CN104881556B - A kind of administering method for blue-green algae problem - Google Patents
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Abstract
A kind of administering method for blue-green algae problem, including step：(1) the improvement algorithm of blue-green algae problem is established；(2) according to fixed time dispensing and the harvest food algae aquatic products that algorithm provides is administered, so as to for blue-green algae problem administer.In step (1), the present invention proposes the three modes biodynamic system launched and harvested with two pulse times.By mathematics model analysis and mathematical derivation, the adequate condition of condition existing for Periodic Solutions and system local stability has been drawn.From the point of view of the result of Matlab emulation, two pulse times that will be launched and harvest is placed in a cycle, it is more preferable than synchronization to obtain effect, namely blue-green algae and silver carp finally all will stabilise at one it is smaller within the scope of, so as to demonstrate the feasibility of mathematical modeling and validity.
Description
Technical Field
The technical scheme mainly researches a fish-algae consumption dynamics model with a pulse control strategy, and proves the global stability of the extinction period solution and the persistence of the system by applying a pulse equation and a small-amplitude disturbance skill. By applying the dynamic model, the biological control method for blue algae treatment problem is obtained.
Background
In recent decades, the control of algal blooms in bodies of water (particularly algae that cause water quality deterioration) has become an increasingly complex problem because of the enormous economic loss that occurs in the event of a sudden loss of control of algal populations. At present, a large amount of cyanobacterial blooms are developed in a plurality of lakes every year, and in the lakes, the ecological balance is destroyed, the water quality is deteriorated, and the human health is threatened. Therefore, the research on how to control the algal bloom has important theoretical significance and practical prospect.
The most common algae control method is to utilize filter-feeding fishes to control the growth of algae population, and the method can prevent the abnormal growth of algae population and further control the eutrophication level of water body. Biological control methods have been used in pest control. Many reservoirs currently use biological methods to control algae reproduction, which has been demonstrated to be effective in preventing algal blooms in east lake Wuhan. However, many scholars suspect that this method is not only expensive, but also not effective in a short time. Another common method is chemical control (usually dilute copper sulfate), which kills most algae populations but has many negative effects. In some cases, this combined effect may lead to a synergistic effect that is better than a simple increase of the individual effects.
Disclosure of Invention
In order to eliminate algae populations, the invention provides the following technical scheme:
a method for treating blue algae problems adopts a biological control method, and comprises the following steps:
establishing a treatment algorithm of the blue algae problem; secondly, algae eating aquatic products are thrown and harvested at a fixed moment given by a treatment algorithm, so that the blue algae problem is treated;
in the step (one):
1) Three-group predation system for establishing pulse control strategy
1.1 The system model is described by the following differential equation:
wherein the content of the first and second substances,
x (t) is the population density of blue algae, y (t) represents the population density of consumers, z (t) is the population density of silver carps, a1 is the population density growth rate of blue algae, a2 is the population density growth rate of consumers, a3 is the population density growth rate of silver carps, alpha is greater than 0, beta is the competition parameter between blue algae and consumers, eta is greater than 0 and mu is the reduction rate of blue algae and consumers respectively, d is greater than 0 and is the conversion rate of silver carps,
Δx＝x(t + )-x(t)，Δy＝y(t + )-y(t)，Δz＝z(t + ) -z (T), T being the pulse period, n =1,2 …, δ>, 0 is the harvest rate of the silver carp at the time t = nT, p&0,p represents the amount of chub delivered at time t = nT (i.e. pulse time);
the competition parameter refers to a coefficient which enables the density of the other party to be reduced by the interaction between the two groups, alpha represents a coefficient which enables the density of the blue-green algae to be reduced by the competition between the consumer and the blue-green algae, beta represents an action coefficient of the blue-green algae to the consumer, and the alpha and beta generally take values between [0,1 ];
eta represents the coefficient that the chub reduces the density of the blue algae by feeding on the blue algae, and mu represents the coefficient that the chub reduces the density of the consumers by feeding on the consumers;
d represents the conversion rate of the silver carp, wherein the conversion rate is a coefficient for increasing the population density of the silver carp by using blue algae and predators as feed;
t is time, T is pulse period, n =1,2,3 …; here, the consumer refers to intermediate predators, also feeding on algae;
the model of the formula (1.1.1) establishes that the chub putting and harvesting occur at a uniform pulse moment;
2) Establishing three groups of biological power systems with two-pulse time throwing and harvesting
1.1 The system model is described as follows:
three group predation system models (1.1.1) for establishing a pulse control strategy are improved: adding blue algae dependent functional functionPulse putting and pulse harvesting are placed at two different moments; introducing a fixed period for harvesting the silver carps to a consumer, chemically controlling algae, releasing the silver carps at a fixed time when t = n T, and obtaining a system model shown as a formula 2.1.1 on the basis of the formula 1.1.1:
under the condition that the population density of consumers and chubs is small, the growth of the blue algae population meets the Logistic function;
r is the intrinsic growth rate, k is the environmental capacity, r/k is the competition coefficient in the population, in the model, the meanings of alpha and beta are the same as in the model 1.1.1,
d 1 、d 2 the mortality rate of consumers and silver carps respectively;
l represents a measure of another pulse instant;
constant E 1 、E 2 、E 3 Respectively the reduction ratio of the population density of blue algae, the population density of consumers and the population density of silver carps due to the adoption of additional control measures, T>0，0<L<1,0≤E 1 、E 2 、E 3 <1；
Andare Holling II type functions, namely functional response functions, depend on the population density of blue-green algae and consumers respectively, and meet the following conditions: f (x) epsilon C 1 ,F(x)∈R + F (0) =0,F (x)/x is monotonically decreasing where C is 1 Is a complex field, R + Is positive.
Drawings
Three group predation systems for pulse control strategies:
FIGS. 1-1-a, b and c show a fixed time harvesting and feeding three population system under pulse control, (p ≧ 18.9752836) where FIG. 1-1-a is a time series diagram for cyanobacteria; 1-1-b are time series diagrams of consumers; FIG. 1-1-c is a time-series chart of silver carp.
1-2-a, b and c fixed-time harvesting and delivery of three population systems (p > 8.905127) under pulse control, wherein FIG. 1-2-a is a time series diagram of cyanobacteria; 1-2-b are time series diagrams of consumers; fig. 1-2-c are time-series graphs of silver carps.
Three groups of biological power systems aiming at two-pulse time release and harvest:
FIGS. 2-1-a, b and c are functional dependent three population systems for pulse delivery and harvesting at different times, wherein FIG. 2-1-a is a time series diagram of blue algae; 2-1-b are time series diagrams of consumers; FIG. 2-1-c is a time-series chart of silver carp.
FIGS. 2-2-a, b and c are functional dependent three-population systems (ln (1-E) for pulsed delivery and harvesting at different times 1 ) + rT ≦ 0), wherein FIG. 2-2-a is a time series diagram for cyanobacteria; 2-2-b are time series diagrams of consumers; fig. 2-2-c are time-series graphs of silver carps.
FIGS. 2-3-a, b and c are functional dependent three-cluster systems (ln (1-E) for pulsed delivery and harvesting at different times 1 )+rT>, 0), wherein FIG. 2-3-a is a time series diagram of blue algae; 2-3-b are time series diagrams of consumers; fig. 2-3-c are time-series graphs of silver carps.
Detailed Description
The technical scheme is further explained by combining the drawings and the detailed implementation mode as follows:
a biological control method for blue algae treatment problem is to treat the blue algae problem according to a finishing algorithm.
In the technical scheme, the design idea of the treatment algorithm is as follows:
1. three-group predation system for establishing pulse control strategy
1.1 mathematical model building
The ecological population dynamics model not only considers the natural growth and interaction of the population, but also improves the understanding of the food chain and food net functions and their dependence on environmental conditions. Pulse differential equations have been widely used to study the mathematical properties of pulse predation and food net models. In addition, although the pulse differential equation theory is not richer than the corresponding differential equation theory, the real ecological problem can be reflected more effectively. The eutrophication of the water body of the reservoir is more and more serious, and the method for effectively controlling the abnormal propagation of the algae population comprises the following steps: generally, people can put certain filter feeding fishes into a reservoir and control the abnormal growth of algae through the food chain relation between the filter feeding fishes and the reservoir. Meanwhile, the manager can harvest the thrown fishes regularly in consideration of cost effectiveness. Whether it is put or harvested, these behaviors are transient processes compared to the period of growth of the species, and are all perturbations from a system perspective. It can therefore be assumed that these disturbances act in the form of pulses. The reference (Jiang G R, lu Q S, luo G L, impulse Control of a stage-structured food management system. Journal of physical Study,2003,36 (4): 331-344) predation-capture system model can be described by the following differential equation:
x (t) is the population density of blue algae, y (t) represents the population density of consumers, z (t) is the population density of silver carp, a i >, 0 (i =1,2,3) is the growth rate, α>0,β> 0 represents a competition parameter between baits, eta>, 0 and mu>, 0 is the reduction rate of food and d> 0 is the conversion rate of predators.
Wherein Δ x = x (t) + )-x(t)，Δy＝y(t + )-y(t)，Δz＝z(t + ) -z (T), T being the pulse period, n =1,2 …, δ>, 0 is the harvest rate of the silver carp at the time t = nT, p&0 is the putting amount of the silver carps at the time of t = nT, and the model builds that the putting and harvesting occur at the time of unified pulse.
1.2 Theoretical analysis of the correctness of the System (1.1.1)
If the blue algae and the consumer are extinct, the system 1.1.1 becomes
Theorem 1.1
(1)Is that the system (1.1.1) isPeriodic solution of time, where T ∈ (nT, (n + 1) T]And n is a positive integer.
(2)Is that the system (1.1.1) is at z 0 ＝ z (0 + ) In which T ∈ (nT, (n + 1) T]And n is a positive integer.
(3) For general and periodic solutions z (t) to the system (1.1.1), when t → ∞ z (t)
Whereas the system (1.1.1) has one blue algae and the semi-trivial periodic solution of the consumer:
theorem 1.2 there is a normal number M, for a time t large enough such that x (t) is less than or equal to M, y (t) is less than or equal to M, and z (t) is less than or equal to M; where X (t) = (X (t), y (t), z (t)) is an arbitrary solution for system (1.1.1).
And (3) proving that: defining the function V (t, x) as
V(t,x)＝dx(t)+dy(t)+z(t)
Thus, when t ≠ nT, the derivative of V (t, x) along the system (2.1.1) is
D + V(t,x)+LV(t)≤d(L+a 1 )x(t)-dx 2 (t)+d(L+a 2 )y(t)-dy 2 (t)+(L-a 3 ) z (t) is 0<L<a 3 ：d(L+a 1 )x(t)-dx 2 (t) and d (L + a) 2 )y(t)-dy 2 (t) are bounded, thus D + V (t, x) + LV (t) is bounded, i.e. D + V(t,x)+LV(t)<K。
Theorem 1.3 if
And
if true, the semi-trivial periodic solution (0, z (t)) of the system (4.1.1) is locally asymptotically stable.
And (3) proving that: the local stability of the periodic solution (0, z (t)) is analyzed by considering small amplitude perturbations of the solution, defining x (t) = u (t), y (t) = v (t), z (t) = w (t) + z (t), substituting the transformation into the system (1.1.1), and then performing a linear approximation, omitting higher order terms, then the system (1.1.1) is linearized as:
therefore, phi (t) is satisfied if phi (t) is the standard base solution matrixWherein
Since φ (t) is a standard basis matrix, φ (0) = I is an identity matrix, and
the local stability of the semi-trivial periodic solution (0, z (t)) of the system (1.1.1) is thus determined by the matrix
Is determined by the characteristic value of (1), wherein
λ 3 ＝(1-δ)exp(-a 3 T)<1；
According to Floquet multiplier theory, if λ 1 <1，λ 2 &And (1) the semi-trivial periodic solution of the system is locally asymptotically stable. Namely that
If the theorem is true, the theorem proves to be complete.
It is demonstrated below that (0,0,z) * (t)) global attractiveness. Selecting epsilon>, 0, such that
Due to the fact thatConsider the following pulse differential equation:
there is z (t) ≧ g (t) due to the pulse differential equation comparative theorem, and when t → z, g (t) → z * (t) when t is sufficientWhen the size of the bag is large enough,
z(t)≥g(t)≥z * (t)-ε。
from equation (1.1.1)
In the interval T = (nT, (n + 1) T ], both sides are integrated simultaneously to obtain
Then x (nT). Ltoreq.x (0) + ) exp (n θ), and n → ∞ time, x (nT) → 0. Thus when n → ∞, x 1 (t) → 0. Using the same method, x can be verified when n → ∞ is reached 2 (t)→0。
It is demonstrated thatAnd isThen when t → ∞ is reached, z (t) → z * (t) of (d). For theExist ofSo that 0<x(t)<ε 1 ,0<y(t)<ε 1 ,Then
By the theorem of comparisong 1 (t)≤z(t)≤g 2 (t), when t → ∞ g 1 (t)→z * (t)，g 2 (t)→z * (t)，g 1 (t) and g 2 (t) are solutions of the following equations, respectively:
and
wherein
So when t is sufficiently large, z * (t)-ε 2 <z(t)<g * 2 (t)+ε 2 ,ε 2 >, 0. Let epsilon 1 → 0, when t → ∞ is reached, g * 2 (t)→z * (t),z(t)→z * (t) of (d). After the syndrome is confirmed.
1.3 Analysis of simulation
At the present theoretical level, the display analytical solution of the system cannot be found, so some of the kinetic properties of the system (1.1.1) can only be studied by numerical simulation. In order to study the impact of impulse disturbances, periodic capture and release on the system dynamics, the long-term dynamics of the system (1.1.1) was numerically simulated.
Given the advance that the system satisfies biological significance, a set of parameter values for the system (1.1.1) and the initial values that it satisfies are as follows:
a1＝1.5；a2＝1；a3＝1；α＝1.05；δ＝0.8；β＝1.05；η＝0.4；μ＝0.2；d＝2.5；
x(1)＝0.1；y(1)＝0.1；z(1)＝0.1；T＝5
FIGS. 1-1-a, b and c are simulations of the dynamic behavior of a dynamic system (1.1.1) with impulsive disturbances with a set of data. The figure shows a fixed time harvest and release three population system under pulse control, p ≧ 18.9752836. From the figure, it can be seen that blue algae and consumers are both extinct, and silver carp is finally stable in a range. FIG. 1-1-a is a time series diagram of cyanobacteria; 1-1-b is a time series diagram of a consumer; FIG. 1-1-c are time series diagrams of silver carp z.
Calculated when p is more than or equal to p max The semi-trivial periodic solution (0, z (t)) is globally asymptotically stable or locally asymptotically stable when =18.9752836, and fig. 1-1-a, b and c further verify the correctness thereof. In fact, when p>, 8.90512, the consumer is exhausted, as can be illustrated in fig. 1-2-a, b and c.
FIGS. 1-2-a, b and c are fixed time harvest and release three population systems (p > 8.905127) under pulsed control. As can be seen from fig. 1-2-a, b and c, when p >8.905127, the consumer is quickly extinct, and blue algae and silver carp coexist, here it can be seen that the consumer is an intermediate predator, and also feeds on blue algae, and when the value of p increases from 0, the density of blue algae increases rapidly, and the density of consumer decreases rapidly, and the density of silver carp increases. These kinetic behaviors show that pulsed perturbations can promote the long-lasting survival of predators, but also accelerate the extinction process of consumers. Therefore, when the value of the released amount p of chub is small, chub z has no negative influence on the maximum density of cyanobacteria x, but it has a great influence on the maximum density of consumer y. This means that when all parameters are determined, the value of the release amount p should be less than a certain threshold value so that the system (1.1.1) is persistent, i.e. the value of p is determined according to theorem 1.3. The above numerical simulations demonstrate the correctness of the theoretical derivation described above and the feasibility of the pulse control strategy employed.
2. Establishing three groups of biological power systems with two-pulse time throwing and harvesting
1.1 ) modeling
The model (1.1.1) is improved: incorporating bait dependent functional functionsPutting pulse putting and pulse harvesting at two different moments; introducing a fixed period for harvesting consumers (fishes), chemically controlling algae, and releasing natural enemies (fishes) at t = nT in a timing mode, wherein a mathematical model of the system is shown as a formula 2.1.1:
in the formula 2.1.1, x, y and z respectively represent the population density of blue algae, consumers and silver carps, the silver carps z prey on the consumers y, and the consumers feed on the blue algae x. Under the condition that the population density of consumers and silver carps is smaller, the growth of the blue algae population meets the Logistic function, r is the intrinsic growth rate, r/k is the competition coefficient in the population, k is the environment capacity, alpha and beta are the conversion rate respectively, a 1 、a 2 Respectively, coefficients of a function of the functional type, d 1 、d 2 For mortality, T is the pulse period, p is the pulse dose, constant E 1 、E 2 、E 3 Respectively the proportion of x, y, z reduced by some control measure, T>0，0<L<1,0≤E 1 、E 2 、E 3 <1；
Andcalled the Holling class ii function (i.e. functional response function), which depends on the population density of the cyanobacteria and the consumer, respectively, and satisfies: f (x) is belonged to C 1 ,F(x)∈R + F (0) =0,F (x)/x is monotonically decreasing.
2.2 System 2.1.1 theoretical analysis of correctness
If consumer y goes out, i.e. y =0, the system (2.1.1) may become two subsystems as follows:
and
theorem 2.2.1
(1) If ln (1-E) 1 )+rT>, 0, there is a periodic solution for the system (2.2.2)And is provided with
(ⅰ)
(ii) System (1.2.2) satisfies the initial value x 0 &X (t) of solution gt, 0 is
(ⅲ)WhereinRespectively taking different values r for r 1 ，r 2 A periodic solution of time;
(2) If ln (1-E) 1 ) + rT is less than or equal to 0, the solution of the system (1.1.2)
It was demonstrated that first, an arbitrary solution v (t) of (2.2.2) is readily available,
and
Then
i is ln (1-E) 1 )+rT>, 0. Derived from the periodicity of the periodic solution
And
obviously, when ln (1-E) 1 )+rT>, 0, there is a unique and strictly periodic solutionIn fact, it is possible to use,
wherein
In addition, by
Can obtain
And
byIs obtained periodically
Conclusion (i) is confirmed.
Let x (t) be (2.2.2) satisfying a positive solution for the initial value, as will be demonstrated below
If it isThenNow suppose thatIf the sign inversion is not equal, then there is a similar analysis.
Note f: R + →R + Is composed of
It is readily apparent that f (x) is strictly increasing, andis strictly in the mean ofReduced, as shown in (2.2.3)
ByCan know the periodicity ofSince f is in R + The upper is strictly increased, then
And
(because off (x)/x is strictly increasing).
The same can be obtained
And
then x ((n + L) T) + ) Is monotonically decreasing and bounded and converges to a certain w 1 >, 0. In addition, the first and second substrates are,
x((n+L+1)T + )-x((n+L)T + )＝f(x((n+L)T + )-x((n+L)T + ) → 0,n → ∞ from above 1 )＝w 1 AndThen
it can be proved by (2.2.3)
Conclusion (ii) is confirmed.
Using the expression of (2.2.5) andcan prove conclusions (iii). In fact, it is possible to use,
similarly, the following conclusions can be drawn:
lemma 2.2.2 system (2.1.3) has a one cycle solutionAnd is
(1)
(2)
(3) WhereinAre respectively d 2 Periodic solutions when taking different values.
Solution to boundary periodAndthe local stability of (a) can be discussed by using Floquet theory and the comparative theorem, and further has the following theorem:
theorem 2.2.1 setting ln (1-E) 1 )+rT>, 0, then
(1) Periodic solutionIs unstable;
(2) If it is
Then the periodic solutionIs locally asymptotically stable.
Consider the followingAndglobal stability of (3).
Theorem 2.2.2 (1) ln (1-E) 1 ) + rT is less than or equal to 0, thenIs globally asymptotically stableDetermining;
(2) If ln (1-E) 1 )+rT>, 0 and
wherein
Wherein
ThenIs globally asymptotically stable.
Proving (1) ln (1-E) 1 ) + rT is less than or equal to 0. Let epsilon 1 >, 0, such that α g 1 (ε 1 )<d 1 (this is always possible becauseWhereinOrder toNote 0<η<, 1, then
By the theory of comparison(t.gtoreq.0) whereinIs the solution of (1.2.1). Because when t → ∞ is reachedWhen the temperature of the water is higher than the set temperature,since x (T) → 0 is known from 2.2.1, T is present 1 >, 0, making x (t) less than or equal to epsilon 1 (t≥T 1 ) For simplicity, x (t) is set to be less than or equal to epsilon 1 (t>, 0) are then
Integrating the formula on ((n + L-1) T, (n + 1) T)
ln(y((n+L)T))-ln(y((n+L-1)T + ))≤(αg 1 (ε 1 )-d 1 )T,n≥1
And
ln(y((n+L)T))-ln(y((n+L-1)T))-ln(1-δ 1 )≤(αg 1 (ε 1 )-d 1 )T,n≥1，
and y ((n + L) T) ≦ y ((n + L-1) T) η and y ((n + L) T) ≦ y (LT) η n This means that when n → ∞ y ((n + L) T) → 0. In addition, by
It can be known that
y(t)≤y((n+L+1)T + ) t∈((n+L-1)T,(n+1)T]
And when t → ∞ is reached, y (t) → 0.
It will be demonstrated below that when t → ∞ timeFor this purpose, let 0<ε 2 <d/(βT 2 ). Since y (T) → 0 when T → infinity, there is T 2 >, 0, so that T is more than or equal to T for all T 2 All have y (t) less than or equal to epsilon 2 . For simplicity, let y (t) be ≦ ε 2 (t>, 0), then
≤βL 2 y(t)z(t)-d 2 z(t)
≤-(d 2 -βL 2 ε 2 )z(t)t t≠(n+L-1)T,t≠nT
By the theorem of comparison
WhereinIs a solution of 2.2.2),is the parameter d in (2.2.2) 2 By d 2 -βL 2 ε 2 The solution obtained thereafter.
Since these solutions are close to → infinity when t → infinity, respectivelyAndso by the lemma of 2.2.2, when t is large enough,
the reason 2.2.2 shows that the conclusion (1) is true.
(2) Let ln (1-E) 1 )+rT>, 0, it will be demonstrated that y (t) → 0 when t → ∞. For this purpose, epsilon is selected 3 >, 0, such that
Because of the fact thatAnd (2.2.7) are true, so ε 3 The choice of (2) is feasible. Note the book
Easy to see 0< ξ <1. By
And comparative theoremWhereinIs a solution of (1.2.1) because any solution is close to when t → ∞ timesAll known from the introduction 2.2.1 that T is present 3 >, 0, such thatFor the sake of simplicityThen
By the theory of comparisonWhereinThe solution (with z (t)) of (2.2.2) has the same initial value. Because when t → ∞ the arbitrary solution is close toTherefore, the existence of T is known from 2.2.2 4 >, 0, such thatFor the sake of simplicity, let
From the final bounded by y (T), there is T 5 >, 0, so that y (t)<M(t≥T 5 ) Where M is a bounded constant. For simplicity, let y (t)&And (t is more than or equal to 0). Notice that g is 2 (y(t))≥c g2 y (t) (t.gtoreq.0), then
And
integrating the formula in the interval ((n + L-1) T, (n + 1) T)
And
so that it is known from the periodicity
y ((n + L) T) is less than or equal to y ((n + L-1) T) xi and y ((n + L) T) is less than or equal to y (LT) xi n
This means that when n → ∞ is reached, y ((n + L) T) → 0, and further
WhereinIs a bounded constant, therefore
This means that
And when t → ∞, y (t) → 0.
It will be demonstrated below that when t → ∞,for this purpose, let 0<ε 4 ≤r/L 1 From y (t) → 0 (t → ∞), it is known that y (t) exists<ε 4 (t≥T 6 ). For simplicity, let y (t)<ε 4 (t.gtoreq.0). When T ≠ (n + L-1) T, T ≠ (n + 1) T,
by the theorem of comparisonWhereinIs (2.2.1)To solve the problem that the reaction solution is not stable,to change r in (2.2.1) to r-L 1 ε 4 The solution of (same initial value as x). Since these solutions are close to → ∞ when t → respectivelyAndso that when t is sufficiently large,
similarly, it can be demonstrated that when t → ∞,after the syndrome is confirmed.
2.3 Simulation analysis of model 2.1.1
On the premise of meeting the biological significance of the system, a group of data is selected to simulate the system (2.1.1) in Matlab, and the numerical values are selected as follows:
T＝3；L＝0.07；r＝1.6；k＝5；α＝0.5；a1＝1.5；η＝0.4；a2＝0.5；
β＝0.5；d1＝0.2；d2＝0.5；δ1＝1.65；δ2＝1.45；δ3＝1.2；x(1)＝0.5；y(1)＝0.3；z(1)＝0.5；
the selection results in figures 2-1-a, b and c, which show that at the time, blue algae and consumers are extinct, and silver carps are gradually stable, thus proving the correctness of conclusion 1 of theorem 2.2.2.
From FIGS. 2-1-a, b and c, when the selected data satisfies ln (1-E) 1 )+rT>, 0, it can be seen that the system
(2.1.1) periodic solutionIs unstable.
When ln (1-E) is satisfied 1 ) When + rT is less than or equal to 0, the density of blue algae and consumers is rapidly reduced to 0 as shown in figures 2-2-a and b, and the chub is finally locally stable as shown in figures 2-2-c, so that the conclusion (1) in the theorem (2.2.2) is correct.
When ln (1-E) is satisfied 1 )+rT>0、
It can be seen from FIGS. 2-3-a, b and c that the conclusions in theorem (2.2.2) are correct.
The model is based on a mathematical model of a formula 1.1.1, three group models of releasing and harvesting at a fixed pulse time are improved into three group models at two pulse times, and the conditions of the existence of a system periodic solution and the sufficient conditions of the local stability of the system are obtained by aiming at the analysis and the mathematical derivation of the mathematical model on the basis of the model. From the Matlab simulation result, the putting and harvesting are carried out at two pulse moments in one period, the obtained effect is better than that at the same moment, namely, the blue algae and the silver carp are finally stabilized in a smaller range, and therefore the feasibility and the effectiveness of the mathematical model are proved.
Claims (1)
1. A method for treating blue algae problems adopts a biological control method, and is characterized by comprising the following steps:
establishing a treatment algorithm of the blue algae problem; (II) putting and harvesting algae-eating aquatic products at a fixed moment given by a treatment algorithm, thereby treating the blue algae problem;
in the step (one):
1) Three-group predation system for establishing pulse control strategy
1.1 The system model is described by the following differential equations:
wherein the content of the first and second substances,
x (t) is the population density of blue algae, y (t) represents the population density of consumers, z (t) is the population density of silver carps, a1 is the population density growth rate of blue algae, a2 is the population density growth rate of consumers, a3 is the population density growth rate of silver carps, alpha is more than 0, beta is more than 0 and represents the competition parameter between blue algae and consumers, eta is more than 0 and mu is the reduction rate of blue algae and consumers respectively, d is more than 0 and is the conversion rate of silver carps,
Δx＝x(t + )-x(t)，Δy＝y(t + )-y(t)，Δz＝z(t + ) -z (T), T being the pulse period, n =1,2 …, δ&0 is the harvesting rate of the silver carp at the time t = nT, and p >0,p represents the putting amount of the silver carp at the time t = nT, namely the pulse time;
the competition parameter refers to a coefficient which enables the density of the other party to be reduced by the interaction between the two groups, alpha represents a coefficient which enables the density of the blue-green algae to be reduced by the competition between the consumer and the blue-green algae, beta represents an action coefficient of the blue-green algae to the consumer, and the alpha and beta generally take values between [0,1 ];
eta is the coefficient that blue algae feed on silver carp to reduce the density of blue algae, and mu is the coefficient that silver carp feed on consumer to reduce the density;
d represents the conversion rate of the silver carp, wherein the conversion rate is a coefficient for increasing the population density of the silver carp by using blue algae and predators as feed;
t is time, T is pulse period, n =1,2,3 …; here, the consumer refers to intermediate predators, also feeding on algae;
the model of the formula (1.1.1) sets up the unified pulse time of silver carp throwing and harvesting;
2) Establishing three groups of biological power systems with two-pulse time throwing and harvesting
1.1 The system model is described as follows:
three kinds of group predation system models (1.1.1) for establishing a pulse control strategy are improved: adding blue algae dependent functional functionPulse delivery and pulse harvesting are arranged inTwo different times; introducing a fixed period for harvesting the silver carp to a consumer, chemically controlling the algae, releasing the silver carp at a fixed time when t = nT, and obtaining a system model shown as a formula 2.1.1 on the basis of the formula 1.1.1:
under the condition that the population density of consumers and chubs is small, the growth of the blue algae population meets the Logistic function;
r is the intrinsic growth rate, k is the environmental capacity, r/k is the competition coefficient in the population, alpha, beta are the conversion rate respectively,
d 1 、d 2 the mortality rate of consumers and silver carps respectively;
l represents a measure of another pulse instant;
constant E 1 、E 2 、E 3 The proportion of the reduction of the population density of blue algae, the population density of consumers and the population density of silver carps is respectively that T is more than 0,0 and more than L is more than 1,0 and less than or equal to E 1 、E 2 、E 3 ＜1；
Andthey are Holling class ii functions, which depend on the population density of the cyanobacteria and the consumer, respectively, and satisfy: f (x) is belonged to C 1 F (x) ∈ R +, F (0) =0,F (x)/x is monotonically decreasing, where C 1 Is a complex field, R + Is positive.
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Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101712511A (en) * | 2009-04-21 | 2010-05-26 | 北京普仁生态技术有限公司 | Method for controlling cyanobacterial bloom by utilizing composite strain and trace element |
| CN103109868A (en) * | 2013-03-11 | 2013-05-22 | 唐山市开平区玉生微生物科技有限公司 | Mixture for killing blue-green algae and method for comprehensively treating blue-green algae by using same |
| CN104355412A (en) * | 2014-11-15 | 2015-02-18 | 北京紫岚风生态环境技术研究院 | Method for biologically purifying eutrophic water body |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101712511A (en) * | 2009-04-21 | 2010-05-26 | 北京普仁生态技术有限公司 | Method for controlling cyanobacterial bloom by utilizing composite strain and trace element |
| CN103109868A (en) * | 2013-03-11 | 2013-05-22 | 唐山市开平区玉生微生物科技有限公司 | Mixture for killing blue-green algae and method for comprehensively treating blue-green algae by using same |
| CN104355412A (en) * | 2014-11-15 | 2015-02-18 | 北京紫岚风生态环境技术研究院 | Method for biologically purifying eutrophic water body |
Non-Patent Citations (2)
| Title |
|---|
| 具有脉冲控制的蓝藻治理模型的研究;李胜光;《中国优秀硕士学位论文全文数据库 基础科学辑》;20131215;全文 * |
| 几种我国常见鱼类对经典生物操纵的影响研究;王娣娟;《中国优秀硕士学位论文全文数据库 工程科技Ⅰ辑》;20121115;全文 * |
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