The symbiotic energy model integrating ecological carrying capacity.

Odum defines "generalized ecological carrying capacity " as the maximum number of human activities that can be accommodated by the natural, economic and social complex ecosystem under the premise of coordinated and sustainable development of natural resources, ecological environment, and economic and social sub-items within a certain region [9–11]. Yafen He and Hualin Xie use the ecological footprint method to evaluate the ecological carrying capacity (ECC) of Nanchang city at a city, county and grid scale, respectively [12]. Jiang Peiwen and Rees W et al. used the ecological footprint method to calculate the ecological footprint and ecological carrying capacity of agriculture-related land use in Shaanxi Province in 2015 [13,14]. Cao Zhi believed that the measurement of the ecological footprint is the consumption footprint of agricultural production, that is, the materials needed for agricultural production [15]. This paper refers to the concept of generalized ecological carrying capacity, defines the ecological footprint as the consumption footprint of agricultural resources, and introduces the ecological footprint measurement model into the non-symbiotic energy part of the symbiotic energy model to improve the symbiotic energy model.

Logistic symbiotic environmental model considering the change of environmental tolerance.

The logistic model, also known as the retarded growth model, is considered to be an optimal mathematical model to describe the population growth rule under limited resource conditions, with rich connotations in ecology, anthropology, zoology and economics [16]; Yingying Tao and Lihong Han described the development law of the green building industrial chain symbiotic system under different symbiosis relationships through a logistic model [17]; Ding Y, Zhou B and Ling D proposed an enterprise symbiosis mechanism and symbiosis stability monitoring model of supply chain alliances based on logistic and Lotka-Volterra equations [18]. In population ecology, environmental capacity is also known as maximum environmental capacity, which is consistent with the concept of the logistic model. In essence, this refers to limited growth in a limited environment. With the help of variable environmental capacity, this paper not only reflects the constraints of environment on species but also reflects the dynamic variables of its own changes. Based on the Logistic growth function, it establishes the symbiosis environmental model that conforms to the relationship between the fresh agricultural product supply chain and environmental capacity, as a criterion of the stability analysis of the quantitative basis. Thus, the variable environmental tolerance characterizes the uncertain fresh quality or higher value of agricultural products that farmers and enterprises are concerned with.

Lyapunov indirect method and its application.

The Lyapunov stability indirect method determines the stability of nonlinear systems by studying the distribution of characteristic roots of linearized equations of state [19]. Chunmei Liu proposed that the Lyapunov method can be used for the stability analysis of systems without delay terms [20]. Taking nonlinear time-invariant systems as an example, a linearization method for nonlinear systems was expounded. HAMZA, Alaa and ORABY, Karima obtained new sufficient conditions for the multiple stability of nonlinear dynamic equations by using the Lyapunov indirect method [21]. In this paper, the Lyapunov indirect method is applied to the analysis of the stability of symbiotic systems. Through an approximate transformation, the environmental model is transformed into a systematic evolutionary model, and the evolutionary singularity and phase transition boundary of symbiotic systems are judged by the positive and negative characteristics of characteristic roots.

This paper proposes an improved symbiosis system stability analysis method for criteria quantification. This method takes the symbiosis system of the supply chain of fresh agricultural products as the research object and uses systematic reasoning based on criteria logic for the characteristics of fresh agricultural products, i.e., the uncertain quality and value of fresh agricultural products in supply chain circulation.

To summarize, three improvement points we contribute for criterion quantification are shown as follows: First, based on the Logistic growth function, the symbiotic energy model that conforms to the interaction relationship of the fresh agricultural products supply chain and the change of environmental tolerance is reconstructed. Second, the ecological footprint measurement method of the generalized ecological carrying capacity is used to quantify the non-symbiotic energy model. Considering the efforts and profit distribution of fresh agricultural product enterprises and farmers, the new symbiosis energy is measured by the change of expected income before and after the formation of the symbiosis relationship. Last not the least, according to the self-similar approximation theory, the Lyapunov characteristic exponential function is constructed through an approximate transformation to quantify the dynamic system of the symbiotic environment.

The eight basic criteria from the symbiosis theory involved can reflect the interaction of the corresponding symbiosis units in a certain symbiosis mode.

Criterion 1: For the quality parameter compatibility criterion, there must be some factors that can express each other or influence each other between symbiotic units, so that interactions can be formed to affect the symbiotic system;

Criterion 2: The symbiotic tissue model criterion is mainly determined by the frequency of interaction of the symbiotic units;

Criterion 3: The symbiotic behavioural model criterion is mainly determined by the interaction mode and intensity of the symbiotic unit. In different symbiotic behavioural modes, symbiotic energy of different intensities will be generated between the two symbiotic units;

Criterion 4: For the symbiosis energy generation criterion, symbiotic energy is formed by the interaction of symbiotic units, which is the basic condition for the existence and development of symbiotic systems;

Criterion 5: The symbiotic interface selectivity criterion is the choice of the environment in which the symbiotic interaction occurs;

Criterion 6: For the symbiotic phase change criterion, the symbiotic interaction phase change determines the system evolution singularity and boundary value;

Criterion 7: For the symbiotic evolutionary criterion, the system changes phase at a singular point and evolves to a critical state at the boundary;

Criterion 8: For the symbiosis stability criterion, when the symbiotic unit develops at zero speed, the system is in a stable state.

Symbiotic unit.

In the symbiotic relationship between agricultural product enterprises and farmers, the quality parameters are defined as the interaction between two symbiotic units in the supply and demand of fresh agricultural products, the distribution of profits, and the asset and technology combination in production and processing. From the three aspects of the characteristics of fresh agricultural products, the willingness and efforts of farmers and enterprises, the following three kinds of mass parameter functions are designed to reflect the internal dynamic mechanism of the symbiosis unit.

⑴The fresh loss rate function

Fresh agricultural products have perishable and fragile characteristics. It is assumed that the time elapsed from production to sale to the market during the life cycle T is t(0≤t≤T) and that the degree of loss will increase with the increase of time t. To describe the attenuation law of fresh agricultural products in terms of loss, the loss rate function of fresh agricultural products is introduced [22]: (1)

⑵The benefit distribution function

To obtain a stable source of agricultural production, the fresh agricultural product management enterprise signs a rebate contract with the farmers to enhance their willingness to work together. The market demand for fresh agricultural products is x: due to the loss of fresh agricultural products in transport, the actual sales volume of fresh agricultural products is q = (1−λ(t))Q, in which the management enterprises should sign the order with the farmers as Q(Q≥x), the purchase price is p₁, the sales price is p₂. According to the profit situation, the fresh agricultural products management enterprises will repay the profits to the farmers in proportion as (1−φ) to the sales. The expected income of the fresh agricultural product management enterprises and farmers under symbiotic conditions are:

The expected income of farmers is: (2) (3)

The expected income of the enterprise is: (4)

and are the production cost of farmers and the sales cost of enterprises in the symbiotic mode, respectively. Q* is the actual sales volume of fresh agricultural products. x refers to the customer demand. f(x) refers to the probability density function of random variable x.

⑶The effort cost function

The qualified output of fresh agricultural products can be increased by means of input into production efforts, according to Xie Gang and Ji-ca Li [23,24]. Assume that when the production effort of fresh agricultural product farmers is x_f(x_f>0), the sales volume of qualified fresh agricultural products is Q(1+x_f), and the production effort cost is . When the sales effort is x_r(x_r>0), the market price is p = p₂(1+x_r), and the sales effort cost is .

In the joint efforts of the fresh agricultural product management enterprises and farmers, the enterprise and the farmers are allocated the production effort cost. Assume that the proportion of the enterprises in the production effort cost is z, and the price of the purchased fresh agricultural products is p₁. Under the symbiotic state, the expected income of farmers who produce fresh agricultural products under the production effort cost is: (5)

The expected return of the enterprises under the cost of sales and the cost of effort allocated is: (6)

Among them: (7)

and are the total cost functions of the fresh agricultural product management enterprises and the farmers in the symbiotic state, considering the effort costs of production and sales.

Symbiotic mode.

The symbiotic interface between farmers and enterprises is the market of fresh agricultural products. On this interface, farmers are interdependent and mutually beneficial, exchanging materials, information and energy to form new symbiotic energy. It is consistent with the basic feature of natural symbiosis that "new energy is generated by mutual influence, without affecting its own structure and nature". Due to the complex and changeable market, it is necessary to ensure the continuous operation of the symbiosis system by the close interaction between the two symbiosis units, which is dominated by the core enterprise and relies on the credit mechanism. By referring to the symbiosis model of insect species in nature, which is parasitic, partial symbiosis and mutualistic, the two symbiosis units in the supply chain of fresh agricultural products are dominated by a mutualism and parasitism model.

Symbiotic environment.

The symbiotic units interact with each other in the system environment until they reach a steady state. Due to an abrupt change of the system environment or other influencing factors, the system will be shaken. When the critical value of a certain condition is reached, the system state will evolve into a different state. In view of the variable environmental capacity, it not only reflects the restriction of the environment on species but also reflects the dynamic variable characteristic of its own change. The symbiotic system based on variable environmental capacity has undergone an evolutionary process of generation, development and stability, and it conforms to the logistic growth function, which can be expressed as: (8)

N₁ and N₂ represent the economic output indicators of fresh agricultural products management enterprises and farmers within a symbiotic relationship, respectively; γ_i represents the phylogenetic rate of the two symbiotic units; and K_i is the environmental capacity of the fresh agricultural product management enterprises and farmers in the symbiosis system, which serves as their symbiosis energy E_i.

Symbiotic energy model.

Symbiotic energy E_i can be defined as follows: without symbiosis, their respective energy is A_i and newly added energy in symbiosis is S_i, as shown in Eq (9)

Without symbiosis, system energy A_i is subdivided into A₁ and A₂ with the help of the concept of functional ecological carrying capacity under natural conditions and the measurement method of agricultural production and consumption footprint. Set μA to convert ecological carrying capacity into economic indicators to unify dimensions. S_i({N₁,N₂,⋯,N_i}) represents the new energy created by fresh agricultural products enterprises and farmers under the symbiotic conditions, S_i({N₁,N₂})≥0. B_i∈(−∞,+∞) is the intensity of the species' acceptance of new energy.

(9)

Among them: (10) (11) (12)

(1) Ecological footprint of fresh agricultural inputs of farmers: (13)

Where C_i represents inputs of farmers of i type of fresh agricultural products, and Y_i represents the regional average output of inputs of i type.

(2) Ecological footprint of fresh agricultural products enterprises: (14)

Where l is the amount of labour force in the enterprise, C_i is the per capita labour value of i fresh agricultural products, and Y_i is the average regional output of i fresh agricultural products.

From the perspective of the economic interest relationship between fresh agricultural products enterprises and farmers, the newly increased symbiotic energy S_i is quantified as the change amount of expected earnings before and after the formation of a symbiotic relationship. Let i = 1,2 represent farmers and enterprises, respectively, i = o,s represents non-symbiotic and symbiotic conditions, respectively, then the following expression represents the expected benefits of fresh agricultural products management enterprises and farmers under non-symbiotic conditions: (15)

Among them: (16) (17) (18)

(16) and (17) are the expected returns of farmers and enterprises under non-symbiotic conditions, respectively; (18) q is the actual shipment amount of the operating enterprises' procurement; here, the loss ratio function λ(t) is Eq (1).

Under the symbiotic condition considering the efforts and profit distribution of fresh agricultural products enterprises and farmers, the expected benefits of the two are: (19) (20) (21) (22)

In summary, combined with the above definition of S_i({N₁,N₂}), we can obtain: (23)

Symbiotic system evolution model.

The evolution direction of the symbiotic system depends on the change of the symbiotic dynamic system and its energy. It is necessary to construct a set of ordinary differential equations of the dynamic system to describe the symbiotic environment based on Eq (8), to realize the quantification of the symbiotic phase change criterion, symbiotic evolution criterion and symbiotic stability criterion. Assume the following:

(1) x and z are defined as the economic output of farmers and enterprises, respectively. Then, and represent the trend indicators of the economic output of farmers and enterprises, respectively.

(2) y_i is defined as the total symbiotic energy of farmers and enterprises, where a_i is set as the ecological carrying capacity of farmers and enterprises under non-symbiosis conditions, and B₁ = b. B₂ = g is set as symbiosis parameters of farmers and enterprises, respectively. It can be known from the symbiosis mode of fresh agricultural product management enterprises and farmers that when b>0,g>0, the two symbiosis units are in a mutual-symbiosis mode. When b>0,g<0 or b<0,g>0 or b<0,g<0, the two symbiotic units are in parasitic mode, and other situations are inconsistent with reality, which will not be discussed in this paper.

(3) set xz to represent the new energy added by farmers and enterprises under the symbiosis condition, then bxz represents the effect of farmers affected by symbiosis, and gxz represents the effect of operating enterprises affected by symbiosis.

In conclusion, the symbiotic energy model of fresh agricultural farmers and management enterprises can be expressed as follows: (24)

The symbiotic environmental model between fresh agricultural products management enterprises and farmers can be expressed as follows: (25) where in order to make the ordinary differential equation solvable, it should be set that x = x(t),z = z(t), in Eq (25) has the initial solution x(0) = x₀,z(0) = z₀ and the changes of variables in the system are symmetric.

The paper is based on the self-similar approximation theory [25]. When n→+∞, there are , the two equations have the same mathematical characteristics and evolutionary direction. Therefore, the symbiotic system environment model (25) of the fresh agricultural product management enterprises and the farmers is transformed into a symbiotic system evolution model (26), which can be used as the Lyapunov characteristic exponential function to be constructed: (26)

Among them, the symbiotic parameter in the symbiotic system is b,gϵ(−∞. +∞); the initial solution of the symbiotic system evolution is x(0) = x₀,z(0) = z₀; the requirement is x(t)≥0,z(t)≥0.

Singularity distribution and phase change boundary determination of the symbiotic system. Based on the evolution model of the symbiotic system (26) and criteria 6, 7 and 8, the singularity distribution and phase transition boundary determination method of the symbiotic system based on Lyapunov are presented in this section. According to criterion 8, in order to maintain the stability of the symbiotic system between fresh agricultural products enterprises and farmers, the rate of change of economic output of the system is zero. Set and set the non-zero stable solution {x*≠0,z*≠0} as the solution of the system evolution model (26): (27)

From , formula (27) can be expressed as: (28)

A Taylor expansion of the system Eq (26) is used to obtain an approximate Lyapunov system of equations, and the two characteristic root expressions are obtained as follows: (29)

As shown in Fig 1, the positive and negative values of λ₁ and λ₂ are used to determine the types of singularities of the symbiotic system between fresh agricultural enterprises and farmers.

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Fig 1. Evolution singularity and boundary of symbiosis system between fresh agricultural product management enterprises and farmers.

https://doi.org/10.1371/journal.pone.0236334.g001

The boundary of phase transition is the red line and the blue line b+g = 0, which divides the evolution of the symbiotic system into three domains.

(1) There is no stable point on the upper side of the phase transition boundary along the symbiotic system. At this time, the evolution of the symbiotic system is in an infinite exponential growth trend.

(2) There is a stable point along the lower boundary of the phase transition boundary b+g = 0 of the symbiotic system, that is, there is a unique solution when the evolution rate of the symbiotic system is zero, indicating that the system converges to steady state at this time.

(3) There are two stable points in the middle of the two-phase transition boundary, namely, stable degenerate points and saddle points. At this time, there are two solutions when the system evolution rate is zero, indicating that the symbiotic system is in an evolutionary state.

Urban rail transit network and basic data.

The evolutionary state of the symbiotic system of Eq (26) is divided into two cases: infinite exponential growth and system convergence to steady state. Based on the 3.3 singular point distribution and phase change boundary determination method, this section compares and analyses the evolutionary direction of the system of different symbiotic parameters and symbiotic modes. It provides quantitative analysis for promoting the evolution of the symbiotic unit to the mutualism symbiotic model.

Numerical Simulation Tool: MathWorks MATLAB 2014b

Simulation running environment: windows 64 bit for windows 7

Simulation step: t = 0.5

Initial conditions: {x₀ = 0.01,z₀ = 5};

Experiment 1: infinite exponential growth: ;

Contrast parameters: {b = 2,g = 1}₁ and {b = −0.8,g = 1}₂;

Experiment 2: convergence to steady state:

Symbiotic contrast parameter:{b = 0.25,g = 0.1}₁; {b = −0.75,g = 1}₂; {b = −1,g = −2}₃.

Through experiments 1 and 2, the logarithmic form of solution x(t)≥0,z(t)≥0 of two symbiotic units is lnx(t),lnz(t), and the evolutionary direction of lnx(t),lnz(t) at time t→∞ is compared.

Evolutionary trend of the symbiotic system under infinite exponential growth.

The simulation results of experiment 1 are shown in Fig 2. Under the same initial conditions, with time t→∞, the development speed of (a) the symbiotic unit shows an infinite growth phenomenon and eventually exceeds the initial value. (b) the development speed of (b) symbiotic units shows unlimited growth, while the other side shows a negative growth direction after a certain moment and will eventually be lower than the initial value.

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Fig 2. The infinite exponential growth of symbiotic systems.

(a) b>0,g>0, (b) b<0,g>0.

https://doi.org/10.1371/journal.pone.0236334.g002

Experimental data show that the symbiosis parameter b,g>0, which is the mutualism mode state at this time, will effectively drive and maintain the infinite growth state of the whole symbiosis system. In contrast, b<0,g>0 is a parasitic mode state, which will eventually lead to a decrease in the rate of evolution of the symbiotic system, or even negative growth, leading to the collapse of the symbiotic relationship between the two.

Evolutionary trend of symbiotic systems converge to steady state.

The simulation results of experiment 2 are shown in Fig 3. When the symbiotic parameter satisfies , Fig 3A reaches steady state at t = 20 and symbiotic parameter b,g>0; in Fig 3B, steady state is reached at t = 15, and symbiotic parameter b<0,g>0 is obtained. Compared with the system in Fig 3B, the system in Fig 3A has a slower convergence speed, and the difference between the initial state and the stable point is small. The overall benefit of the symbiotic system in Fig 3A is greater than that in Fig 3B. when the symbiotic parameter satisfies b+g<0 and b<0,g<0, Fig 3C reaches steady state at t = 10. The convergence rate of the system in Fig 3C is significantly higher than that in Fig 3A and 3B, and there is a big difference between the initial state and the stable point, and the overall system income is the least in the three scenarios.

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Fig 3. The symbiotic system converges to a steady-state trend.

(a) b,g>0, (b) b<0,g>0, (c) b<0,g<0.

https://doi.org/10.1371/journal.pone.0236334.g003

Effectiveness analysis of the symbiotic energy model of the symbiotic system.

The paper takes the A-enterprise of an ecological technology company in Jilin Province as a symbiotic unit that cooperates with farmers. Company A is a fresh fruit enterprise at Changbai Mountain in the field of blueberry processing and product sales. It has 1800 acres of blueberry as its own specialty planting base, 9500 acres of regional chain planting base, and drives more than 1,230 farmers around the country. No symbiosis is involved, and the operation mode of enterprise A and surrounding farmers is shown in Fig 4.

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Fig 4. The operation mode of fresh agricultural products enterprises and farmers without symbiosis mode.

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Farmers planted blueberries at Changbai Mountain at the cost of for the planting base, and sold an amount of blueberries Q to enterprise A at the price of p₁. After the lean processing of blueberries at the cost of , enterprise A sells blueberries to consumers at the price of p₂. Where consumer demand is x, then the sales volume of enterprise A is an integral variable related to x.

Under the symbiotic condition, the interaction between enterprise A and the surrounding farmers is shown in Fig 5.

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Fig 5. The symbiosis mode of fresh agricultural product management enterprises and farmers under the symbiosis mode.

https://doi.org/10.1371/journal.pone.0236334.g005

To maintain a good symbiotic relationship, in terms of production and planting, farmers will share the effort cost of z to the processing cost of the enterprise, that is, farmers will produce Changbai Mountain blueberries at the cost of in the planting base, and increase the order quantity of blueberries of enterprise A to Q(1+x_f) through production efforts, and sell them at the price of p₁. Enterpris’ A's efforts in sales will also form part of the cost. Therefore, enterprise A reasonably increases the market price p₂(1+x_r) at the cost of , and the consumer demand is still x. Enterprise A sells the amount of blueberries to consumers within the period of less than the life cycle T of blueberries, and after making profits, it allocates profits with farmers at the proportion of (1−φ).

Under non-symbiotic conditions, Changbai Mountain blueberry management farmer's planting cost and enterprise A’s processing cost can be obtained from the basic data and formula (7) in the previous article, as shown in Table 1:

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Table 1. Related cost data of experimental case.

https://doi.org/10.1371/journal.pone.0236334.t001

The expected revenue of blueberry farmers and the expected revenue of enterprise A at Changbai Mountain before and after the establishment of the symbiotic relationship can be obtained according to the basic data and relevant models in Table 2.

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Table 2. Related benefit data for experimental case.

https://doi.org/10.1371/journal.pone.0236334.t002

Solutions: (30)

From Eq (2.16) and (4.1), we can obtain: (31)

It is easy to know from 0≤t≤T that the new symbiotic energy S_i({N₁,N₂})>0 of the two symbiotic units of Changbai mountain blueberry enterprise A and planting farmers is in line with the practical significance. Moreover, the expected benefits and before and after the symbiosis relationship are both greater than zero, that is to say, by establishing a symbiotic system, both symbiotic units gain benefits in the interaction relationship, which shows that the symbiotic energy equation established in this paper is effective.

Analysis of the singularity distribution of the symbiotic system.

In the phase diagram of the symbiosis system between enterprise A and farmers, the distribution of the MATLAB numerical simulation track line reflects the distribution of characteristic roots of ordinary differential equation of the dynamic system of this symbiosis system. Under different symbiotic parameters, the phase diagram of the symbiotic system is shown in Fig 6:

①when ;

②when ;

③when .

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Fig 6. Phase diagrams of symbiotic systems with different symbiotic parameters.

(a) Stable degradation node and saddle point phase diagram. (b) Stable degradation node phase diagram. (c) Stable nodal phase diagram.

https://doi.org/10.1371/journal.pone.0236334.g006

In Fig 6A, the symbiotic parameter satisfies when ; at this time, the trajectory of xz in the phase diagram of the symbiotic system shows two stable points: the first one is equal to the real root stable degenerate node, and the phase diagram is solved. The stable point is {x* = 1.39,z* = 1.178}₁; the other is the side where the saddle point has a stable point, and the stable point of the phase diagram is {x* = 3.28,z* = 1.811}₂. Substituting the conclusions into Eq (29), when b+g = 0, the eigenvalues are λ₁ = −1,λ₂ = −1, which is consistent with the definition of stable degenerate nodes; when {b = 0.25,g = 0.1}, the eigenvalues are λ₁ = −1,λ₂ = 1.079, which is consistent with the saddle point definition.

In Fig 6B, when the symbiotic parameters satisfy b+g = 0,{b = 0.2,g = −0.2}₂, the trajectories of x,z in the symbiotic system phase diagram are all tangent to a straight line, and the stability point of the phase diagram is {x* = 1.39,z* = 1.178}, which is the stable degradation node in Fig 6B. This phenomenon indicates that there is no other stable point in the region of , that is, the state is the side with no stable point in the symbiotic saddle point.

In Fig 6C, when the parameter satisfies b+g<0,{b = −1,g = −2}₃, then the trajectory of x,z in the phase diagram of the symbiotic system presents a stable point, and the stable point of the phase diagram of the symbiotic system is {x* = 0,785,z* = 0.6164}. By substituting the obtained conclusion into Eq (29), the characteristic roots of Lyapunov of stable nodes are solved as λ₁ = −1,λ₂ = −2.4516, which conforms to the definition of stable nodes.

Analysis of the evolution direction of the symbiotic system under different symbiotic modes.

The basic data of the symbiosis system are shown in Table 3. By Eqs (9), (13) and (14), the total symbiosis energy E₁ of blueberry farmers and the total symbiosis energy E₂ of enterprise A can be obtained.

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Table 3. Basic data of experimental case.

https://doi.org/10.1371/journal.pone.0236334.t003

Solutions: (32)

When , there is no singularity in the symbiosis system, and under the mutual-symbiosis mode, the economic output indicatorsN₁ and N₂ of enterprise A and planting farmers are in the infinite exponential growth mode, and the symbiosis energy and expected benefits formed by the two symbiosis units in the system are all greater than the non-symbiosis state, which is the ideal state in reality. When or b+g<0, The initial value is {N₁,N₂}₀ = {0.01,5}. When different symbiotic parameters converge to a stable state, the data are shown in Table 4, and the analysis conclusion is as follows:

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Table 4. The economic indicators data table of Changbai Mountain blueberry management enterprise A and planting farmers.

https://doi.org/10.1371/journal.pone.0236334.t004

1. Expected return and
   • Both and are greater than 0, and the economic profit of both enterprise A and farmers has been improved; Stable sales channels drive farmers' economic returns, and farmers' expected returns increase with time t;
   • To improve the quality of fresh agricultural products and qualified production, enterprise A needs to pay part of the income, production and sales input cost, then the expected income of enterprise A decreases with the growth of time t. This helps maintain stable, long-term symbiosis.
2. Symbiosis energy E₁ and E₂
   • In the mutual-symbiosis model (b>0,g>0), the symbiosis energy E₁ and E₂ produced by enterprise A and farmers are higher than the two parasitic models, which proves the conclusion that mutual-symbiosis is the optimal development state of the symbiosis system. E₁ and E₂ decrease with the increase of time, indicating that the shorter the time, the higher the market value of fresh agricultural products, reflecting the direct impact of the characteristics of fresh agricultural products on the income of farmers and enterprises.
   • Under the parasitic mode (b<0,g>0), enterprise A and farmers need to maintain a stable and sustainable cooperation state to obtain more symbiotic energy, which promotes their development into a mutualistic symbiotic mode with a slower convergence rate. This phenomenon confirms the conclusion that mutualism is the ultimate direction of symbiotic system development.
3. Index of economic output of symbiotic system ΔN₁ and ΔN₂
   • In the mutualistic symbiosis model (b>0,g>0), when observing t = 20,t = 15, and t = 10, the symbiosis system is convergent to a steady state, {N₁,N₂} has no solution in t = 15 and t = 10, and a long and stable cooperative relationship can help the mutualistic symbiosis model play a role.
   • Under different symbiosis modes, the convergence speed of the symbiosis system between enterprise A and farmers is different. The mutualism symbiosis model converges slowly, the economic output index is good; The parasitic model converges quickly, and the index of economic output is poor. The initial economic output index is set as {N₁,N₂}₀ = {0.01,5}. Under the mutualistic symbiosis mode, the symbiosis system converges to stability at t = 20, at this time {N₁,N₂}_f = {0.342,4.177}. In the parasitic mode (b<0,g>0), the symbiotic system converges to stability at t = 15, at this time {N₁,N₂}_f = {1668,1.227}. In the other parasitic mode (b<0,g<0), the symbiotic system converges to stability at t = 10, at this time {N₁,N₂}_f = {0.788,0.303}.
   • The change of the economic output index ΔN₁ of farmers increases under the three modes, while the change of economic output index ΔN₂ of enterprise A decreases. To obtain long-term high-quality and reliable supply channels, enterprises need to give assistance to farmers, resulting in the decline of their economic output index. However, the long-term symbiotic relationship brings more symbiotic energy to the enterprise, which can gain more profits in the fresh agricultural products market and improve the overall competitiveness of the symbiotic system.