to realize. The existing data driving model adopts various intelligent methods, such as an artificial neural network, a support vector machine, a fuzzy theory, a decision tree and the like, so that the subjective rejection of the water bloom influence factors can bring unstable factors to accurate prediction of the water bloom, and the problems of local minimum, poor generalization capability, limited prediction precision and the like can be caused.
Mechanism-driven modeling and data-driven modeling have respective advantages and limitations, so how to combine the mechanism-driven modeling and the data-driven modeling to establish a water bloom prediction model with high prediction precision is an urgent problem to be solved in the research field of cyanobacterial water bloom, and has important significance for promoting water environment protection and technical progress.
Disclosure of Invention
The invention aims to solve the problems that the existing blue algae bloom prediction precision is not high, the influence of partial influence factors on a blue algae growth time-varying system is neglected, the influence of the influence factors such as the light and the water temperature on the blue algae growth is considered according to the actual measurement data such as the chlorophyll a concentration, the nitrogen and phosphorus nutrient salt concentration, the water temperature and the light collected in the actual blue algae growth time-varying system, the influence of the influence factors such as the light and the water temperature on the blue algae growth along with the time change is considered, the blue algae growth rate is used as the time-varying parameter, a blue algae growth nonlinear dynamics time sequence model with double nutrient salt circulation is established, the optimization and calibration are carried out on the fixed parameters in the blue algae growth nonlinear dynamics time sequence model by combining the numerical algorithm and the intelligent evolution algorithm, the time-varying parameters and the blue algae biomass are predicted by carrying out the multivariate time sequence modeling on the influence factors such as the light and the water, obtaining the condition of the blue algae water bloom outbreak, and further realizing the early warning of the water bloom outbreak behavior. The blue algae bloom forecasting method based on the blue algae growth nonlinear dynamics time sequence model not only determines the blue algae bloom outbreak condition, but also improves the bloom forecasting precision, provides effective reference basis for environmental protection departments, and provides treatment decision for water environment treatment.
The invention provides a cyanobacterial bloom prediction method based on a cyanobacterial growth nonlinear dynamics time sequence model, which comprises the following five steps:
step one, modeling the blue algae growth double-nutrient salt circulation nonlinear dynamics;
the interaction and relationship between the growth of the blue algae and the load of nitrogen and phosphorus nutritive salts are main ways for disclosing the forming mechanism of the water bloom, and the forming process of the water bloom can be expressed by establishing the relationship between the nitrogen and phosphorus nutritive salts and the biomass of the blue algae. The chlorophyll a concentration is the most direct index for representing the blue algae biomass in the water body, the chlorophyll a concentration can be used as a representation factor for reflecting the blue algae water bloom generation, the change rate of the chlorophyll a concentration is closely related to the blue algae growth rate and the death rate, wherein the blue algae growth rate is mainly determined by water bloom influence factors such as water temperature and illumination, and a nonlinear dynamics model between the chlorophyll a concentration and the nitrogen-phosphorus double nutrient salt is established according to the relation between the nitrogen-phosphorus double nutrient salt and the blue algae growth rate obtained by a Monod equation (Moro equation), so that the change between the chlorophyll a concentration and the nitrogen-phosphorus double nutrient salt in the blue algae growth process is simulated.
Step two, modeling the time sequence of the nonlinear dynamics of the blue algae growth;
considering that the growth rate of the blue algae is not a constant parameter which is not constant in an actual environment but a time-varying parameter which varies with time, the growth rate of the blue algae in the nonlinear dynamics model established in the step one is used as a time-varying parameter which varies with time under the influence of factors such as illumination, water temperature and the like, and a blue algae growth nonlinear dynamics time sequence model is established.
Step three, optimizing and calibrating parameters of a blue algae growth nonlinear dynamics time sequence model;
the blue algae growth nonlinear dynamics time sequence model comprises a fixed constant parameter and a time-varying parameter. And (4) carrying out optimization calibration on the stable parameters by combining the data collected in the actual water body. Because the established nonlinear dynamics time sequence model contains three nonlinear differential equations, belongs to a three-variable nonlinear differential equation set, and has more fixed parameters, the fixed parameters of the three-variable nonlinear differential equation set are optimized and calibrated by combining a numerical algorithm and a genetic algorithm in an intelligent evolution algorithm. For the time-varying parameters, the growth rate of the blue-green algae is related to the influence factors such as illumination, water temperature and the like, the numerical values of the influence factors such as the water temperature and the illumination are changed according to the time sequence, the correlation among various influence factors is considered at the same time, and the monitoring data is regarded as a multivariate time sequence. Thereby realizing the prediction of the blue algae growth nonlinear dynamics time sequence model.
Step four, determining the nonlinear dynamic conditions of the cyanobacterial bloom outbreak;
aiming at the cyanobacteria growth nonlinear dynamics time sequence model with fixed-time parameters and time-varying parameters determined, a bifurcation theory is adopted to research the balance point and the stability of the cyanobacteria growth time-varying system. Because the blue algae growth nonlinear dynamics time sequence model is a three-variable nonlinear differential equation set, the nonlinear dynamics analysis is complex, and the dimensionality reduction treatment is carried out by adopting a central manifold theory on the basis of the bifurcation theory research so as to further research the bifurcation behavior of the critical point of the blue algae growth time-varying system, determine the stability of the blue algae growth time-varying system in each range, obtain the range of the blue algae growth rate when the system is stable, and determine the nonlinear dynamics condition of the blue algae bloom outbreak.
Fifthly, carrying out outbreak early warning on the cyanobacterial bloom;
and obtaining the change range of the growth rate of the blue-green algae when the time-varying system for the growth of the blue-green algae is stable according to the nonlinear dynamics conditions of the water bloom outbreak in the step four, and judging whether the growth rate of the blue-green algae exceeds the stable growth rate range of the blue-green algae according to the influence of the predicted time change of the influence parameters such as water temperature, illumination and the like on the growth rate of the blue-green algae, namely whether the risk of the water bloom outbreak of the blue-green algae exists or not, thereby carrying out risk early warning on the. And the time of the blue algae bloom outbreak can be predicted by combining the disaster point of the bloom occurrence in the actual water environment and comparing the disaster point with the predicted value of the chlorophyll a concentration, so that the early warning of the bloom outbreak disaster is realized.
The invention has the advantages that:
1. the invention establishes a blue algae growth double-nutrient salt circulation nonlinear dynamics model, on the basis, the influence of the change of the influence factors such as water temperature, illumination and the like in the actual water body along with the time on the blue algae growth rate is considered, the blue algae growth nonlinear dynamics time sequence model is constructed, and the description of the blue algae growth time-varying system is more practical.
2. The method adopts a numerical method and an intelligent evolution algorithm to combine to obtain the optimal parameters when the optimization rate of the fixed parameters of the blue algae growth nonlinear dynamics time sequence model is fixed, so that the model has applicability and practical applicability. The problem that the existing research is not suitable for parameter calibration of a three-variable nonlinear differential equation set is solved.
3. In the invention, when the cyanobacterial bloom is predicted, the advantages of mechanism driving and data driving prediction are combined, two prediction methods are combined, the time sequence modeling of the influence factors is firstly performed, the change of the influence factors along with the time is predicted, and the change is reflected in the time-varying parameters of the cyanobacterial growth nonlinear dynamics time sequence model, so that the chlorophyll a concentration of the cyanobacterial is predicted. The method can reflect the influence of various influencing factors on the formation of algal blooms under the complex water environment, can also exert the advantages of data-driven search modeling, can be suitable for the requirements of different lake and reservoir water environments, and has expandability.
4. The invention provides a method for early warning the outbreak of the cyanobacterial bloom by combining a dynamic critical point and an actual water environment disaster point. According to the bifurcation theory and the central manifold theory, the critical point of the system stability transition during the blue algae growth is researched, and when the maximum value of the chlorophyll a concentration is larger than or equal to the disaster point of the actual water environment due to the oscillation action of the system, the critical point shows that the water bloom outbreak occurs. The method of the invention can predict the sudden water quality disaster of the water bloom outbreak in time and provide basis for decision departments to make control strategies for the water bloom.
Drawings
FIG. 1 is a flow chart of a cyanobacterial bloom prediction method based on a nonlinear dynamics time sequence model;
FIG. 2 is a parameter calibration flow of a genetic algorithm in combination with a numerical algorithm;
FIG. 3 is a comparison of the original modeling data and the model fitting results after parameter calibration;
FIG. 4 is a bifurcation diagram of the blue algae growth rate near G0-29.92;
FIG. 5 is a graph of second equilibrium point as a function of growth rate (G > G0);
fig. 6 to 9 are time charts of the system phase trajectory, chlorophyll a concentration, nitrogen concentration, and phosphorus concentration when G is 38;
fig. 10 to 13 are time charts of the system phase trajectory, chlorophyll a concentration, nitrogen concentration, and phosphorus concentration at G25;
fig. 14 to 17 are time charts of the system phase trajectory, chlorophyll a concentration, nitrogen concentration, and phosphorus concentration at G50;
FIG. 18 is a water temperature time series prediction chart;
FIG. 19 is a graph of illumination time series prediction;
FIG. 20 is a graph of chlorophyll, nitrogen and phosphorus concentration time-varying phase traces;
FIG. 21 is a time varying chlorophyll prediction value;
FIG. 22 is a time varying nitrogen nutritive salt prediction value;
FIG. 23 is a time varying phosphorus nutritive salt prediction;
FIG. 24 is a comparison graph of time-varying kinetic predictions, source data, time series predictions;
FIG. 25 is a graph showing the growth rate of blue algae as a function of time.
Detailed Description
The present invention will be described in further detail below with reference to the accompanying drawings.
The invention provides a cyanobacterial bloom prediction method based on a nonlinear dynamics time sequence model, which comprises the following specific steps as shown in a flow chart of figure 1:
step one, modeling the blue algae growth double-nutrient salt circulation nonlinear dynamics;
the blue algae biomass can be measured by the content index of the chlorophyll a, and the growth process of the blue algae is expressed by establishing the relation between the nitrogen and phosphorus nutrient salt and the concentration of the chlorophyll a. The concentration change rate of the chlorophyll a is closely related to the growth rate and the net loss rate of the blue-green algae, and the analysis of the influence factors on the growth rate of the blue-green algae shows that the growth rate of the blue-green algae is mainly determined by the influence factors such as illumination, water temperature and the like, so that the growth rate of the blue-green algae is expressed by the action function of the illumination and the water temperature. According to the relation of the growth rate of the algae under the nutrient salt limiting condition, the growth function of the nutrient salt is established to be in accordance with the Monod equation, and the established blue algae growth double-nutrient salt circulation nonlinear kinetic model is as follows:
wherein, caThe concentration of chlorophyll a at the time t, N, P are the concentrations of total nitrogen and total phosphorus of nutritive salt at the time t respectively; n is a radical of0、P0The concentrations of total nitrogen and total phosphorus of the nutritive salt at the initial moment are respectively; g is the growth rate of blue algae, T1Is the environmental loss rate of the algae; t is2Is the loss rate of nitrogen nutritive salt, gNIs the absorption rate of the blue algae on nitrogen nutrient salt dNThe contribution rate of the metabolic decomposition of the algae to the nitrogen nutritive salt; t is3Is the loss rate of phosphorus nutritive salt, gPIs the absorption rate of the blue algae on the phosphorus nutrient salt, dPThe contribution rate of the metabolic decomposition of the algae to the phosphorus nutritive salt is shown. g (N), g (P) are the growth functions of total nitrogen and total phosphorus of nutrient salts respectively, and the expression is as follows:
wherein, KN、KPRespectively representing the half-saturation concentration constants of the growth of blue algae to nitrogen and phosphorus. In an actual water body, the growth rate G of the blue algae is determined by the influence factors such as water temperature T, illumination I and the like, and taking the water temperature T and the illumination I as an example, the relationship is as follows:
wherein G isT(T),GI(I) Is the influence function of water temperature and illumination on the growth rate of blue algae, gmaxIs the maximum growth rate of blue algae, kIIs the light half-saturation concentration.
Step two, modeling the time sequence of the nonlinear dynamics of the blue algae growth;
in an actual water body, since the influence factors such as water temperature and illumination change with time, the growth rate of blue algae determined by the influence factors also changes with time, and is not constant as defined in the formula (1). Therefore, the blue algae growth double nutrient salt circulation kinetic system is a time-varying system, and the time-varying blue algae growth rate G (t) is introduced on the basis of the formula (1) to construct a blue algae growth double nutrient salt circulation time-varying parameter nonlinear kinetic model:
wherein, the formula of the time-varying blue algae growth rate G (t) changing along with time is as follows:
wherein, T (t), I (t) are the values of the water temperature and the illumination at the time t respectively.
Step three, optimizing and calibrating parameters of a blue algae growth nonlinear dynamics time sequence model;
for the fixed-constant parameter, according to the actual water body monitoring data, combining the genetic algorithm and the numerical algorithm, carrying out the fixed-constant parameter (T) in the blue algae growth double-nutrient salt circulation nonlinear dynamics model, namely the formula (1)1、T2、gN、dN、T3、gP、dP、KN、KP、gmax、kI) And (6) carrying out optimization calibration. The steady parameter optimization calibration flow is shown in fig. 2, and the specific steps are as follows:
(1) setting initialization conditions and initializing population individuals. Determining the number of individuals, the maximum genetic algebra and the number of parameters to be optimizedNumber, generation of gully, and fitness threshold. And multi-parameter cascade floating-point number coding is adopted. And randomly generating a plurality of different parameter combinations as initial populations to form a parameter space to be optimized. The parameter to be optimized comprises T1、T2、gN、dN、T3、gP、dP、KN、KP、gmax、kI。
(2) And (5) evaluating the fitness value. In the evolution process of the population, the advantages and disadvantages of individuals are evaluated through fitness values, and a fitness function is established:
wherein F is a fitness value, and n is the total days for collecting water environment data; c. CatThe actual value of the chlorophyll a concentration of the chlorophyll a at the time t, ca(t) is a chlorophyll a concentration function value of chlorophyll a at the time t; n is a radical oftThe real value of the total nitrogen concentration of the nutritive salt at the time t, and N (t) is the value of the total nitrogen concentration function of the nutritive salt at the time t; ptThe real value of the total phosphorus concentration of the nutrient salt at the time t, and P (t) is the value of the total phosphorus concentration function of the nutrient salt at the time t.
The complex structure of the equation (1) in the multivariate differential equation system can not obtain caThe analytical expressions of (t) and N (t) need to be calculated by adopting a numerical algorithm, and the numerical integral expressions of the 4-order Runge Kutta algorithm in the numerical algorithm are respectively as follows:
wherein the parameter qi(i ═ 1,2,3,4) specific expressions are as follows:
q1＝f1(ca(t),N(t),P(t))
q2＝f1(ca(t)+q1×h/2,N(t)+q1×h/2,P(t)+q1×h/2)
q3＝f1(ca(t)+q2×h/2,N(t)+q2×h/2,P(t)+q2×h/2)
q4＝f1(ca(t)+q3×h,N(t)+q3×h,P(t)+q3×h)
f1(x,y,z)＝T1x+Gg(y)g(z)x
wherein h is the step length, T1Is the environmental loss rate of algae, and g (y), g (z) are the same as the function forms of g (N), g (P), respectively. x, y, z being a function f1(x, y, z).
Similarly, the parameter ri(i ═ 1,2,3,4) specific expressions are as follows:
r1＝f2(ca(t),N(t),P(t))
r2＝f2(ca(t)+r1×h/2,N(t)+r1×h/2,P(t)+r1×h/2)
r3＝f2(ca(t)+r2×h/2,N(t)+r2×h/2,P(t)+r2×h/2)
r4＝f2(ca(t)+r3×h,N(t)+r3×h,P(t)+r3×h)
f2(x,y,z)＝T2(N0-y)-gNg(y)g(z)ca+dNx
parameter si(i ═ 1,2,3,4) specific expressions are as follows:
s1＝f3(ca(t),N(t),P(t))
s2＝f3(ca(t)+s1×h/2,N(t)+s1×h/2,P(t)+s1×h/2)
s3＝f3(ca(t)+s2×h/2,N(t)+s2×h/2,P(t)+s2×h/2)
s4＝f3(ca(t)+s3×h,N(t)+s3×h,P(t)+s3×h)
f3(x,y,z)＝T3(P0-z)-gPg(y)g(z)ca+dPx
and calculating and obtaining chlorophyll a, nitrogen and phosphorus concentration values at all times according to a numerical method, and substituting the obtained chlorophyll a, nitrogen and phosphorus concentration values and actual monitored chlorophyll a, nitrogen and phosphorus concentration values into a fitness function formula (6) to obtain the fitness value F of each individual of the population. According to the formula (6), when the fitness value is smaller, the calculated value of the numerical method is closer to the actual monitoring value, which shows that the individual is excellent and participates in the propagation of the next generation of population.
(3) Selection, crossover, mutation, reinsertion operations.
(4) The genetic algorithm terminates. And (3) continuously repeating the steps (2) and (3) to form new individuals according to the fitness value of the individuals, wherein the termination condition is that the minimum fitness value of the genetic algebra or the parameter combination with the maximum repetition times is less than or equal to the fitness threshold value. And taking the minimum value of the parameter combination adaptability values when the minimum value reaches the termination condition as the adaptability value of the blue algae growth nonlinear dynamics time sequence model, wherein the parameter combination is the optimal parameter combination of the blue algae growth nonlinear dynamics time sequence model.
For the time-varying parameters, the function values of the water temperature T (t) and the illumination I (t) and other influencing factors are changed in time sequence, and meanwhile, the correlation among the influencing factors is considered, and the monitoring data is regarded as a multivariate time sequence. And (3) predicting values of the influence factors such as water temperature T (t) and illumination I (t) changing along with time according to the multivariate time series model, and substituting the prediction results into a formula (5) to obtain the predicted value of the time-varying parameter blue algae growth rate G (t). Substituting the predicted value of the blue algae growth rate G (t) into a blue algae growth nonlinear dynamics time sequence model, and predicting the blue algae growth nonlinear dynamics time sequence model according to a calculation formula (7) of predicting the chlorophyll a concentration, the nitrogen and the phosphorus nutrient salt concentration by a Longge Kuta algorithm.
Step four, determining the nonlinear dynamic conditions of the cyanobacterial bloom outbreak;
the method for determining the blue algae bloom outbreak nonlinear dynamics condition comprises the steps of taking the blue algae growth rate as a time-varying parameter, firstly determining a bifurcation set and a bifurcation point of a blue algae growth nonlinear dynamics time sequence model by adopting a bifurcation theory, then reducing the dimension of the blue algae growth nonlinear dynamics time sequence model by adopting a central manifold method to obtain a reduction equation, specifically researching the critical local stability and the bifurcation type, determining the critical point of the blue algae bloom outbreak, and providing a formula condition met by the critical point.
(1) Firstly, determining a bifurcation set and a bifurcation point of a cyanobacteria growth nonlinear dynamics time sequence model by adopting a bifurcation theory. The specific method comprises the following steps: making the right sides of the three equations in the equation set (4) equal to zero, solving the equation set to obtain the balance point Q (c) of the blue algae growth nonlinear dynamics time sequence modela *,N*,P*)，ca *、N*、P*The chlorophyll a concentration, the total nitrogen concentration and the total phosphorus concentration at the equilibrium point are respectively, and in order to facilitate the analysis to carry out coordinate translation,
the system of equations after coordinate translation is transformed into:
separating the linear terms and the higher order terms yields:
wherein
k is the jacobian matrix of the linear part,
is a high-order term.
Solving an equation (8) to obtain a Jacobian matrix of a linear part of a blue algae growth nonlinear dynamics time sequence model equation:
high-order terms:
wherein h.o.t represents a high-order term of cubic degree or more in the expansion.
From equation (1), the equilibrium point Q (c) of the time-varying system for the growth of blue algaea *,N*,P*) There are two cases:
(I)(0,N0,P0),N0,P0the concentrations of the two nutrient salts at the initial moment are respectively;
(II)(ca *,N*,P*) Wherein c isa *,N*,P*Are not 0.
And considering the stability condition of the blue-green algae growth time-varying system according to the eigenvalue of the Jacobian matrix under different conditions.
(I) For the first balance point (0, N)0,P0) The physical meaning of the method corresponds to the initial state, the concentration of the blue algae biomass is 0, and the Jacobian matrix at the moment is as follows:
from the bifurcation theory, the stability of the time-varying system for the growth of blue algae depends on the eigenvalues of the corresponding Jacobian matrix (10). The three characteristic values of the first balance point are respectively
-T2，-T3. Due to T2、T3The loss rates of the nitrogen and phosphorus nutrient salts are positive numbers respectively, and when the first characteristic value is
The eigenvalues are positive numbers and the state is stable; when the characteristic value
Is negative, the state is unstable, so when
When the characteristic value is 0, the time-varying system for the growth of blue-green algae is in a critical state, and the system is enabled to be in a critical state
Then G is0As a bifurcation point.
(II) for a second equilibrium point (c)a *,N*,P*) This is the case when all three variables are present. Wherein c isa *,N*,P*G (t), the relation satisfied is:
substituting the last two formulas in the relational expression (11) into the first formula to obtain a quadratic equation containing only chlorophyll a concentration and blue algae growth rate, solving the quadratic equation to obtain two roots related to the chlorophyll a concentration, wherein the corresponding equilibrium state is (c)a1 *,N1 *,P1 *) And (c)a2 *,N2 *,P2 *) The corresponding Jacobian matrix is:
because under the parameter representation, three eigenvalues of the Jacobian matrix (12) cannot be expressed explicitly. But may be in accordance with JackThe characteristic value equation of the ratio matrix is used for solving the bifurcation point of the blue algae growth time-varying system, namely the corresponding blue algae growth rate when the characteristic value has zero root is set as G1. The eigenvalue equation obtained by calculation is as follows: lambda [ alpha ]3+aλ2+bλ+c＝0
Wherein
When 0 is present in the feature root, c is satisfied as 0, and in conjunction with equation (11), c can be obtaineda *,N*,P*G (t), the value of G (t) at this time is represented by G1Then G is1Also a branch point.
The analysis shows that the time-varying system for the growth of the blue algae has two bifurcation points, so that a bifurcation set can be divided into three ranges (0, G)0)、(G0，G1) And is greater than G1When the growth rate of the blue algae takes different values, the first balance point of the blue algae growth time-varying system is unchanged, and the second balance point can be changed along with the growth rate of the blue algae.
(2) After the branch point of the time-varying system for the growth of the blue-green algae is determined, in order to determine the range of the growth rate of the blue-green algae when the system is stable and the critical point of the burst of the blue-green algae, the stability of the time-varying system for the growth of the blue-green algae and the branch situation of the critical point system in each range need to be researched, and the specific analysis is as follows:
first range (0, G)0): according to the analysis of the blue algae growth double-nutrient salt nonlinear kinetic model on the first balance point, when the blue algae growth rate is (0, G)0) In the range, the system is in a stable state when the blue algae grows, and is stabilized at a first balance point. When growth rate
When the characteristic value is 0, the time-varying system for the growth of the blue-green algae is in a critical state, and the time-varying system for the growth of the blue-green algae needs to be researched at a bifurcation point G0Local stability in the vicinity. When the blue-green algae growth time-varying system is in a critical state, the local stability of the nonlinear systemCannot be analyzed by adopting the traditional linearization stability principle, but depends on the nonlinear term of the system, and the critical local stability of the nonlinear system is often closely related to the bifurcation behavior of the dynamic behavior of the system. For a high-dimensional dynamics system, bifurcation behaviors are complex, dimension reduction measures are needed to be adopted to be changed into a low-dimensional equation set for research, the dimension reduction of the blue-green algae growth time-varying system is carried out by adopting a central manifold method to obtain a reduction system, and the blue-green algae growth rate is determined to be G0The method comprises the following specific steps of the bifurcation behavior of a nearby system:
to obtain a reduced system, let
By conversion, G is knowncBifurcation behavior near 0. At the moment, the Jacobian matrix of the blue algae growth time-varying system is as follows:
the three characteristic values are 0, -T2，-T3The linear transformation matrix composed of the corresponding feature vectors is:
by linear transformation (u, v, w)T＝M(x,y,z)TThen the original differential equation (8) can be converted into:
φ'＝Dφ+M-1f(Mφ)。
wherein D ═ diag (0-T)2-T3),φ＝(x,y,z)TAnd when the growth rate of the blue algae is taken as a bifurcation variable, developing to obtain:
according to the central manifold theory, the local dynamic characteristics of the blue algae growth time-varying system are mainly determined by the flow on the central manifold, the system has a two-dimensional central manifold,
wherein, O ((x + G)c)3) Is a high order term of the function expansion.
For determining the coefficient a in the system of equations (15)1,a2,a3,b1,b2,b3Substituting (15) into the last two equations for solving y and z in the differential equation set (14), obtaining coefficient values after comparing the coefficients of powers of the same degree, then substituting (14) into the first equation for solving x, obtaining an equation on the central manifold, namely a reduction equation, and converting the growth rate parameter into G (t), obtaining the reduction equation as follows:
wherein the content of the first and second substances,
for the sake of analytical convenience, the reduction equation obtained above is expressed by x' ═ f (x, g (t)) ═ lx + mx2+nx3Denotes the coefficients by the letters l, m, n, respectively, since a1＜0,b1If the coefficient is less than 0, m is more than 0 and n is less than 0 by judging the positive and negative properties of the coefficient of the reduced equation (16). Analysis of the reduction equation shows that: equation f (x, g (t)) 0 has two solution curves, and for solution curve x 0, the derivative value Dxf (x, G (t)) a, so that when a < 0, the growth rate is determined
When G (t) > G, the equilibrium point becomes progressively more stable0It is not stable; for another solution curve is on the axis of symmetry
When a solution exists, the following needs to be satisfied:
The growth rate of blue algae by central manifold is G0Analysis of bifurcation behavior of nearby System, Explanation G0The point is a supercritical fork-shaped bifurcation point and is not a burst point of the cyanobacterial bloom. Analyzing to obtain the growth rate of blue algae when the growth rate is (0, G)0) When the growth rate of the blue algae is within the range, the system is stabilized at a first balance point when the blue algae grows, and when the growth rate of the blue algae is G0And is greater than G0The time varying system for the growth of blue algae is stable at the second equilibrium point.
Continuing the analysis for the second equilibrium point, for a meaningful equilibrium point (c)a1*,N1*,P1The eigenvalue of the Jacobian matrix corresponding to the second balance point can not be expressed explicitly when the bifurcation set is solved, but the corresponding blue-green algae growth rate G when 0 root appears in the eigenvalue can be obtained1I.e. G1Satisfies the following conditions:
when the growth rate is in (G)0，G1) Range, eigenvalues are all less than 0, equilibrium point will stabilize at the second plateauBalance point, when greater than G1Then, the condition that the characteristic value is larger than 0 occurs, and the system is unstable when the blue algae grows. Description of the invention G1The point is the critical point of the outbreak of the cyanobacterial bloom. Critical point G1Satisfies equation set (17).
Through the analysis, when the growth rate of the blue algae is (0, G)1) In this range, the time-varying system is stable during the growth of cyanobacteria, i.e., at (0, G)0) When the growth rate of the blue algae is in (G), the system is stabilized at a first balance point when the growth rate of the blue algae is in (G)0，G1) When the blue algae grows, the time varying system is stabilized at a second balance point; when the growth rate is larger than G1In time, the time-varying system for the growth of the blue algae is unstable, and the blue algae bloom outbreak behavior occurs.
Fifthly, carrying out outbreak early warning on the cyanobacterial bloom;
when the growth rate of the blue algae is changed along with time as a time-varying parameter, the first balance point of the blue algae growth time-varying system is unchanged, and the second balance point is changed along with time. Since the equilibrium point is no longer a fixed value, but varies over time, the stability of the system also varies over time. According to the analysis of the system bifurcation set and the bifurcation points in the step three, the following results are obtained: when the growth rate of the blue algae is more than G1In time, a time-varying system for the growth of the blue algae is unstable, and the risk of water bloom outbreak is caused, wherein the growth rate of the blue algae is determined by the influence factors such as illumination, water temperature and the like, so the condition formula of the water bloom outbreak risk of the blue algae is as follows:
G(t)＝GT(T(t))·GI(I(t))＞G1(18)
wherein G is1Solved by equation set (17). Predicting the water temperature and the illumination value through a time sequence model, substituting the predicted water temperature and illumination value into a blue algae growth rate function, and predicting the outbreak point G when the blue algae growth rate is greater than the blue algae growth rate1Sometimes, there is a risk of water bloom outbreaks.
And then according to the predicted growth rate of the blue algae and by combining a blue algae growth nonlinear dynamics time sequence model, predicting the biomass of the blue algae and the concentration value of nutrient salt, and when the predicted biomass of the blue algae is larger than the water bloom disaster point threshold value (set as c) of the actual water environmentaT) When the condition is met:
ca(t+1)＝ca(t)+h×(q1+2q2+2q3+q4)/6＞caT(19)
indicating the occurrence of the water bloom outbreak behavior, and early warning the water bloom outbreak behavior of the blue algae in advance according to the predicted occurrence time of the water bloom of the blue algae.
Example 1:
taking the data of the Taihu lake basin of Jiangsu province as an example, the method provided by the invention is adopted to predict the cyanobacterial bloom and warn the cyanobacterial bloom. The data mainly comprises chlorophyll a concentration, total nitrogen concentration, total phosphorus concentration, water temperature, illumination and other data. For convenient analysis, the raw monitoring data of chlorophyll a concentration, total nitrogen, total phosphorus and the like are subjected to pretreatment such as standardization, abnormal point elimination and the like, as shown by an unmarked curve in fig. 3.
According to the measured data of 1 characterization factor (chlorophyll a concentration) and four influence factors (total nitrogen concentration, total phosphorus concentration, water temperature and illumination) in 400 days from 2010 to 2011 of Taihu lake, the types and units of the measured data are specifically shown in Table 1.
TABLE 1 measured data types and units
Step one, modeling the blue algae growth double-nutrient salt circulation nonlinear dynamics;
see formulas (1) - (3) in the detailed description.
Step two, modeling the time sequence of the nonlinear dynamics of the blue algae growth;
see formulas (4) - (5) in the detailed description.
Step three, optimizing and calibrating parameters of a blue algae growth nonlinear dynamics time sequence model;
for the constant parameters, the constant parameters in the formula (1) are calibrated by adopting a parameter optimization calibration method combining the numerical algorithm and the intelligent optimization algorithm, and the calibration result is shown in table 2.
TABLE 2 blue algae growth non-linear dynamics time sequence model fixed parameter calibration result
| Parameter(s) | Constrained range | Calibration results |
| Maximum growth rate g of blue algaemax | 0.01～100 | 24.7 |
| Illumination half-saturation concentration kI | 0.01～10 | 0.08 |
| Environmental loss rate of algae T1 | 0.01～30 | 21.6 |
| Depletion rate of nitrogen nutritive salt T2 | 0.01～20 | 10.3 |
| Depletion rate T of phosphorus nutritive salt3 | 0.01～1 | 0.09 |
| Absorption rate g of blue algae to nitrogenN | 0.01～110 | 2.68 |
| Contribution rate of algal metabolic decomposition to nitrogen dN | 0.01～100 | 57 |
| Absorption rate g of blue algae to phosphorusP | 0.01～1 | 0.014 |
| Contribution rate of algal metabolic decomposition to phosphorus dP | 0.01～1 | 0.23 |
| Half-saturation constant K of blue algae growth to nitrogenN | 0.01～10 | 3 |
| Half-saturation constant K of blue algae growth to phosphorusP | 0.01～5 | 0.51 |
| Initial value N of nitrogen concentration0 | 0.01～50 | 23 |
| Initial value P of phosphorus concentration0 | 0.01～10 | 3 |
The fitting result of the blue algae growth nonlinear dynamics time sequence model with the fixed parameters to the chlorophyll a concentration is shown as an × -shaped labeled curve in figure 3, wherein the actual monitoring data adopted by the nutrient salt concentration, the water temperature and the illumination are ideal, and the fixed parameters are compared with the actual data and the fitting data in figure 3.
For time-varying parameters, in order to predict the concentration of blue algae and nitrogen and phosphorus nutrient salts, firstly, the influence factors such as illumination, water temperature and the like need to be predicted, so that a binary time series model is established for the illumination and the water temperature, the binary time series model is subjected to characteristic item decomposition according to a trend item, a period item and a random item, the illumination and the water temperature are predicted, and the prediction results of the water temperature and the time series data of the illumination are shown as the labeled curves with the shape of × in fig. 4 and fig. 5.
And substituting the predicted values of the time sequence data of the water temperature and the illumination into a time-varying parameter blue-green algae growth rate formula, wherein the phase locus of a time-varying system for the blue-green algae growth is shown in fig. 6, and the time courses of the chlorophyll a concentration and the nitrogen and phosphorus nutrient salt concentration are shown in fig. 7, 8 and 9.
Comparing the chlorophyll a concentration time history (predicted value) of the time-varying system for blue-green algae growth with the monitoring data thereof and directly adopting a time sequence prediction method, the curve is shown in fig. 10, which shows that the time-varying system for blue-green algae growth has higher precision of the prediction result than the time sequence method directly adopted.
Step four, determining the nonlinear dynamic conditions of the cyanobacterial bloom outbreak; according to theoretical analysis, first the fork branch point G under the first equilibrium point is calculated0The value of the carry-in calibration parameter can be calculated G029.92. Then, according to the central manifold theory and the bifurcation theory, research G is carried out0Bifurcation situation of points. After linear transformation, Gc＝G(t)-G0G (t) -29.92, shift to study GcThe blue algae grows near zero and becomes a time-varying system. At this time, the three eigenvalues are 0, -10.3, -0.09, and the eigenvector corresponding to the three eigenvalues form an eigenvector matrix as follows:
when the time-varying parameter blue-green algae growth rate is taken as a variable, the original differential equation is converted into:
wherein f is1,f2,f3Are functions related to f (-0.619x,0.965x + y, x + z).
According to the central manifold theorem, the method is provided,
in order to determine the coefficient values of the two functions, the two functions are respectively substituted into the last two equations of the system of differential equations,
obtaining:
the blue-green algae growth time-varying system is known to have two-dimensional central manifold, and the equation on the central manifold, namely the reduction equation, is obtained after calculation:
converting the growth rate parameter to G, one can obtain:
the bifurcation of the state variables of the reduction equation with the parameters is shown in fig. 11. According to the growth rate G of blue algae0The bifurcation diagram near 29.92 reveals that: when the growth rate of the blue algae is 29.92 less, the blue algae is stable for the first balance point, and when the growth rate of the blue algae is more than 29.92, the first balance point is unstable, and when the system has slight fluctuation during the growth of the blue algae, the blue algae is stable at the second balance point, so that the growth rate of the blue algae belongs to fork-shaped branches at 29.92.
When the growth rate of the blue algae is more than 29.92, continuing to analyze the second balance point, and solving the branch point G of the blue algae growth time-varying system1By solving the equation set (9), G is obtained148.85. With variation in growth rate, secondThe change chart of the equilibrium points is shown in fig. 12, when the growth rate of the blue algae is 48.05, 0 characteristic roots appear, namely the critical point of the system for converting the blue algae growth time from the stable state to the unstable state under the second equilibrium point. When the growth rate of the blue algae is more than 48.85, the system is unstable when the blue algae grows, and the phenomenon of water bloom outbreak can occur. Therefore, the overall analysis of the time-varying system for the growth of the blue algae shows that when the growth rate of the blue algae is in the range of (0, 29.92), the system is stable at a first balance point, when the growth rate of the blue algae is in the range of (29.92, 48.05), the system is stable at a second balance point, and when the growth rate of the blue algae is more than 48.05, the system is unstable, and the water bloom outbreak behavior can occur.
The stability of the time varying system for the growth of blue algae under different growth rates is illustrated by selecting different growth rates. When the growth rate is 38, analyzing a first balance point (0,23,3) of the system, wherein three eigenvalues corresponding to the Jacobian matrix are respectively 7.13, -10.3 and-0.09, and the eigenvalue contains a positive number, so that the first balance point is unstable; and analyzing the second equilibrium point analysis, when the growth rate is 38 and the concentration value of chlorophyll a is a positive value, calculating to obtain an equilibrium point (2.86,22.85 and 0.92), wherein the three characteristic values of the corresponding Jacobian matrix are-0.2093 +1.1882i, -0.2093-1.1882i and-11.1336, and indicating that the time-varying system for the growth of the blue algae is stable in oscillation. The phase locus of the time-varying system for blue-green algae growth at G-38 is shown in FIG. 13, and the time courses of the corresponding chlorophyll a concentration and nitrogen and phosphorus nutrient salt concentration are shown in FIGS. 14, 15 and 16.
As can be seen from fig. 13 to 16, when the nutritive salt is accumulated to a certain degree and the influence factors such as water temperature and light meet certain conditions, the growth speed of the blue-green algae (chlorophyll a concentration) is fast, that is, the system oscillates during the growth of the blue-green algae. With the large consumption of nutrient salts for the growth of the blue algae, the chlorophyll a concentration also begins to decrease when the nutrient salts are consumed to a degree that is not enough to maintain the growth of the blue algae, and the concentration of the nutrient salts begins to increase again due to the degradation of dead algae and the metabolism of the algae, and the steps are repeated, and finally, the concentration is stabilized at a certain level. Therefore, the formula (1) can better explain the growth condition of the blue algae and accords with the actual condition to a certain extent.
When the growth rate of the blue algae is 25, the three characteristic values of the matrix corresponding to the first balance point (0,23 and 3) are-2.69, -10.3 and-0.09 respectively, and all the three characteristic values are negative values, which indicates that the blue algae growth time-varying system is stable at the first balance point and the blue algae bloom outbreak behavior cannot occur. The phase locus of the time-varying system for the growth of the blue algae is shown in fig. 17, the time courses of the corresponding chlorophyll a concentration and the nitrogen and phosphorus nutrient salt concentration are shown in fig. 18, fig. 19 and fig. 20, and the chlorophyll a concentration curve is maintained near zero, which indicates that the system is stable.
When the growth rate of the blue algae is 50, the three characteristic values of the matrix corresponding to the first balance point (0,23,3) are respectively 16.2, the meaningful balance point value in the second balance point is (45,34,0.3), the three characteristic values of the corresponding matrix are respectively-7.5530, 0.6598 and 20.1382, and the characteristic root contains a positive number, which indicates that the time-varying system for the growth of the blue algae is unstable and the sudden action of the blue algae bloom occurs. The phase locus of the time-varying system for the growth of the blue algae is shown in fig. 21, the time courses of the corresponding chlorophyll a concentration and the nitrogen and phosphorus nutrient salt concentration are shown in fig. 22, fig. 23 and fig. 24, and the chlorophyll a concentration curve is mutated to a larger value, which indicates that the system is unstable and directly generates a water bloom outbreak phenomenon.
Through the time history graphs of the blue algae biomass and the nutrient salts under different growth rates selected above, different dynamic behaviors of the time-varying system for the blue algae growth are shown.
Fifthly, carrying out outbreak early warning on the cyanobacterial bloom;
because the growth rate of the blue algae changes along with time along with the influence factors such as illumination, water temperature and the like, the second balance point of the system changes along with time when the blue algae grows, and the stability of the system changes along with time when the blue algae grows. According to the discussion of the branch points and the branch points in the third step, it is known that when the growth rate is more than 48.05, the risk of the outbreak of the cyanobacterial bloom exists, and the influencing factors are as follows:
G(t)＝GT(T(t))·GI(I(t))＞48.05
the graph of the change of the growth rate of the blue algae with time obtained by substituting the illumination and water temperature values predicted by the time series into the growth rate function is shown in FIG. 25, the growth rate of the blue algae corresponding to the circle mark in the graph is 48.05, which shows that the water bloom outbreak phenomenon of the blue algae can occur at 313 th day.
Substituting the predicted growth rate of the blue algae into a blue algae growth nonlinear dynamics time sequence model, predicting the change of the biomass of the blue algae according to the numerical method in the step two, and when the predicted biomass of the blue algae is greater than or equal to the actual water environment water bloom disaster point (the threshold value is 40), indicating that the water bloom outbreak phenomenon can occur. According to the time-dependent change chart of the chlorophyll concentration of the blue algae in the figure 7, the predicted value of the chlorophyll concentration of the blue algae at 313 th day is 43 which is more than 40, and the outbreak behavior of the blue algae water bloom is really happened.
The established blue algae growth nonlinear dynamics time sequence model can predict the blue algae biomass and the concentration of nutrient salt in the water environment, can also judge the risk and occurrence of the water bloom outbreak, realizes the prediction and early warning of the water bloom outbreak, and provides a new method for doing water bloom prevention work in advance.
Claims (1)
1. The cyanobacterial bloom forecasting method based on the nonlinear dynamics time sequence model comprises the following five steps,
step one, establishing a blue algae growth double-nutrient salt circulation nonlinear dynamics model:
wherein, caThe concentration of chlorophyll a at the time t, N, P are the concentrations of total nitrogen and total phosphorus of nutritive salt at the time t respectively; n is a radical of0、P0The concentrations of total nitrogen and total phosphorus of the nutritive salt at the initial moment are respectively; g is the growth rate of blue algae, T1Is the environmental loss rate of the algae; t is2Is the loss rate of nitrogen nutritive salt, gNIs the absorption rate of the blue algae on nitrogen nutrient salt dNThe contribution rate of the metabolic decomposition of the algae to the nitrogen nutritive salt; t is3Is the loss rate of phosphorus nutritive salt, gPIs the absorption rate of the blue algae on the phosphorus nutrient salt, dPThe contribution rate of the metabolic decomposition of the algae to the phosphorus nutritive salt; g (N), g (P) are total nitrogen and total nutrient saltA phosphorus growth function;
step two, modeling the time sequence of the nonlinear dynamics of the blue algae growth;
introducing a time-varying blue algae growth rate G (t) on the basis of a formula (1) to construct a blue algae growth double-nutrient salt circulating time-varying parameter nonlinear dynamics model:
wherein, the formula of the time-varying blue algae growth rate G (t) changing along with time is as follows:
wherein, T (t), I (t) are the numerical values of the water temperature and the illumination at the time t respectively;
step three, optimizing and calibrating parameters of a blue algae growth nonlinear dynamics time sequence model;
for the fixed-time parameters, optimizing and rating the fixed-time parameters in the blue-green algae growth double-nutrient salt circulation nonlinear kinetic model, namely the formula (1), according to actual water body monitoring data by combining a genetic algorithm and a numerical algorithm;
the optimization calibration of the stable constant parameters comprises the following specific steps:
(1) setting initialization conditions and initializing population individuals;
determining the number of individuals, the maximum genetic algebra, the number of parameters to be optimized, the generation channels and the fitness threshold; encoding by adopting multi-parameter cascade floating point number; randomly generating a plurality of different parameter combinations as initial populations to form a parameter space to be optimized;
(2) and (3) evaluating the fitness value: in the evolution process of the population, the advantages and disadvantages of individuals are evaluated through fitness values, and a fitness function is established:
wherein F is a fitness value, and n is a total value of collected water environment dataDays; c. CatThe actual value of the chlorophyll a concentration of the chlorophyll a at the time t, ca(t) is a chlorophyll a concentration function value of chlorophyll a at the time t; n is a radical oftThe real value of the total nitrogen concentration of the nutritive salt at the time t, and N (t) is the value of the total nitrogen concentration function of the nutritive salt at the time t; ptThe real value of the total phosphorus concentration of the nutrient salt at the time t, and P (t) is the function value of the total phosphorus concentration of the nutrient salt at the time t;
using a 4-step Runge Kutta algorithm in the numerical algorithm, then caThe numerical integrals of (t) and N (t) are expressed as:
wherein the parameter qi1,2,3,4, and the specific expression is as follows:
q1＝f1(ca(t),N(t),P(t))
q2＝f1(ca(t)+q1×h/2,N(t)+q1×h/2,P(t)+q1×h/2)
q3＝f1(ca(t)+q2×h/2,N(t)+q2×h/2,P(t)+q2×h/2)
q4＝f1(ca(t)+q3×h,N(t)+q3×h,P(t)+q3×h)
f1(x,y,z)＝T1x+Gg(y)g(z)x
wherein h is the step length, T1Is the environmental loss rate of algae, g (y), g (z) are the same as the function forms of g (N), g (P), x, y, z are the function f1(x, y, z) independent variables;
similarly, the parameter ri1,2,3,4, and the specific expression is as follows:
r1＝f2(ca(t),N(t),P(t))
r2＝f2(ca(t)+r1×h/2,N(t)+r1×h/2,P(t)+r1×h/2)
r3＝f2(ca(t)+r2×h/2,N(t)+r2×h/2,P(t)+r2×h/2)
r4＝f2(ca(t)+r3×h,N(t)+r3×h,P(t)+r3×h)
f2(x,y,z)＝T2(N0-y)-gNg(y)g(z)ca+dNx
parameter si1,2,3,4, and the specific expression is as follows:
s1＝f3(ca(t),N(t),P(t))
s2＝f3(ca(t)+s1×h/2,N(t)+s1×h/2,P(t)+s1×h/2)
s3＝f3(ca(t)+s2×h/2,N(t)+s2×h/2,P(t)+s2×h/2)
s4＝f3(ca(t)+s3×h,N(t)+s3×h,P(t)+s3×h)
f3(x,y,z)＝T3(P0-z)-gPg(y)g(z)ca+dPx
calculating and obtaining chlorophyll a, nitrogen and phosphorus concentration values at each moment according to a numerical method, and substituting the obtained chlorophyll a, nitrogen and phosphorus concentration values and actual monitored chlorophyll a, nitrogen and phosphorus concentration values into a fitness function formula (6) to obtain fitness values F of each individual of the population;
(3) selecting, crossing, mutating and reinserting;
(4) termination of the genetic algorithm:
continuously repeating the steps (2) and (3) according to the fitness value of the individual to form a new individual, wherein the termination condition is that the minimum fitness value of the genetic algebra or the parameter combination with the maximum repetition times is less than or equal to the fitness threshold value; taking the minimum value of the parameter combination adaptability values when the minimum value reaches the termination condition as the adaptability value of the blue algae growth nonlinear dynamics time sequence model, wherein the parameter combination is the optimal parameter combination of the blue algae growth nonlinear dynamics time sequence model;
for the time-varying parameters, predicting values of influence factors of water temperature T (t), illumination I (t) and … … changing along with time according to a multivariate time series model, and substituting prediction results into a formula (5) to obtain a prediction value of the time-varying parameter blue-green algae growth rate G (t); substituting the predicted value of the blue algae growth rate G (t) into a blue algae growth nonlinear dynamics time sequence model, and predicting the blue algae growth nonlinear dynamics time sequence model according to a calculation formula of the Longge Kuta algorithm for predicting the chlorophyll a concentration, the nitrogen and the phosphorus nutrient salt concentration;
step four, determining the nonlinear dynamic conditions of the cyanobacterial bloom outbreak;
taking the growth rate of the blue algae as a time-varying parameter, firstly determining a bifurcation set and a bifurcation point of a blue algae growth nonlinear dynamics time sequence model by adopting a bifurcation theory, then reducing the dimension of the blue algae growth nonlinear dynamics time sequence model by adopting a central manifold method to obtain a reduction equation, determining a critical point of the blue algae bloom outbreak by combining the critical local stability and the type of bifurcation, and providing a formula condition met by the critical point;
(1) firstly, determining a bifurcation set and a bifurcation point of a cyanobacteria growth nonlinear dynamics time sequence model by adopting a bifurcation theory, wherein the specific method comprises the following steps: making the right sides of the three equations in the equation set (4) equal to zero, solving the equation set to obtain the balance point Q (c) of the blue algae growth nonlinear dynamics time sequence modela *,N*,P*)，ca *、N*、P*The chlorophyll a concentration, the total nitrogen concentration and the total phosphorus concentration at the equilibrium point are respectively, and in order to facilitate the analysis to carry out coordinate translation,
the system of equations after coordinate translation is transformed into:
separating the linear terms and the higher order terms yields:
wherein
k is the jacobian matrix of the linear part,
is a high-order term;
solving an equation (8) to obtain a Jacobian matrix of a linear part of a blue algae growth nonlinear dynamics time sequence model equation:
high-order terms:
wherein h.o.t represents a high-order term of cubic degree or more in the expansion;
from equation (1), the equilibrium point Q (c) of the time-varying system for the growth of blue algaea *,N*,P*) There are two cases:
(I)(0,N0,P0),N0,P0the concentrations of the two nutrient salts at the initial moment are respectively;
(II)(ca *,N*,P*) Wherein c isa *,N*,P*Are not all 0;
(I) for the first balance point (0, N)0,P0) Its physical meaning corresponds to the initial stateThe concentration of the blue algae biomass is 0, and the Jacobian matrix at the moment is as follows:
according to the bifurcation theory, the stability of the blue algae growth time-varying system depends on the eigenvalue of the corresponding Jacobian matrix (10); the three characteristic values of the first balance point are respectively
-T2，-T3Due to T2、T3The loss rates of the nitrogen and phosphorus nutrient salts are positive numbers respectively, and when the first characteristic value is
The eigenvalues are positive numbers and the state is stable; when the characteristic value
Is negative, the state is unstable, so when
When the characteristic value is 0, the time-varying system for the growth of blue-green algae is in a critical state, and the system is enabled to be in a critical state
Then G is0Is a bifurcation point;
(II) for a second equilibrium point (c)a *,N*,P*) Is the case where all three variables are present, where ca *,N*,P*G (t), the relation satisfied is:
substituting the last two equations in relation (11) into the first equation to obtain concentrated solution containing only chlorophyll-aSolving the quadratic equation of the degree and the growth rate of the blue algae to obtain two roots related to the chlorophyll a concentration, wherein the corresponding equilibrium state is (c)a1 *,N1 *,P1 *) And (c)a2 *,N2 *,P2 *) The corresponding Jacobian matrix is:
when the characteristic value has zero root, the corresponding blue algae growth rate is set as G1The eigenvalue equation obtained by calculation is as follows: lambda [ alpha ]3+aλ2+bλ+c＝0，
Wherein the content of the first and second substances,
when 0 root appears in the feature root, c is satisfied as 0, and the c is obtained in conjunction with the equation (11)a *,N*,P*G (t), the value of G (t) at this time is represented by G1Then G is1Is also a bifurcation point;
(2) after the branch point of the time-varying system for the growth of the blue-green algae is determined, in order to determine the range of the growth rate of the blue-green algae when the system is stable and the critical point of the burst of the blue-green algae, the stability of the time-varying system for the growth of the blue-green algae and the branch situation of the critical point system in each range need to be researched, which is specifically as follows:
first range (0, G)0): according to the analysis of the blue algae growth double-nutrient salt nonlinear kinetic model on the first balance point, when the blue algae growth rate is (0, G)0) In the range, the blue algae growth time-varying system is in a stable state and can be stabilized at a first balance point; when growth rate
When the characteristic value is 0, the time varying system for the growth of the blue algae is in a critical state, and the dimensionality reduction of the time varying system for the growth of the blue algae is reduced by adopting a central manifold method to obtain the reductionSystem for determining the growth rate of blue algae as G0The method comprises the following specific steps of the bifurcation behavior of a nearby system:
to obtain a reduced system, let
By conversion, G is knowncThe bifurcation behavior near 0, at which time the Jacobian matrix of the time-varying system for blue-green algae growth is:
the three characteristic values are 0, -T2，-T3The linear transformation matrix composed of the corresponding feature vectors is:
by linear transformation (u, v, w)T＝M(x,y,z)TThen, the original differential equation (8) is converted into:
φ'＝Dφ+M-1f(Mφ)；
wherein D ═ diag (0-T)2-T3),φ＝(x,y,z)TAnd when the growth rate of the blue algae is taken as a bifurcation variable, developing to obtain:
according to the central manifold theory, the local dynamic characteristics of the blue algae growth time-varying system are mainly determined by the flow on the central manifold, the system has a two-dimensional central manifold,
wherein, O ((x + G)c)3) Is a higher order term of the function expansion;
for determining the coefficient a in the system of equations (15)1,a2,a3,b1,b2,b3Substituting (15) into the last two equations for solving y and z in the differential equation set (14), obtaining coefficient values after comparing the coefficients of powers of the same degree, then substituting (14) into the first equation for solving x, obtaining an equation on the central manifold, namely a reduction equation, and converting the growth rate parameter into G (t), obtaining the reduction equation as follows:
wherein the content of the first and second substances,
the reduction equation obtained above is expressed by x' ═ f (x, g (t)) ═ lx + mx2+nx3Denotes the coefficients by the letters l, m, n, respectively, since a1＜0,b1If the value is less than 0, judging the positive and negative properties of the coefficient of the reduced equation (16) to obtain m more than 0 and n less than 0; analysis of the reduction equation shows that: equation f (x, g (t)) 0 has two solution curves, and for solution curve x 0, the derivative value Dxf (x, G (t)) a, so that when a < 0, the growth rate is determined
When G (t) > G, the equilibrium point becomes progressively more stable0It is not stable; for another solution curve is on the axis of symmetry
When a solution exists, the following needs to be satisfied:
the growth rate of blue algae by central manifold is G0Analysis of bifurcation behavior of nearby System, Explanation G0The point is a supercritical fork-shaped bifurcation point and is not a burst point of the cyanobacterial bloom; when the growth rate of blue algae is (0, G)0) When the growth rate of the blue algae is within the range, the system is stabilized at a first balance point when the blue algae grows, and when the growth rate of the blue algae is G0And is greater than G0In a section of range, the system is stabilized at a second balance point when the blue algae grows;
continuing the analysis for the second equilibrium point, for a meaningful equilibrium point (c)a1 *,N1 *,P1 *) When the bifurcation set is solved, the eigenvalue of the Jacobian matrix corresponding to the second equilibrium point can not be expressed explicitly, but the corresponding blue-green algae growth rate G when 0 root appears in the eigenvalue can be obtained1I.e. G1Satisfies the following conditions:
when the growth rate is in (G)0，G1) Range, characteristic values less than 0, equilibrium point stabilized at the second equilibrium point, when greater than G1Then, the condition that the characteristic value is larger than 0 occurs, and the system is unstable when the blue algae grows; description of the invention G1The point is the critical point of the blue algae bloom outbreak; critical point G1Satisfies the equation set (17);
the method is characterized in that:
in the first step, the circulating expression of the double nutrient salts for the growth of the blue algae is as follows:
wherein, KN、KPRespectively representing half-saturation concentration constants of blue algae growth to nitrogen and phosphorus;
in the fourth step, when the growth rate of blue algae is (0, G)1) In this range, the time-varying system is stable during the growth of cyanobacteria, i.e., at (0, G)0) When the growth rate of the blue algae is in (G), the system is stabilized at a first balance point when the growth rate of the blue algae is in (G)0，G1) When the blue algae grows, the time varying system is stabilized at a second balance point; when the growth rate is larger than G1In time, the time-varying system for the growth of the blue algae is unstable, and the blue algae bloom outbreak behavior occurs;
fifthly, carrying out outbreak early warning on the cyanobacterial bloom;
the conditional formula of the cyanobacterial bloom outbreak risk is as follows:
G(t)＝GT(T(t))·GI(I(t))＞G1(18)
when the predicted blue algae biomass is larger than the water bloom disaster point threshold c of the actual water environmentaTWhen the condition is met:
ca(t+1)＝ca(t)+h×(q1+2q2+2q3+q4)/6＞caT(19)
indicating the occurrence of the water bloom outbreak behavior, and early warning the water bloom outbreak behavior of the blue algae in advance according to the predicted occurrence time of the water bloom of the blue algae.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710458468.9A CN107292436B (en) | 2017-06-16 | 2017-06-16 | Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710458468.9A CN107292436B (en) | 2017-06-16 | 2017-06-16 | Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN107292436A CN107292436A (en) | 2017-10-24 |
| CN107292436B true CN107292436B (en) | 2020-07-17 |
Family
ID=60096982
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201710458468.9A Active CN107292436B (en) | 2017-06-16 | 2017-06-16 | Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN107292436B (en) |
Families Citing this family (15)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109740877B (en) * | 2018-05-22 | 2023-04-11 | 中国环境科学研究院 | Lake separating eutrophication nutrition footprint index evaluation method |
| CN109541133B (en) * | 2018-09-26 | 2021-10-15 | 中国农业科学院茶叶研究所 | Early nondestructive identification method for nitrogen absorption capacity of tea tree |
| CN109657200B (en) * | 2018-12-05 | 2020-10-23 | 北京师范大学 | Method for determining burst probability of cyanobacterial bloom in lake reservoir |
| CN109711066B (en) * | 2018-12-28 | 2023-01-31 | 南开大学 | Shallow water type small lake and reservoir water bloom prediction method and prediction model |
| CN109858132B (en) * | 2019-01-24 | 2023-05-26 | 北京工商大学 | Blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm |
| CN110427733B (en) * | 2019-09-09 | 2022-11-29 | 河北工程大学 | Method for obtaining algae concentration based on phosphorus cycle |
| CN111160667B (en) * | 2020-01-02 | 2023-05-30 | 北京工商大学 | Method and device for improving robustness of food safety prediction model |
| CN112215438B (en) * | 2020-11-09 | 2021-05-14 | 广东新禾道信息科技有限公司 | Emergency disaster early warning analysis data processing method and system |
| CN112817949B (en) * | 2021-01-05 | 2023-07-04 | 福建省厦门环境监测中心站(九龙江流域生态环境监测中心) | Water bloom prediction method and device combining double models and bioenergy equivalent energy |
| CN114324796B (en) * | 2021-12-28 | 2023-03-10 | 中国科学院南京地理与湖泊研究所 | Prediction method for attached filamentous algae blooms on aquatic plants |
| CN115195951A (en) * | 2022-07-28 | 2022-10-18 | 上海交通大学 | Unmanned ship and method for blue algae bloom early warning and autonomous dosing algae inhibition |
| CN115099072B (en) * | 2022-08-24 | 2022-11-11 | 自然资源部第一海洋研究所 | Marine ecological dynamics model parameter nonlinear optimization method |
| CN115662497B (en) * | 2022-11-10 | 2023-12-29 | 淮阴师范学院 | Construction method of blue algae growth model based on nutrient salt concentration |
| CN116819024B (en) * | 2023-06-28 | 2024-04-19 | 同济大学 | Simulation optimization method and device for lake and reservoir eutrophication ecological dynamic model |
| CN117035164B (en) * | 2023-07-10 | 2024-03-12 | 江苏省地质调查研究院 | Ecological disaster monitoring method and system |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2005045074A2 (en) * | 2003-11-05 | 2005-05-19 | Stichting Voor De Technische Wetenschappen | Means and methods for classifying cyanobacteria |
| JP2008214942A (en) * | 2007-03-02 | 2008-09-18 | Gunma Prefecture | Prediction of generation of water-bloom and method for preventing its generation |
| CN102141517A (en) * | 2011-01-05 | 2011-08-03 | 中国科学院南京地理与湖泊研究所 | Method for predicting water area where water bloom of blue algae occurs first next year in large shallow lake |
| CN103984996A (en) * | 2014-05-26 | 2014-08-13 | 北京工商大学 | Lake-reservoir algal bloom generating mechanism time varying model optimization and prediction method based on taboo searching algorithm and genetic algorithm |
| CN104899653A (en) * | 2015-06-02 | 2015-09-09 | 北京工商大学 | Lake and reservoir cyanobacterial bloom prediction method based on expert system and cyanobacterial growth mechanism timing model |
-
2017
- 2017-06-16 CN CN201710458468.9A patent/CN107292436B/en active Active
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2005045074A2 (en) * | 2003-11-05 | 2005-05-19 | Stichting Voor De Technische Wetenschappen | Means and methods for classifying cyanobacteria |
| JP2008214942A (en) * | 2007-03-02 | 2008-09-18 | Gunma Prefecture | Prediction of generation of water-bloom and method for preventing its generation |
| CN102141517A (en) * | 2011-01-05 | 2011-08-03 | 中国科学院南京地理与湖泊研究所 | Method for predicting water area where water bloom of blue algae occurs first next year in large shallow lake |
| CN103984996A (en) * | 2014-05-26 | 2014-08-13 | 北京工商大学 | Lake-reservoir algal bloom generating mechanism time varying model optimization and prediction method based on taboo searching algorithm and genetic algorithm |
| CN104899653A (en) * | 2015-06-02 | 2015-09-09 | 北京工商大学 | Lake and reservoir cyanobacterial bloom prediction method based on expert system and cyanobacterial growth mechanism timing model |
Non-Patent Citations (2)
| Title |
|---|
| 基于非线性误差补偿旳蓝藻水华时序综合预测方法;王凌斌等;《计算机与应用化学》;20150328;第32卷(第3期);论文第2节 * |
| 蓝藻生长时变系统非线性动力学分析及水华预测方法;王立等;《化工学报》;20170305;第68卷(第3期);论文第1-3节 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN107292436A (en) | 2017-10-24 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN107292436B (en) | Cyanobacterial bloom prediction method based on nonlinear dynamics time sequence model | |
| Chen et al. | Artificial neural network modeling of dissolved oxygen in reservoir | |
| Karul et al. | Case studies on the use of neural networks in eutrophication modeling | |
| CN103218669B (en) | A kind of live fish cultivation water quality comprehensive forecasting method of intelligence | |
| Terseleer et al. | Trait‐based representation of diatom functional diversity in a plankton functional type model of the eutrophied southern North Sea | |
| Ranković et al. | Prediction of dissolved oxygen in reservoirs using adaptive network-based fuzzy inference system | |
| Yulianto et al. | Application of fuzzy inference system by Sugeno method on estimating of salt production | |
| Aldrees et al. | Prediction of water quality indexes with ensemble learners: Bagging and Boosting | |
| Yang et al. | Prediction and control of water quality in Recirculating Aquaculture System based on hybrid neural network | |
| CN105069220A (en) | Back-propagation (BP) neural network immune genetic algorithm based microbial fermentation optimization method | |
| CN109858132B (en) | Blue algae bloom outbreak early warning method based on mutation theory and improved cuckoo algorithm | |
| Yi et al. | Suitable habitat mathematical model of common reed (Phragmites australis) in shallow lakes with coupling cellular automaton and modified logistic function | |
| Aldrees et al. | Evolutionary and ensemble machine learning predictive models for evaluation of water quality | |
| Zhang et al. | The long-term changes in food web structure and ecosystem functioning of a shallow lake: Implications for the lake management | |
| Ellina et al. | Research of fuzzy implications via fuzzy linear regression in data analysis for a fuzzy model | |
| Coleman et al. | Bayesian parameter estimation with informative priors for nonlinear systems | |
| CN114611336A (en) | Circulating water aquaculture dissolved oxygen prediction control method, device, equipment and medium | |
| CN116884523B (en) | Multi-parameter prediction method for water quality of marine pasture | |
| Vázquez-Burgos et al. | An Analytical Hierarchy Process to manage water quality in white fish (Chirostoma estor estor) intensive culture | |
| Miao et al. | A hybrid neural network and genetic algorithm model for predicting dissolved oxygen in an aquaculture pond | |
| Wang et al. | Nonlinear dynamic numerical analysis and prediction of complex system based on bivariate cycling time stochastic differential equation | |
| Galezan et al. | Evaluating the rearing condition of rainbow trout (Oncorhynchus mykiss) using fuzzy inference system | |
| Zhang et al. | An echo state network based adaptive dynamic programming approach for time-varying parameters optimization with application in algal bloom prediction | |
| Xia et al. | Environmental factor assisted chlorophyll-a prediction and water quality eutrophication grade classification: A comparative analysis of multiple hybrid models based on a SVM | |
| Wang | Application of carbon emission prediction based on a combined neural algorithm in the control of coastal environmental pollution in China |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |