Analysis method of plankton model branch with Allee effect of algae

Abstract

The invention relates to an analysis method for branches of a plankton model with an Allee effect of algae, and belongs to the field of prediction and simulation of an algae-zooplankton model in a marine ecosystem. According to the method, the influence of the Alle effect and toxin release of algae is considered, a biological mathematical model about algae-zooplankton is established, and the actual biological significance of the number of model research object populations is determined; sufficient conditions for existence of an internal equilibrium state and local asymptotic stability of the model are given; and taking the growth rate of the algae as a branch parameter, and providing sufficient conditions for the model to generate saddle node branches and transcritical branches. According to the branch analysis method of the algae-zooplankton model based on the Alle effect and toxin release influence of the algae, the periodic evolution law of the population number of the algae and the zooplankton in the aquatic environment can be obtained, and the influence of the growth rate of the algae on the population number is known; and the method has positive guiding significance on the treatment of the marine ecological environment.

Description

Technical Field

[0001] The invention relates to the field of algae and zooplankton population prediction in marine ecological environment, and in particular to an analysis method of a plankton model branch in which algae have Allee effect. Background Art

[0002] In recent years, with the continuous enhancement of human activities, the marine ecological environment has deteriorated day by day, especially since 2023, a certain country has successively discharged sewage into the sea, making the marine ecosystem more and more fragmented. This will seriously affect the livelihoods and health of people around the world. Therefore, more and more scholars at home and abroad have invested in the research of marine species protection and sustainable development of marine ecology.

[0003] As early as the 1930s, ecologist Allee discovered that each population has its own optimal survival density. When the density is lower than a certain critical value, the population will face the crisis of extinction; when the density is too high, the population will also face adverse effects such as reproduction. This is the Allee effect.

[0004] For the Allee effect, scholars place it in various ecological environments to study it. For example, in 2023, Gao Caiwen and others discussed the role of various uncertain noises, combined Allee with uncertain disturbances, and established an uncertain Logistic population model, and used U.S. population data to illustrate the inverse uncertain distribution of parameter estimation and integration. Liu Qi and Chang Xiaoyuan analyzed the diffusion single population model with time lag and Allee effect, explored the existence of the Hopf bifurcation experienced by the model near the positive equilibrium point, and found that the asymptotic behavior of the model's solution is related to the initial conditions, and when the hysteresis is maximum, the model's solution eventually converges to a trivial equilibrium point. Cao Qi and Yang Wensheng considered a predator-prey model with Allee effect and general clustering behavior, and analyzed the conditions for the existence of transcritical bifurcations and Hopf bifurcations. They found that the properties of the long-term solution of the model are related to the initial conditions, and that the equilibrium points will not be globally stable. Dey et al. assumed that the population was affected by the Allee effect of reproduction, and used the Beddington-DeAngelis functional response to represent the predator-prey interaction, and established a predator-prey model. Through research, they found that the steady-state pattern was related to the intensity of species interactions, diffusion rate and species abundance.

[0005] Based on the relevant theories and methods of biological models, scholars have conducted many studies on the problem of algal blooms in the marine ecological environment, promoting the development of ecological dynamics.

[0006] During the research process, the inventors found that there is still room for further research on the method of branching problem of plankton model with Allee effect and toxin release of algae: (1) With the help of branching theory, there are relatively few literatures that consider the Allee effect of algae growth in algae and zooplankton models, and also ignore the effect of the intrinsic growth rate of algae on the periodic changes in population size at this time; (2) For biological models with Holling II algae toxin release function, theoretical deduction and evolutionary numerical simulation are carried out by combining the positivity, boundedness, stability and branching problem analysis methods of the solutions in the biological models, so as to more vividly reflect the evolution process of algae and zooplankton populations in the marine ecological environment and predict the occurrence of algal blooms. 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 an analysis method for the branching of the plankton model with Allee effect and toxin release of algae. This method rationally utilizes the relevant theories of stability of differential equations and branching problems, and is more suitable for the simulation of the evolution of algae-zooplankton population in a two-dimensional model aquatic environment.

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

[0009] The invention is an analysis method for the plankton model branch of algae with the Allee effect, and its specific steps are as follows:

[0010] A. Considering the Allee effect and the effect of the release of toxins, establish a biological mathematical model about algae-zooplankton to determine the actual biological significance of the population of the model's research object;

[0011] B. Given sufficient conditions for the existence of internal equilibrium solutions and local asymptotic stability of the algae-zooplankton model;

[0012] C. Taking the growth rate of algae as branching parameters, sufficient conditions are given for the algae-zooplankton model to generate saddle node branches and transcritical branching phenomena.

[0013] The above-mentioned method for analyzing the branch of the plankton model with the Allee effect of algae has a further preferred technical solution: the construction of the biological model between the algae and the zooplankton described in the above step (A) has the following specific steps:

[0014] A1. In the marine ecological environment, the Allee effect of algae and the toxin release effect of phytoplankton is considered between algae and zooplankton, and the toxin release function is a Holling type II function. The model is established as follows:

[0015]

[0016] Here, t represents time, P(t) and Z(t) represent the population density of algae and zooplankton in the ocean at time t; the remaining parameters are normal numbers, and the specific biological significance is shown in Table (1).

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

[0018] P(0)=P0≥0, Z(0)=Z0≥0 (2)

[0019] A3. Under the initial condition (2), it can be determined that the solution of the model (1) at any time t>0 is maintained non-negative boundary.

[0020] A4. If the parameter of model (1) satisfies θ<s(1+α)<θ+m, then

[0021] For the sake of convenience, here is a simplified notation:

[0022]

[0023] Table 1. Biological significance of parameters in model (1)

[0024]

[0025] The above-mentioned method for analyzing the branch of the plankton model with Allee effect of algae, and its further preferred technical solution is: the sufficient conditions for the existence and local asymptotic stability of the internal equilibrium solution of the algae-zooplankton model given in the above step (B) are as follows:

[0026] B1. First, discuss the existence of the balance point of model (1). Obviously, no matter how the parameters are valued, model (1) always has a unique trivial equilibrium point E0 = (0,0). Let E j =(P j ,0),j=1,2, represents the axial equilibrium point. According to the value of the parameter, there may be zero, one, and two axial equilibrium points in model (1). Among them, the axial equilibrium point E j P satisfies the equation

[0027] G(P)≡rP(1-P)-c=0 (4)

[0028] Available,

[0029] c=rP j (1-P j ) (5)

[0030] From this, we can find the discriminant equation Δ=r of the root of equation (4)2 -4rc=r(r-4c), when Δ>0, i.e. r>4c, model (1) has two axial equilibrium points E1=(P1,0), E2=(P2,0), where,

[0031]

[0032] When Δ = 0, i.e. r = 4c, model (1) has only one axial equilibrium point When Δ<0, that is, r<4c, model (1) has no axial equilibrium point.

[0033] B2. Discuss the existence of internal equilibrium points. Let E * =(P * ,Z * ) represents the internal equilibrium point of model (1). Internal equilibrium point E * =(P * ,Z * ) must satisfy the equation system

[0034] The solution. So the internal equilibrium point E * The component P * Satisfy the equation

[0035] Q(P)=αsP 2 +(s-αm-θ)Pm=0 (8)

[0036] When P = 0, Q(P) = -m < 0, when P → +∞, Q(P) → +∞, so there must be a P * , so that Q(P) = 0. Therefore, model (1) must exist P * , and we can obtain

[0037]

[0038] From the above equations (7), we can get

[0039]

[0040] If Δ>0 and That is, r>4c and When the internal equilibrium point E of model (1) * If Δ≤0, that is, r≤4c, the internal equilibrium point of model (1) does not exist.

[0041] B3. Study the stability of the equilibrium point of model (1). The Jacobian matrix at is

[0042]

[0043] Obviously, the ordinary balance point E0=(0,0) is locally asymptotic. The Jacobian matrix at the trivial equilibrium point E0=(0,0) of the model (1) is

[0044]

[0045] The characteristic equation at E0 is

[0046] (c+λ)(m+λ)=0 (9) Obviously, the two characteristic roots of the characteristic equation (9) are λ1 = -c <0 and λ2 = -m <0 respectively.

[0047] When r>4c, model (1) is at the axial equilibrium point E j =(P j ,0),j=1,2, the Jacobian matrix is

[0048]

[0049] At the equilibrium point E j The characteristic equation at is

[0050]

[0051] So λ1=rP j (1-2P j ), when When λ1<0, λ2<0, the characteristic root of the characteristic equation (11) has a strict negative real part, and the equilibrium point E1 is locally asymptotic and stable; When λ1<0, λ2>0, equilibrium point E1 is a saddle point. When When λ1>0, λ2>0, the characteristic root of the characteristic equation (11) does not have a strict negative real part, and the equilibrium point E2 is unstable; When λ1>0, λ2>0, equilibrium point E2 is a saddle point.

[0052] Model (1) at the internal equilibrium point E * =(P * ,Z * ) is the Jacobian matrix

[0053]

[0054] in,

[0055] a1=rP * (1-2P * ),a2=-sP *

[0056]

[0057] Here, it is obvious that a2<0, a4=0, E * The characteristic equation corresponding to the Jacobian matrix at is

[0058] a1a4-a2a3>0 (12)

[0059] in,

[0060] Tr(J E* )=λ1+λ2=a1,Det(J E* )=λ1λ2=-a2a3

[0061] Therefore, the characteristic root of the characteristic equation (12) is

[0062]

[0063] Then, the roots of the characteristic equation (12) have strictly negative real parts if and only if a1+a4<0 and a1a4-a2a3>0.

[0064] Right now and

[0065] The above-mentioned method for analyzing the branching of the plankton model with the Allee effect of algae has a further preferred technical solution: in the above step (C), the growth rate of algae is used as the branching parameter to give the sufficient conditions for the algae-zooplankton model to produce saddle node branching and transcritical branching phenomena, and the specific steps are as follows:

[0066] C1. When (2+α)s-(2α+4)m-2θ>0 and r=4c, model (1) undergoes saddle node branching.

[0067] Since G(P) in equation (4) has a double root P=Ps at r=4c, that is, G(Ps)=G'(Ps)=0. From G(Ps)=G'(Ps)=0, we can get r=4c. From formula (4), we can see that when r>4c Because (2+α)s-(2α+4)m-2θ>0, E1 is always a saddle point, and E2 is an unstable node or saddle point; these two axial equilibrium points merge into one axial equilibrium point when r=4c It disappears when r<4c. At the equilibrium point E SN At r = 4c, its Jacobian matrix is

[0068]

[0069] Let φ=(1,0) T , They correspond to the matrices and two eigenvectors of zero eigenvalues. We can verify that:

[0070]

[0071] Where F = [F1(P, Z), F2(P, Z)] T . Therefore, the model (1) undergoes a saddle node branch at r=4c.

[0072] C2. When (2+α)s-(2α+4)m-2θ>0 and s(1+αP TC ) 2 ≠θ, model (1) in E TC =(P TC ,0) undergoes a degenerate transcritical branch, where P TC =A1.

[0073] At point E TC =(P TC ,0), when r=r TC When , its Jacobian matrix is

[0074]

[0075] Let φ=(p,1) T ,ψ=(0,1) T Corresponding to and The two eigenvectors of the zero eigenvalue of

[0076]

[0077] We can verify that:

[0078]

[0079]

[0080] Therefore, model (1) has TC =(P TC ,0)=(A1,0) undergoes a degenerate transcritical branch.

[0081] Compared with the prior art, the method of the present invention has the following beneficial effects:

[0082] The method of the present invention more reasonably considers the Allee effect of algae growth, and also considers the impact of some algae toxins released on zooplankton. It can better reflect the actual prey-predator relationship between algae and zooplankton through the model itself, and determine the law of the change of algae and zooplankton population over time, which is more suitable for analyzing the evolution process of the algae and zooplankton population period in the marine ecological environment. In this method, on the one hand, the model parameters are given to meet certain suitable conditions so that the two types of populations reach a stable equilibrium state; on the other hand, the growth rate of algae is considered as a branch parameter, and its change affects the population, and the threshold of the growth rate is determined; when the growth rate exceeds the threshold, the algae and zooplankton population changes from the original stable equilibrium state to periodic fluctuations, so that the periodic evolution law of the algae and zooplankton population in the aquatic environment can be obtained, and the impact of the algae growth rate on the population can be understood. Then, when the algae bloom phenomenon occurs, how to control the growth of algae, inhibit the excessive growth of algae, and promote the balanced and harmonious development of the marine ecology is clear, which has a positive guiding significance for the governance of the marine ecological environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 It is a flow chart of an analysis method of a plankton model branch in which algae have Allee effect of the present invention;

[0084] Figure 2 yes Figure 1 The flowchart of step 101 of establishing a biomathematical model of algae-zooplankton based on the specific marine environment and considering the Allee effect and the impact of toxin release of algae to determine the actual biological significance of the population number of the model research object;

[0085] Figure 3 yes Figure 1 A flowchart of sufficient conditions for the existence and local asymptotic stability of the internal equilibrium solution of the algae-zooplankton model as described in step 102;

[0086] Figure 4 yes Figure 1 The flow chart of the sufficient conditions for the algae-zooplankton model to produce saddle node branching and transcritical branching phenomena is given by taking the growth rate of algae as the branching parameter in step 103;

[0087] Figure 5 This is the numerical simulation diagram of model (1) when the parameters are r = 0.84, c = 0.16, s = 0.66, m = 0.259, θ = 0.35, α = 0.0048 and the initial conditions are (0.56, 0.00037);

[0088] Figure 6It is the numerical simulation diagram of model (1) when the parameters are r = 0.7785, c = 0.06, s = 0.66, m = 0.259, θ = 0.15, α = 0.0048 and the initial condition is (0.2056, 0.0045);

[0089] Figure 7 This is the numerical simulation diagram of model (1) when the parameters are r = 0.7785, c = 0.06, s = 0.66, m = 0.259, θ = 0.2, α = 0.0048 and the initial condition is (0.2056, 0.0045);

[0090] Figure 8 This is the numerical simulation diagram of model (1) when the above group (4) parameters are r = 0.66, c = 0.06, s = 0.66, m = 0.259, θ = 0.15, α = 0.0048 and the initial conditions are (0.2056, 0.0045). DETAILED DESCRIPTION

[0091] The implementation process of the present invention is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0092] Reference Figure 1 , an analysis method for the branching problem of a plankton model with an Allee effect of algae, the method comprising the following steps:

[0093] Step 101: Based on the specific marine ecological environment, considering the Allee effect of algae and the impact of toxin release, establish a biomathematical model of algae-zooplankton, determine the actual biological significance of the population size of the model research object, and refer to Figure 2 , the specific steps are as follows:

[0094] Step 201: In the marine ecological environment, the Allee effect of algae and the impact of toxin release by phytoplankton are considered, and the toxin release function is a Holling II type function. The model is established as follows:

[0095]

[0096] Here, t represents time, P(t) and Z(t) represent the population density of algae and zooplankton in the ocean at time t respectively; the other parameters are all positive numbers, and their specific biological meanings are shown in Table (1).

[0097] Step 202: According to the actual biological significance of the model, the initial conditions of the model are given to satisfy:

[0098] P(0)=P0≥0, Z(0)=Z0≥0 (2)

[0099] Step 203: Under the initial condition (2), it can be determined that the solution of the model (1) remains non-negatively bounded for any time t>0.

[0100] Step 204: If the parameters of model (1) satisfy θ<s(1+α)<θ+m, then

[0101] For the sake of convenience, here is a simplified notation:

[0102]

[0103] Table 1. Biological significance of parameters in model (1)

[0104]

[0105] Step 102: Provide sufficient conditions for the existence and local asymptotic stability of the internal equilibrium solution of the algae-zooplankton model, refer to Figure 3 , the specific steps are as follows:

[0106] Step 301: First, discuss the existence of the equilibrium point of model (1). Obviously, no matter how the parameters are taken, model (1) always has a unique trivial equilibrium point E0 = (0, 0). Let E j =(P j ,0),j=1,2,represents the axial equilibrium point. According to the parameter values, model (1) may have zero, one, or two axial equilibrium points. Among them, the axial equilibrium point E j P satisfies the equation

[0107] G(P)≡rP(1-P)-c=0 (4)

[0108] Available,

[0109] c=rP j (1-P j ) (5)

[0110] From this, we can find the discriminant of the root of equation (4): Δ = r 2 -4rc=r(r-4c), when Δ>0, i.e. r>4c, model (1) has two axial equilibrium points E1=(P1,0), E2=(P2,0), where,

[0111]

[0112] When Δ = 0, i.e. r = 4c, model (1) has only one axial equilibrium point When Δ<0, that is, r<4c, model (1) has no axial equilibrium point.

[0113] Step 302: Discuss the existence of the internal equilibrium point. Let E* =(P * ,Z * ) represents the internal equilibrium point of model (1). Internal equilibrium point E * =(P * ,Z * ) must satisfy the equation system

[0114] The solution. So the internal equilibrium point E * The component P * Satisfy the equation

[0115] Q(P)=αsP 2 +(s-αm-θ)Pm=0 (8)

[0116] When P = 0, Q(P) = -m < 0, when P → +∞, Q(P) → +∞, so there must be a P * , so that Q(P) = 0. Therefore, model (1) must exist P * , and we can obtain

[0117]

[0118] From the above equations (7), we can get

[0119]

[0120] If Δ>0 and That is, r>4c and When the internal equilibrium point E of model (1) * If Δ≤0, that is, r≤4c, the internal equilibrium point of model (1) does not exist.

[0121] Step 303: Study the stability of the equilibrium point of model (1). The Jacobian matrix at is

[0122]

[0123] Obviously, the trivial equilibrium point E0 = (0,0) is locally asymptotically stable. The Jacobian matrix of model (1) at the trivial equilibrium point E0 = (0,0) is

[0124]

[0125] The characteristic equation at E0 is

[0126] (c+λ)(m+λ)=0 (9) Obviously, the two characteristic roots of the characteristic equation (9) are λ1=-c<0, λ2=-m<0.

[0127] When r>4c, model (1) is at the axial equilibrium point E j =(P j ,0),j=1,2, the Jacobian matrix is

[0128]

[0129] At the equilibrium point E j The characteristic equation at is

[0130]

[0131] So λ1=rP j (1-2P j ), when When λ1<0,λ2<0, the characteristic root of the characteristic equation (11) has a strictly negative real part, and the equilibrium point E1 is locally asymptotically stable; when When λ1<0, λ2>0, the equilibrium point E1 is a saddle point. When λ1>0, λ2>0, the characteristic root of the characteristic equation (11) does not have a strictly negative real part, and the equilibrium point E2 is unstable; when When λ1>0, λ2>0, the equilibrium point E2 is a saddle point.

[0132] Model (1) at the internal equilibrium point E * =(P * ,Z * ) is the Jacobian matrix

[0133]

[0134] in,

[0135] a1=rP * (1-2P * ),a2=-sP *

[0136]

[0137] Here, it is obvious that a2<0, a4=0, E * The characteristic equation corresponding to the Jacobian matrix at is

[0138] a1a4-a2a3>0 (12)

[0139] in,

[0140] Tr(J E* )=λ1+λ2=a1,Det(J E* )=λ1λ2=-a2a3

[0141] Therefore, the characteristic root of the characteristic equation (12) is

[0142]

[0143] Then, the roots of the characteristic equation (12) have strictly negative real parts if and only if a1+a4<0 and a1a4-a2a3>0.

[0144] Right now and

[0145] Step 103: Taking the growth rate of algae as the branching parameter, sufficient conditions for the algae-zooplankton model to produce saddle node branching and transcritical branching phenomena are given, referring to Figure 4 , the specific steps are as follows:

[0146] Step 401: When (2+α)s-(2α+4)m-2θ>0 and r=4c, the model (1) undergoes a saddle node branch.

[0147] Since G(P) in equation (4) has a double root P=Ps at r=4c, that is, G(Ps)=G'(Ps)=0.

[0148] From G(Ps)=G'(Ps)=0, we can get r=4c. From formula (4), we can see that when r>4c Because (2+α)s-(2α+4)m-2θ>0, E1 is always a saddle point, and E2 is an unstable node or saddle point; these two axial equilibrium points merge into one axial equilibrium point when r=4c It disappears when r<4c. At the equilibrium point E SN At r = 4c, its Jacobian matrix is

[0149]

[0150] Let φ=(1,0) T , They correspond to the matrices and The two eigenvectors of the zero eigenvalue of . We can verify that:

[0151]

[0152] Where F = [F1(P, Z), F2(P, Z)] T Therefore, model (1) undergoes a saddle-node branch at r = 4c.

[0153] Step 402: When (2+α)s-(2α+4)m-2θ>0 and s(1+αP TC ) 2 ≠θ, model (1) in E TC =(P TC ,0) undergoes a degenerate transcritical branch, where P TC =A1.

[0154] At point E TC =(P TC ,0), when r=r TC When , its Jacobian matrix is

[0155]

[0156] Let φ=(p,1) T ,ψ=(0,1) T Corresponding to and The two eigenvectors of the zero eigenvalue of

[0157]

[0158] We can verify that:

[0159]

[0160] Therefore, model (1) has TC =(P TC ,0)=(A1,0) undergoes a degenerate transcritical branch.

[0161] Matlab is used to conduct simulation experiments on the algae-zooplankton model. By experimenting with the parameter algae growth rate r, the evolution law of the algae and zooplankton populations is obtained. The experiment includes the following three parts.

[0162] 1. Select four sets of appropriate values ​​for all parameters of the algae-zooplankton model. The specific values ​​are as follows:

[0163] (1)r=0.84, c=0.16, s=0.66, m=0.259, θ=0.35, α=0.0048;

[0164] (2)r=0.7785, c=0.06, s=0.66, m=0.259, θ=0.15, α=0.0048;

[0165] (3)r=0.7785, c=0.06, s=0.66, m=0.259, θ=0.2, α=0.0048;

[0166] (4) r = 0.66, c = 0.16, s = 0.66, m = 0.259, θ = 0.35, α = 0.0048.

[0167] And verify that the parameters satisfy the conditions for the local asymptotic stability of the equilibrium point and the existence of the positive equilibrium solution branch.

[0168] 2. Compare the effects of different algal growth rates on the change in the number of research objects;

[0169] 3. Compare the simulation experiment verifications of the effects on the algal population quantity and zooplankton quantity when the different algal toxicity release rates are 0.35, 0.15, and 0.2 respectively.

[0170] Figure 5 It is the numerical simulation of model (1) with the above-mentioned first set of parameters r = 0.84, c = 0.16, s = 0.66, m = 0.259, θ = 0.35, α = 0.0048 and the initial condition (0.56, 0.00037). At this time, P1 = 0.7740, A1 = 0.8317, satisfying r > 4c, 1 / 2 < P1 < A1, and the axial equilibrium point E1 = (0.7740, 0) is locally asymptotically stable.

[0171] Figure 6 It is the numerical simulation of model (1) with the above-mentioned second set of parameters r = 0.7785, c = 0.06, s = 0.66, m = 0.259, θ = 0.15, α = 0.0048 and the initial condition (0.2056, 0.0045). At this time, P2 = 0.0842, A1 = 0.5075, P1 = 0.9158, satisfying r > 4c, 1 / 2 < A1 < P1, The interior equilibrium point is E * = (0.5075, 0.2039) is locally asymptotically stable.

[0172] Figure 7 It is the numerical simulation of model (1) with the above-mentioned third set of parameters r = 0.7785, c = 0.06, s = 0.66, m = 0.259, θ = 0.2, α = 0.0048 and the initial condition (0.2056, 0.0045). At this time, P2 = 0.0842, A1 = 0.5624, P1 = 0.9158, satisfying r > 4c, 1 / 2 < A1 < P1, The interior equilibrium point is E * = (0.5624, 0.1994) is locally asymptotically stable.

[0173] Figure 8This is the numerical simulation of model (1) with the parameters r = 0.66, c = 0.06, s = 0.66, m = 0.259, θ = 0.15, α = 0.0048 and the initial condition (0.2056, 0.0045). At this time, P2 = 0.1011, A 1 = 0.5075, P1 = 0.8989, Satisfying r>4c,1 / 2 <A1<P1, The positive equilibrium point is E * =(0.5075,0.1590) is locally asymptotically stable.

[0174] It can be seen that when other parameters remain unchanged, as θ increases, P * Increase, and Z * In other words, the increase in toxin release rate increases the number of phytoplankton and reduces the number of zooplankton individuals. At the same time, the impact of phytoplankton growth rate on the ecosystem is observed. When other parameters remain unchanged, as r increases, Z * Increase, and P * That is to say, the increase in the growth rate of phytoplankton has increased the number of zooplankton, and the number of individual zooplankton has remained stable.

Claims

1. A method for analyzing the branch of a plankton model in which algae have an Allee effect, characterized in that: The specific steps are as follows: A. Establish a biomathematical model of algae-zooplankton; B. Determine sufficient conditions for the existence and local asymptotic stability of the internal equilibrium solution of the algae-zooplankton model; C. Using the growth rate of algae as the branching parameter, determine the sufficient conditions for the algae-zooplankton model to produce saddle node branching and transcritical branching phenomena.

2. The method for analyzing the branch of the plankton model with Allee effect of algae according to claim 1, characterized in that: The specific steps of constructing the biological model between algae and zooplankton described in step (A) above are as follows: A1. Considering the Allee effect of algae and the impact of phytoplankton toxin release, and the toxin release function is a Holling II type function, the model (1) is established as follows: Where: t represents time, P(t) and Z(t) represent the population density of algae and zooplankton in the ocean at time t respectively; the other parameters are positive numbers, and the specific meanings of the parameters are: A2. According to the actual biological significance of the model, the initial conditions of the model are given to meet the following requirements: P(0)=P0≥0, Z(0)=Z0≥0 (2) A3. Under the initial condition (2), determine that the solution of model (1) remains non-negative and bounded for any time t>0; A4. If the parameters of model (1) satisfy θ<s(1+α)<θ+m, then The simplified notation is:

3. The method for analyzing the branch of the plankton model with Allee effect of algae according to claim 1, characterized in that: The above step (B) provides sufficient conditions for the existence and local asymptotic stability of the internal equilibrium solution of the algae-zooplankton model, and the specific steps are as follows: B1. Determine the equilibrium point of model (1). No matter how the parameters are taken, model (1) always has a unique trivial equilibrium point E0 = (0, 0). Let E j =(P j ,0),j=1,2,represents the axial equilibrium point; according to the value of the parameter, model (1) may have zero, one, or two axial equilibrium points; among them, the axial equilibrium point E j The P satisfies the equation: G(P)≡rP(1-P)-c=0 (4) Available, c=rP j (1-P j ) (5) From this, we can find the discriminant of the root of equation (4): Δ = r 2 -4rc=r(r-4c), when Δ>0, i.e. r>4c, model (1) has two axial equilibrium points E1=(P1,0), E2=(P2,0), where, When Δ = 0, i.e. r = 4c, model (1) has only one axial equilibrium point When Δ<0, i.e., r<4c, model (1) has no axial equilibrium point; B2. Determine the existence of an internal equilibrium point, let E * =(P * ,Z * ) represents the internal equilibrium point of model (1); the internal equilibrium point E * =(P * ,Z * ) must satisfy the equation The solution; therefore, the internal equilibrium point E * The component P * Satisfy the equation Q(P)=αsP 2 +(s-αm-θ)Pm=0 (8) When P = 0, Q(P) = -m < 0, when P → +∞, Q(P) → +∞, so there must be a P * , so that Q(P) = 0; therefore, model (1) must exist P * , and we can obtain From the above equations (7), we can get If Δ>0 and That is, r>4c and When the internal equilibrium point E of model (1) * exists; if Δ≤0, that is, r≤4c, the internal equilibrium point of model (1) does not exist; B3. Determine the stability of the equilibrium point of model (1): Model (1) at the equilibrium point The Jacobian matrix at is: The trivial equilibrium point E0 = (0,0) is locally asymptotically stable; the Jacobian matrix of model (1) at the trivial equilibrium point E0 = (0,0) is: The characteristic equation at E0 is: (c+λ)(m+λ)=0 (9) The two characteristic roots of the characteristic equation (9) are λ1=-c<0, λ2=-m<0; When r>4c, model (1) is at the axial equilibrium point E j =(P j ,0),j=1,2, the Jacobian matrix is At the equilibrium point E j The characteristic equation at is: So λ1=rP j (1-2P j ), when When λ1<0,λ2<0, the characteristic root of the characteristic equation (11) has a strictly negative real part, and the equilibrium point E1 is locally asymptotically stable; when When λ1<0, λ2>0, the equilibrium point E1 is a saddle point; when When λ1>0,λ2>0, the characteristic root of the characteristic equation (11) does not have a strictly negative real part, and the equilibrium point E2 is unstable; when When λ1>0, λ2>0, the equilibrium point E2 is a saddle point; Model (1) at the internal equilibrium point E * =(P * ,Z * ) is: in, a1=rP * (1-2P * ),a2=-sP * It can be seen that a2<0, a4=0, E * The characteristic equation corresponding to the Jacobian matrix at is: a1a4-a2a3>0 (12) in, Tr(J E* )=λ1+λ2=a1,Det(J E* )=λ1λ2=-a2a3 Therefore, the characteristic root of the characteristic equation (12) is: Then, the roots of the characteristic equation (12) have strictly negative real parts if and only if a1+a4<0 and a1a4-a2a3>0; that is, and 4. The method for analyzing the branch of the plankton model with Allee effect of algae according to claim 1, characterized in that: In the above step (C), the growth rate of algae is used as the branching parameter to give the sufficient conditions for the algae-zooplankton model to produce saddle node branching and transcritical branching phenomena. The specific steps are as follows: C1. When (2+α)s-(2α+4)m-2θ>0 and r=4c, model (1) undergoes saddle node branching; Since G(P) in equation (4) has a double root P=Ps at r=4c, that is, G(Ps)=G'(Ps)=0; From G(Ps)=G'(Ps)=0, we can get r=4c; From formula (4), we can see that when r>4c Because (2+α)s-(2α+4)m-2θ>0, E1 is always a saddle point, and E2 is an unstable node or saddle point; these two axial equilibrium points merge into one axial equilibrium point when r=4c It disappears when r<4c; at the equilibrium point E SN At r = 4c, its Jacobian matrix is Let φ=(1,0) T , They correspond to the matrices and The two eigenvectors of the zero eigenvalue of ; verification yields: Where F = [F1(P, Z), F2(P, Z)] T ; Therefore, model (1) undergoes a saddle node branch at r = 4c; C2. When (2+α)s-(2α+4)m-2θ>0 and s(1+αP TC ) 2 ≠θ, model (1) in E TC =(P TC ,0) undergoes a degenerate transcritical branch, where P TC =A1; At point E TC =(P TC ,0), when r=r TC When , its Jacobian matrix is: Let φ=(p,1) T ,ψ=(0,1) T Corresponding to and The two eigenvectors of the zero eigenvalue of Verified: Therefore, model (1) in E TC =(P TC ,0)=(A1,0) undergoes a degenerate transcritical branch.

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