Ecological analysis method of plankton-fish model based on interval value parameters

Abstract

The invention relates to an ecological analysis method of a plankton-fish model based on interval value parameters, and belongs to the field of prediction and simulation of the number of plankton and fish populations in a marine ecosystem. According to the method, a plankton-fish biological mathematical model is established by considering the condition that parameters are interval values; determining the positive property and the bounded property of the number of the model research objects; analyzing existence and stability of balance points of the model; meanwhile, an ecological balance point of the model is given under the condition of considering ecological and economic benefits; the optimal solution of the model is obtained, and the optimal fishing strategy is determined. By means of the analysis method, the evolution law of the phytoplankton, zooplankton and fish population in the ocean can be obtained, and the method has positive guiding significance on species coexistence and ecological balance of coastal water areas.

Description

Technical Field

[0001] The invention belongs to the technical field of prediction and simulation of the number of plankton and fish populations in marine ecosystems, and in particular, relates to an ecological analysis method of a plankton-fish model based on interval value parameters. Background Art

[0002] Many scholars have been building mathematical models and studying the dynamic behavior of ecosystems under precise biological parameter conditions. In fact, in reality, biological parameters are not fixed. At the same time, information asymmetry, data loss, measurement errors and other problems may occur during the modeling process, resulting in a certain error between the biological parameter value and the true value. Therefore, fuzzy mathematical modeling has important application value in the study of real-world problems such as ecological environment, biological systems and disease transmission.

[0003] Phytoplankton is an autotrophic plankton that is the main creator of marine, ocean and freshwater ecosystems. Zooplankton is a microscopic animal that lives in water and eats phytoplankton. Due to its global presence and importance, the study of the dynamic properties of phytoplankton, zooplankton and fish is of vital importance to our society. Mathematical models are used to describe the dynamics of the interaction between phytoplankton and zooplankton. In 2002, Chattopadhyayay et al. studied the mathematical model of the TPP-zooplankton system, in which the predatory ability of zooplankton is reduced due to the release of toxic substances. In addition to zooplankton, some fish and other aquatic organisms also eat phytoplankton. Many researchers have become interested in phytoplankton models, for example, Das and Ray, Saha and Bandyopadhyay, Zhang and Rehim, etc. have studied models based on phytoplankton and zooplankton. In 2008, Das and Ray studied the effect of time lag on nitrogen cycle in the interaction between phytoplankton and zooplankton. In 2000, Barros and Bassanezi et al. studied population dynamics and fuzzy dynamics models and analyzed the stability of fuzzy differential equations. In 2003, Guo et al. first considered fuzzy impulses in the Logistic growth model. In 2008, Bassanezi et al. studied the traditional deterministic predator-prey population model based on fuzzy rule methods. In addition, in 2015, Zhao and Wang used interval values ​​to study the Lotka–Volterra model to determine the optimal harvesting method and bioeconomic equilibrium point.

[0004] Based on the relevant theories and methods of biological models, scholars have conducted many studies on the plankton and fish problems in the marine ecological environment, which has promoted the development of ecological dynamics. In the process of implementing the invention, the inventor found that there is still room for further research on the ecological problem analysis method of the plankton-fish model with interval value parameters:

[0005] (1) There is relatively little literature on the temporal evolution of plankton and fish, and the uncertain effects of parameters have been neglected in the problems of plankton and fish in aquatic environments;

[0006] (2) In order to balance the ecological balance of the ocean and maximize economic benefits, this paper combines the theory of well-posedness, stability, and optimal control of differential equations to conduct theoretical analysis and numerical simulation evolution simulation, so it can more vividly reflect the evolution process of the number of phytoplankton, zooplankton, and fish populations in the marine ecological environment. Summary of the invention

[0007] In view of the problems and shortcomings of the prior art mentioned above, the problem to be solved by the present invention is to provide a method for analyzing the stability and optimal strategy problems of a plankton-fish model with interval value parameters. This method more reasonably utilizes the theories related to the stability and optimization of differential equations, and is more suitable for the simulation of population quantity evolution and optimal control strategies of three-dimensional ecological models.

[0008] In order to solve the above problems, the present invention adopts the following technical solutions:

[0009] The present invention is an ecological analysis method for a plankton-fish model with interval value parameters, and the specific steps are as follows:

[0010] A. Considering the case where the parameters are interval values, establish a biomathematical model of plankton-fish; determine the positivity and boundedness of the number of research objects in the model;

[0011] B. Analyze the existence and stability of the model's equilibrium point; at the same time, give the model's ecological equilibrium point while considering the relationship between ecological and economic benefits;

[0012] C. Obtain the optimal solution of the model and determine the best fishing strategy.

[0013] The above-mentioned ecological analysis method of a plankton-fish model based on interval-valued parameters has a further preferred technical solution: the parameters considered in step (A) are interval-valued, a biomathematical model of plankton-fish is established, and the positivity and boundedness of the number of research objects of the model are determined, and the specific steps are as follows:

[0014] A1. Considering the impact of nutrients on fish populations, the Holling III functional response function of fish to animals, and the main biological parameters in the model are not accurate, the biological model studied is

[0015]

[0016] Where x1 represents the number of phytoplankton, x2 represents the number of zooplankton, and x3 represents the number of fish populations. The biological significance of the parameters of model (1) is shown in Table 1, and all parameter values ​​are positive numbers.

[0017] A2. According to the actual biological significance of the model, the initial conditions of the model are given to meet the following requirements:

[0018] x1(0)=x 10 ≥0, x2(0)=x 20 ≥0, x3(0)=x 30 ≥0 (2)

[0019] In order to introduce the interval value algorithm, the following definition is given first.

[0020] A closed interval [c1,c u ] is defined as an interval number, which satisfies {z:c1≤z≤c u ,z∈R}, we call c1 and c u are the lower and upper bounds of the interval value respectively.

[0021] From a mathematical point of view, in order to better describe real numbers, the interval-valued function is defined below.

[0022] Let g(q) = c 1-q d q , q∈[0,1], c, d>0, then g(q) is called an interval-valued function of the closed interval [c,d].

[0023] A3. Under the initial condition (2), the solution of model (1) for any t>0 is non-negative and uniformly bounded.

[0024] Table 1 Biological significance of parameters in model (1)

[0025]

[0026] C. Obtain the optimal solution of the model and determine the best fishing strategy.

[0027] The above-mentioned ecological analysis method of a plankton-fish model based on interval value parameters has a further preferred technical solution: the existence and stability of the equilibrium point of the analysis model in step (B), while giving the ecological equilibrium point of the model under the condition of considering the balance between ecological and economic benefits, and the specific steps are as follows:

[0028] B1. Determine the existence and stability of the boundary equilibrium point of model (1).

[0029] The equilibrium point of model (1) satisfies the following equations

[0030]

[0031] Obviously, model (1) has four possible equilibrium points: E0 = (0, 0, 0), E1 = (K, 0, 0), It is easy to see that two trivial equilibrium points, E0 = (0,0,0) and E1 = (K,0,0), must exist. At the equilibrium point E0 = (0,0,0), the population is in an extinction state, and this situation always exists. The equilibrium point E1 = (K,0,0) is the axial equilibrium point where only the phytoplankton population survives.

[0032] The trivial equilibrium point E0 = (0,0,0) is unstable. When , the equilibrium point E1 = (K, 0, 0) of model (1) is locally asymptotically stable. When , then model (1) has a semi-trivial equilibrium point Furthermore, when G 33 <0,G 11 <0,G 12 G 21 When <0, the semi-trivial equilibrium point E2 is locally asymptotically stable. Among them:

[0033]

[0034] B2. Determine the existence and stability of the positive equilibrium point of model (1).

[0035] when When , then model (1) has a positive trivial equilibrium point In fact, from model (1), we can see that the positive equilibrium point E3 should satisfy the following equation:

[0036]

[0037] From the third equation of equation group (4), we get Will Substituting the first and second equations of equation (4), we can obtain

[0038]

[0039] In the first equation of equation (5), let We get make then, And further,

[0040]

[0041] From the initial conditions, we know that In the second equation of the equation system (5), let get

[0042]

[0043] make get

[0044]

[0045] Furthermore, we get

[0046]

[0047] because when hour,

[0048]

[0049] At this time, the positive equilibrium point E3 exists.

[0050] In addition, when κ1>0,κ3>0,κ1κ2-κ3>0, the positive equilibrium point is locally asymptotically stable. Among them,

[0051] κ1=-(A 11 +A 22 +A 33 )

[0052] κ2=A 11 A 22 +A 11 A 33 +A 22 A 33 -A 12 A 21 -A 13 A 31 -A 23 A 32

[0053] κ3=A 11 A 23 A 32 +A 12 A 21 A 33 +A 13 A 31 A 22 -A 11 A22 A 33 -A 12 A 22 A 33 -A 12 A 31 A 23 -A 13 A 21 A 32

[0054] here,

[0055]

[0056] B3. Determine the existence of the ecological equilibrium point of model (1).

[0057] When the model reaches both biological and economic balance, it is said to reach the ecological balance point. When the model reaches biological equilibrium, and when the benefits and costs are equal, the model reaches economic equilibrium. Assuming p is the benefit per unit of fish caught, c is the cost per unit of fish caught, the net economic income can be expressed as Θ = (pq1x3-c)E. The ecological equilibrium point is E4 = (x 1∞ ,x 2∞ ,x 3∞ ,x 1∞ ,E ∞ ), where x 1∞ ,x 2∞ ,x 3∞ and E ∞ are the solutions to the following equation, and all of them are positive.

[0058]

[0059] From the last equation of equation (7), we get x 3∞ Substituting the first two equations into equation (7) yields

[0060]

[0061] In the first equation of (8), let x 1∞ =0, we get Let x 2∞ = 0, we get x 1∞ =K=x 11∞ , and meet

[0062] In the second equation of (8), let x 1∞ =0, we get Let x 2∞ =0, we get in And assume

[0063] From the above analysis, it can be seen that when Two equal slope lines are at the point (x 1∞ ,x 2∞ ) intersects at a point, and an equilibrium point exists. 1∞ and x 2∞ Substituting into the third equation of (8), we get

[0064]

[0065] when When E ∞ > 0, then there is a non-trivial ecological balance point E4 = (x 1∞ ,x 2∞ ,x 3∞ ,E ∞ ).

[0066] C. Obtain the optimal solution of the model and determine the best fishing strategy.

[0067] The above-mentioned ecological analysis method of the plankton-fish model based on interval value parameters has a further preferred technical solution: in step (C), the optimal solution of the model is obtained and the optimal fishing strategy is determined, and the specific steps are as follows:

[0068] C1. Use a time-dependent control variable H as the fishing effort to optimize the overall profit. Construct a Hamilton function,

[0069]

[0070] Among them, λ1, λ2 are accompanying variables, e -δt is the line-fitting factor, and δ is the line-fitting rate. The optimal control condition is:

[0071]

[0072] Here μ(t) is a sign function. And E(t) satisfies the following conditions

[0073]

[0074] in is the net economic income per unit of fish caught, λ3e δt is the shadow price. When the net economic income per unit is greater than the shadow price, E(t) takes E max , otherwise E(t) is 0.

[0075] C2. Determine the best fishing strategy, using the Pontryagin maximum principle, and the accompanying equation is

[0076]

[0077] Therefore, from equations (9) and (11), we can get:

[0078]

[0079] Combined with the positive equilibrium point satisfying (7), we get

[0080]

[0081] Similarly, we get

[0082]

[0083] Taking the derivative of equation (14) with respect to t, we obtain

[0084]

[0085] Combining equations (14) and (15), we get

[0086] λ2=B1e -δt (16)

[0087] in, Substituting (16) into (12), we obtain

[0088]

[0089] in B3=B1ωx2. Therefore, The shadow price is bounded if and only if s0 = 0.

[0090]

[0091] Substituting λ1 and λ3 into equation (13), we obtain

[0092]

[0093] in,

[0094]

[0095] therefore, If and only if s1 = 0, the shadow price λ2e δt is bounded, at this time

[0096]

[0097] From equations (16) and (17), we can get Substituting further into B1, C1 and C2, we get

[0098]

[0099] Therefore, for any q∈[0,1], choosing a specific value of δ, we obtain the optimal balance E δ =(x 1δ ,x 2δ ,x 3δ ,E δ ).

[0100] Compared with the prior art, the method of the present invention has the following beneficial effects: the ecological analysis method of the plankton-fish model of the present invention more reasonably considers the influence of the interval value of the parameter, and overcomes all parameter uncertainty problems caused by different real-world situations, information gaps, data shortages, measurement errors and establishment foundations. From this perspective, models with imprecise parameters are more applicable to overcoming constraints and being more realistic, and are more suitable for analyzing the evolution of population numbers in marine ecological models and optimizing fishing strategies. The present invention appropriately balances current and future benefits in the commercial development of renewable resources. The optimal harvesting strategy given by the present invention not only emphasizes the protection of prey species, but also takes into account quadratic costs. The main advantage of using quadratic costs is that it can create an analytical formula for this optimal harvest, resulting in different results than the Bang-Bang control produced by using a linear cost function. BRIEF DESCRIPTION OF THE DRAWINGS

[0101] Figure 1 is a flow chart of the ecological analysis method of the present invention;

[0102] Figure 2 yes Figure 1 The flowchart of step 101 of establishing a plankton-fish model based on interval value parameters according to the specific marine coastal water environment and taking into account the seasonal periodicity effect, and determining the positivity and boundedness of the number of research objects of the model;

[0103] Figure 3 yes Figure 1 The existence and stability of the equilibrium point of the analysis model described in step 102, and a flow chart of the ecological equilibrium point of the model is given under the condition of considering the balance between ecological and economic benefits;

[0104] Figure 4 yes Figure 1 A flowchart of obtaining the optimal solution of the model and determining the best fishing strategy as described in step 103;

[0105] Figure 5 Simulated graphs of phytoplankton, zooplankton and fish populations with parameters q∈(0,1),ω=0.15,q1E=0.05 selected in model (1);

[0106] Figure 6 Simulated graphs of phytoplankton, zooplankton and fish populations with parameters q∈(0,1),ω=0.32,q1E=0.05 selected in model (1);

[0107] Figure 7 The parameters q∈(0,1), ω=0.32, q1E=0.075 are selected in model (1), and the simulation diagram of the number of phytoplankton, zooplankton and fish populations. DETAILED DESCRIPTION

[0108] The implementation process of the present invention is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0109] Reference Figure 1 The ecological problem analysis method of the plankton-fish model based on interval value parameters of the present invention comprises the following steps:

[0110] Step 101: Based on the specific aquatic environment and considering the interval value of the parameter, a biomathematical model of plankton-fish is established to determine the positivity and boundedness of the number of research objects of the model. Figure 2 , the specific steps are as follows:

[0111] Step 201, considering the effect of nutrients on fish population, the Holling III type functional response function of fish to animals, and the main biological parameters in the model are not accurate, the biological model studied is

[0112]

[0113] Where x1 represents the number of phytoplankton, x2 represents the number of zooplankton, and x3 represents the number of fish populations. The biological significance of the parameters of model (1) is shown in Table 1, and all parameter values ​​are positive numbers.

[0114] Step 202: According to the actual biological significance of the model, the initial conditions of the model are given to satisfy:

[0115] x1(0)=x 10 ≥0, x2(0)=x 20 ≥0, x3(0)=x 30 ≥0 (2)

[0116] In order to introduce the interval value algorithm, the following definition is given first.

[0117] A closed interval [c1,c u ] is defined as an interval number, which satisfies {z:c1≤z≤c u ,z∈R}, we call c1 and cu are the lower and upper bounds of the interval value respectively.

[0118] From a mathematical point of view, in order to better describe real numbers, the interval-valued function is defined below.

[0119] Let g(q) = c 1-q d q , q∈[0,1], c, d>0, then g(q) is called an interval-valued function of the closed interval [c,d].

[0120] Step 203: Under the initial condition (2), the solution of model (1) for any t>0 is non-negative and uniformly bounded.

[0121] Table 1 Biological significance of parameters in model (1)

[0122]

[0123] Step 102: Analyze the existence and stability of the equilibrium point of the model, and give the ecological equilibrium point of the model under the condition of considering the balance between ecological and economic benefits. Figure 3 , the specific steps are as follows:

[0124] Step 301: Determine the existence and stability of the boundary equilibrium point of model (1).

[0125] The equilibrium point of model (1) satisfies the following equations

[0126]

[0127] Obviously, model (1) has four possible equilibrium points: E0 = (0, 0, 0), E1 = (K, 0, 0), It is easy to see that two trivial equilibrium points, E0 = (0,0,0) and E1 = (K,0,0), must exist. At the equilibrium point E0 = (0,0,0), the population is in an extinction state, and this situation always exists. The equilibrium point E1 = (K,0,0) is the axial equilibrium point where only the phytoplankton population survives.

[0128] The trivial equilibrium point E0 = (0,0,0) is unstable. When , the equilibrium point E1 = (K, 0, 0) of model (1) is locally asymptotically stable. When , then model (1) has a semi-trivial equilibrium point Furthermore, when G 33 <0,G 11 <0,G 12 G 21 When <0, the semi-trivial equilibrium point E2 is locally asymptotically stable. Among them:

[0129]

[0130] Step 302: Determine the existence and stability of the positive equilibrium point of model (1).

[0131] when When , then model (1) has a positive trivial equilibrium point In fact, from model (1), we can see that the positive equilibrium point E3 should satisfy the following equation:

[0132]

[0133] From the third equation of equation group (4), we get Will Substituting the first and second equations of equation (4), we can obtain

[0134]

[0135] In the first equation of equation (5), let We get make then, And further,

[0136]

[0137] From the initial conditions, we know that In the second equation of the equation system (5), let get

[0138]

[0139] make get

[0140]

[0141] Furthermore, we get

[0142]

[0143] because when hour,

[0144]

[0145] At this time, the equilibrium point E3 exists.

[0146] In addition, when κ1>0,κ3>0,κ1κ2-κ3>0, the positive equilibrium point is locally asymptotically stable. Among them,

[0147] κ1=-(A 11 +A 22 +A 33 )

[0148] κ2=A 11 A 22 +A 11 A 33 +A 22 A 33 -A 12 A 21 -A 13 A 31 -A 23 A 32

[0149] κ3=A 11 A 23 A 32 +A 12 A 21 A 33 +A 13 A 31 A 22 -A 11 A 22 A 33 -A 12 A 22 A 33 -A 12 A 31 A 23 -A 13 A 21 A 32

[0150] here,

[0151]

[0152] Step 303: Determine the existence of the ecological balance point of model (1).

[0153] When the model reaches both biological and economic balance, it is said to reach the ecological balance point. When the model reaches biological equilibrium, and when the benefits and costs are equal, the model reaches economic equilibrium. Assuming p is the benefit per unit of fish caught, c is the cost per unit of fish caught, the net economic income can be expressed as Θ = (pq1x3-c)E. The ecological equilibrium point is E4 = (x 1∞ ,x 2∞ ,x 3∞ ,x 1∞ ,E ∞ ), where x 1∞ ,x 2∞ ,x 3∞ and E ∞are the solutions to the following equation, and all of them are positive.

[0154]

[0155] From the last equation of equation (7), we get x 3∞ Substituting the first two equations into equation (7) yields

[0156]

[0157] In the first equation of (8), let x 1∞ =0, we get Let x 2∞ = 0, we get x 1∞ =K=x 11∞ , and meet

[0158] In the second equation of (8), let x 1∞ =0, we get Let x 2∞ =0, we get in And assume

[0159] From the above analysis, it can be seen that when Two equal slope lines are at the point (x 1∞ ,x 2∞ ) intersects at a point, and an equilibrium point exists. 1∞ and x 2∞ Substituting into the third equation of (8), we get

[0160]

[0161] when When E ∞ > 0, then there is a non-trivial ecological balance point E4 = (x 1∞ ,x 2∞ ,x 3∞ ,E ∞ ).

[0162] Step 103: Obtain the optimal solution of the model and determine the best fishing strategy. Figure 4 , the specific steps are as follows:

[0163] Step 401: Use a time-dependent control variable H as the fishing effort level to optimize the overall profit. Construct a Hamilton function,

[0164]

[0165] Among them, λ1, λ2 are accompanying variables, e -δt is the line-fitting factor, and δ is the line-fitting rate. The optimal control condition is:

[0166]

[0167] Here μ(t) is a sign function. And E(t) satisfies the following conditions

[0168]

[0169] in is the net economic income per unit of fish caught, λ3e δt is the shadow price. When the net economic income per unit is greater than the shadow price, E(t) takes E max , otherwise E(t) is 0.

[0170] Step 402: Determine the best fishing strategy. Using the Pontryagin maximum principle, the adjoint equation is:

[0171]

[0172] Therefore, from equations (9) and (11), we can get:

[0173]

[0174] Combined with the positive equilibrium point satisfying (7), we get

[0175]

[0176] Similarly, we get

[0177]

[0178] Taking the derivative of equation (14) with respect to t, we obtain

[0179]

[0180] Combining equations (14) and (15), we get

[0181] λ2=B1e -δt (16)

[0182] in, Substituting (16) into (12), we obtain

[0183]

[0184] in B3=B1ωx2. Therefore, The shadow price is bounded if and only if s0 = 0.

[0185]

[0186] Substituting λ1 and λ3 into equation (13), we obtain

[0187]

[0188] in,

[0189]

[0190] therefore, If and only if s1 = 0, the shadow price λ2e δt is bounded, at this time

[0191]

[0192] From equations (16) and (17), we can get Substituting further into B1, C1 and C2, we get

[0193]

[0194] Therefore, for any q∈[0,1], choosing a specific value of δ, we obtain the optimal balance E δ =(x 1δ ,x 2δ ,x 3δ ,E δ ).

[0195] Matlab is used to simulate the plankton-fish model. Through the experiment of the influence of biological parameter q on the model under different zooplankton energy conversion rates and fishing coefficients, the evolution law of phytoplankton, zooplankton and fish populations is obtained. The experiment is as follows:

[0196] For all parameters of the plankton-fish model, appropriate values ​​are selected, and the specific values ​​are as follows:

[0197] r∈[0.2,0.3], γ∈[0.1,0.2], ξ=0.1, η=0.2, δ1=[0.009,0.3]

[0198] And verify that the parameters satisfy the conditions for the existence of axial equilibrium point and positive equilibrium point and local asymptotic stability;

[0199] Compare the population changes with parameter values ​​of q1E=0.05,ω=0.15 and q1E=0.05,ω=0.32, and analyze the impact of predation rate ω;

[0200] The comparison parameter values ​​were selected as q1E=0.05, ω=0.32 and q1E=0.075, ω=0.32, and the impact on the changes in fish catch coefficient was analyzed.

[0201] Results reference Figure 5 , Figure 6 and Figure 7 . It can be seen that under the conditions of different zooplankton energy conversion rates and fishing coefficients, the biological parameter q affects the model. As q increases, the growth rate of phytoplankton, zooplankton mortality and fish mortality, as well as the internal competition rate of fish populations increase. Therefore, when the parameters are selected as q1E=0.05, ω=0.15, when other parameters remain unchanged, when q increases from 0 to 1, the population of phytoplankton increases, the population of zooplankton and fish decreases, and the population of the ecological model has been in a stable state. Further, q1E=0.05, ω=0.32 are selected. When q changes from 0 to 1, the population of the ecological model has been in a stable state. Here, due to the increase in the zooplankton energy predation rate ω, the phytoplankton population decreases and the zooplankton population increases, which indirectly leads to an increase in the fish population. Selecting q1E=0.075, ω=0.32, the population size of the ecological model has been in a stable state. As the fish fishing coefficient increases, the fish population size decreases. Furthermore, due to the decrease in fish intake of zooplankton, the number of zooplankton increases, and the phytoplankton population will indirectly decrease.

Claims

1. An ecological analysis method for a plankton-fish model based on interval value parameters, characterized in that: The specific steps are as follows: A. Considering the case where the parameters are interval values, establish a biomathematical model of plankton-fish; determine the positivity and boundedness of the number of research objects in the model; B. Analyze the existence and stability of the model's equilibrium point; give the model's ecological equilibrium point; C. Calculate the optimal solution of the model and use it to determine the fishing strategy.

2. The ecological problem analysis method of the plankton-fish model based on interval value parameters according to claim 1 is characterized in that: The specific steps of the above step (A) are as follows: A1. Establish a biomathematical model, that is, model (1): Where x1 represents the number of phytoplankton, x2 represents the number of zooplankton, and x3 represents the number of fish populations. The biological meanings of the parameters in model (1) are as follows, and all parameter values ​​are positive numbers: A2. According to the actual biological significance of the model, the initial conditions of the model are given to meet the following requirements: x1(0)=x 10 ≥0,x2(0)=x 20 ≥0,x3(0)=x 30 ≥0 (2) Introduce the interval value algorithm and give the following definition: A closed interval [c1,c u ] is defined as an interval number, which satisfies {z:c1≤z≤c u ,z∈R}, we call c1 and c u are the lower and upper bounds of the interval value respectively; Define an interval-valued function: Let g(q) = c 1-q d q , q∈[0,1], c, d>0, then g(q) is called an interval-valued function of the closed interval [c,d]; A3. Under the initial condition (2), the solution of model (1) for any t>0 is non-negative and uniformly bounded.

3. The ecological analysis method of a plankton-fish model based on interval value parameters according to claim 1, characterized in that: The specific steps of step (B) are as follows: B1. Determine the existence and stability of the boundary equilibrium point of model (1); The equilibrium point of model (1) satisfies the following equations: Model (1) has four possible equilibrium points: E0 = (0, 0, 0), E1 = (K, 0, 0), Two trivial equilibrium points, E0 = (0,0,0) and E1 = (K,0,0), must exist; at the equilibrium point E0 = (0,0,0), the population is in an extinction state, and this situation always exists; the equilibrium point E1 = (K,0,0) is the axial equilibrium point where only the phytoplankton population survives; The trivial equilibrium point E0 = (0,0,0) is unstable; when When , the equilibrium point E1 = (K, 0, 0) of model (1) is locally asymptotically stable; when When , then model (1) has a semi-trivial equilibrium point Furthermore, when G 33 <0,G 11 <0,G 12 G 21 <0, the semi-trivial equilibrium point E2 is locally asymptotically stable; where: B2. Determine the existence and stability of the positive equilibrium point of model (1); when When , then model (1) has a positive trivial equilibrium point The positive equilibrium point E3 satisfies the following equation: From the third equation of equation group (4), we get Will Substituting the first and second equations of equation group (4) yields: In the first equation of equation (5), let get make further, From the initial conditions, we know that Heng established; In the second equation of equation (5), let get: make get: Then we get: because when hour, At this time, the equilibrium point E3 exists; When κ1>0,κ3>0,κ1κ2-κ3>0, the positive equilibrium point is locally asymptotically stable; where κ1=-(A 11 +A 22 +A 33 ) κ2=A 11 TO 22 +A 11 TO 33 +A 22 TO 33 -TO 12 TO 21 -TO 13 TO 31 -TO 23 TO 32 κ3=A 11 THE 23 THE 32 +A 12 THE 21 THE 33 +A 13 THE 31 THE 22 -THE 11 THE 22 THE 33 -THE 12 THE 22 THE 33 -THE 12 THE 31 THE 23 -THE 13 THE 21 THE 32 here, B3. Determine the existence of the ecological equilibrium point of model (1); When model (1) reaches both biological and economic equilibrium, the model reaches the ecological equilibrium point; When the benefit and cost are equal, the model reaches biological equilibrium; when p is the benefit per unit of fish caught, c is the cost per unit of fish caught, and the net economic income is expressed as Θ = (pq1x3-c)E; the ecological equilibrium point is E4 = (x 1∞ ,x 2∞ ,x 3∞ ,x 1∞ ,E ∞ ), where x 1∞ ,x 2∞ ,x 3∞ and E ∞ is the solution of the following equation, and all of them are positive; From the last equation of equation group (7), we get x 3∞ Substituting the first two equations into equation (7) yields: In the first equation of (8), let x 1∞ =0, we get Let x 2∞ = 0, we get x 1∞ =K=x 11∞ , and meet In the second equation of (8), let x 1∞ =0, we get Let x 2∞ =0, we get in And assume From the above analysis, it can be seen that when Two equal slope lines are at the point (x 1∞ ,x 2∞ ) intersects, and the equilibrium point exists; 1∞ and x 2∞ Substituting into the third equation of (8), we obtain: when When E ∞ > 0, then there is a non-trivial ecological balance point E4 = (x 1∞ ,x 2∞ ,x 3∞ ,E ∞ ).

4. The ecological analysis method of the plankton-fish model based on interval value parameters according to claim 1, characterized in that: The specific steps of step (C) are as follows: C1. Use a time-dependent control variable H as the fishing effort to optimize the overall profit; construct a Hamilton function: Among them, λ1, λ2 are accompanying variables, e -δt is the line-fitting factor, δ is the line-fitting rate; the optimal control condition is: Here μ(t) is a sign function; and E(t) satisfies the following conditions: in is the net economic income per unit of fish caught, λ3e δt is the shadow price; when the net economic income per unit is greater than the shadow price, E(t) takes E max , otherwise E(t) takes 0; C2. Determine the best fishing strategy, using the Pontryagin maximum principle, and the accompanying equation is Therefore, from equations (9) and (11), we can get: Combined with the positive equilibrium point satisfying (7), we get Similarly, we get: Taking the derivative of equation (14) with respect to t, we obtain Combining equations (14) and (15), we get λ2=B1e -δt (16) in, Substituting (16) into (12), we obtain: in B3=B1ωx2; therefore, The shadow price is bounded if and only if s0 = 0; Substituting λ1 and λ3 into equation (13), we obtain in, therefore, If and only if s1 = 0, the shadow price λ2e δt is bounded, at this time From equations (16) and (17), we can get Substituting further into B1, C1 and C2, we get Therefore, for any q∈[0,1], choosing a specific value of δ gives the optimal balance E δ =(x 1δ ,x 2δ ,x 3δ ,E δ ).

Images