Analysis of 3D Cobweb Economic Differential Dynamic System Based on Supply-Demand and Price Relationships

. When the supply and demand of commodities are out of equilibrium, the dynamic changes in supply and demand are mainly manifested in the dynamic ﬂuctuation of commodity prices and the change of price will drive the change of supply and demand. This paper proposes a method for constructing a three-dimensional cobweb economic model using the “predator-prey” theory. Firstly, based on the intrinsic driving relationships among supply, price, and demand, a diﬀerential dynamic model of the commodity and a pulse diﬀerential model when productivity increases are established. Secondly, the diﬀerential dynamics theory is used to analyze the existence of the model’s positive equilibrium and to prove its global asymptotic stability, uniform boundedness, persistence, and existence of the periodic solution. Finally, the correctness of the conclusions of the model is veriﬁed by numerical simulation. The relevant conclusions of model (1) reveal the evolution laws of supply, price, and demand that ﬂuctuate around the positive equilibrium and tend to balance. Model (32) reveals that with the continuous improvement of production technology, supply quantity, price, and demand will ﬂuctuate, but all three will change periodically. These conclusions can provide valuable help for people in the production, sale, and purchase of goods.


Introduction
It has always been an important subject for scholars and businessmen to explore the dynamic operation law of the commodity economy.Because it is very difficult to do experimental research on economic operation law, the mathematical model has become an important tool to analyze economic operation law.For example, the statistical model [1] is an indispensable mathematical tool in the econometric theory [2].e separation theorem [3] of the investment theory is expressed by the probability model and the optimization model [4].e application and research of mathematical model in economics have many specific elaborations in [5], which will not be repeated here.
e demand, supply, and price are essential elements in all commodity economy systems.e cobweb economy models [6][7][8][9] are a mathematical system to study the relationships among them.Most cobweb economic systems only study the law of commodity operation from the external relationship between demand and price or between supply and price.Few scholars set out to construct a mathematical system from the internal driving force between them to explore the relationships among demand, price, and supply.
Demand is the driving force of the commodity economy.Without demand, there will be no supply.e purpose of supply is to maximize the utility of goods.
e marginal utility of goods [10] determines the price, and the total utility determines the supply of goods.When the quantity of supply changes, it will drive the change of price, and the change of price will drive the change of demand.Conversely, when demand changes, it also drives price and supply changes.e external relations among them are as follows: when the quantity of goods exceeds the demand, the price will decrease; when supply is less than demand, the price will rise.
e change of commodity quantity or price is a process changing with time.When the price rises, it will drive the increase of commodity quantity, and the increase of commodity quantity will hinder the rise of price.Generally, the change of price will also affect the demand: when the price increases, the demand will decrease.When the price decreases, the demand will increase moderately.
According to the above analysis, the relationship between supply and price or between price and demand is the relationship between predator and prey [11][12][13][14] in biomathematics [15].More specifically, prices exist for predator demand and supplies prey on price.e quantity of supply (the number of predators) increases (or decreases) with the increase (or decrease) of the price (the number of prey).In this paper, the differential dynamics theory between predator and prey is applied to construct the differential dynamic system among supply, price, or demand and then to explore the dynamic operation law among them.
By revealing the internal driving relationship among supply, price, and demand, it can help people predict the development trend of supply and demand scientifically and provide the basis for making stable price policy.Businesses can adjust production, circulation, or procurement to effectively avoid risks according to the development trend of supply and demand.Under the condition of ensuring that the price does not fluctuate obviously, using the development trend of supply and demand to promote the flexible change of price can create a benign business competition environment.

Model Construction and Analysis
e price and supply of goods are based on the demand of consumers.Many commodities, such as pork, eggs, or seafood, will decrease in demand when prices rise.When the price decreases, the demand will increase.Under the condition that other conditions remain unchanged, the demand and price of goods change in the opposite direction.Because the number of consumers is limited, there is an upper limit on demand.e supply of goods also varies with the price.When the price rises, the supply will increase.When the price decreases, the supply will decrease.In addition, when the supply increases to "oversupply" with the increase of price, the goods will be sold at a lower price, resulting in some decreases in the price of goods.e decline in commodity prices will stimulate the increase in demand.When the demand increases to "demand exceeds supply," commodity prices will rise again, leading to an increase in supply.
Before establishing the differential dynamics model of the relationship among demand, price, and supply, some following assumptions need to be given about the model.
(1) Limited by consumption capacity, this article assumes that the demand has an upper limit, and in other words, the demand is density dependent [16][17][18].
(2) e increase in demand has the characteristics of rapid and direct functional response to price, and it can be assumed that the functional response is the linear relation.
(3) Price is the external manifestation of the game between demand and supply.It can be assumed that the change rate of price is inversely proportional to supply and directly proportional to demand.(4) Because most commodities require production time, the increase in supply does not often have a linear relationship with the functional response of price.Assume that the functional response is a type II functional response [17][18][19].
Let D(t), P(t), and S(t) be the dimensionless functions of demand, price, and supply of goods at time t, respectively.Based on the above analysis and assumptions, the threedimensional cobweb economic differential dynamics system based on the relationship among supply, price, and demand is as follows.
where e 1 , e 2 , e 3 , k, α, β, c, δ, ω are all positive real numbers.eir specific economic meanings are shown in Table 1.e initial conditions of system (1) are given as D(0) > 0, P(0) > 0, S(0) > 0. ( Known from the practical applicability of system (1), its feasible domain is It is easy to know that the right end of system (1) is smooth and continuous in R 3 + .erefore, system (1) has a unique solution in R 3 + .System (1) fully considers the functional response of demand, price, supply, and their mutual interference, expresses the differential dynamic relationships among demand, price, and supply, and has practical significance and theoretical value.Based on the economic significance, this article only conducts a qualitative analysis of system (1) in R 3 + .

Property Analysis of System (1)
Theorem 1.All solutions of system (1) with the initial conditions (2) are positive for all t ≥ 0.
In practice, the price, demand, and supply of commodities hardly have negative values, so the conclusion of eorem 1 is true.

□
Lemma 1 (see [20]).Consider the scalar differential equation where f(t, u) is continuous in t and locally Lipschitz in u, for all t ≥ 0 and all u ∈ Js ⊂ R. Let [t 0 , T) (T could be infinity) be the maximal interval of existence of the solution u(t), and suppose u(t) ∈ J for all t ∈ [t 0 , T).Let v(t) be a continuous function whose upper right-hand derivative D + v(t) satisfies the differential inequality Lemma 1 is called the comparison principle.

V(D, P, S)
en, According to Lemma 1, it can be seen that erefore, the conclusion of eorem 2 holds.e proof is finished.
In practice, limited by production capacity, the supply is bound to have an upper limit.When the price of a product is higher than the consumer's ability to bear, the consumer will not be able to buy the product, so the price must be within the range of consumers' acceptance.Limited by the number of consumers, the demand must be limited.e results are basically consistent with those of eorem 2.
If D ≠ 0, P ≠ 0, S � 0 and βe 1 − ke 2 > 0, then system (1) has a non-negative equilibrium If D ≠ 0, P ≠ 0, S ≠ 0, by solving the third equation in the system of equation ( 12), we can get By substituting the above formula into the first equation, we can obtain By substituting the above formula into the second equation, then e boundary equilibrium E 0 indicates the state in which goods are eliminated, such as tape recorder, BPS, and so on, consumers have no demand for them and will not buy them, and then there will be no commodity supply.e boundary equilibrium E 1 indicates that the product is in a state of "selfsufficiency," the product is only used to satisfy the self, the product has no price, and there is no external supply.e non-negative equilibrium E 2 is also a boundary equilibrium, which is a state of commodity with "price without market": there is demand and there is price, but there is no supply.
e only positive equilibrium E 3 represents the relative equilibrium state when supply and demand play games with each other.
For exploring the stability of system (1) at the above equilibriums, the Jacobi matrix [18] of system (1) needs to be calculated, and its specific expression is (1) e boundary equilibrium E 0 is unstable.
Proof. e following only proves conclusion (3), and ( 1) and (2) can be proved by the same method.Let I be the identity matrix, and then the expression of the characteristic equation |J(E 2 ) − λI| � 0 is Let λ 1 , λ 2 , λ 3 be the three roots of the above equation; then,

4
Discrete Dynamics in Nature and Society If δ − e 3 < 0 and H 1 : βe 1 − ke 2 > 0 hold, it can be seen that λ 1 is a negative number, and λ 2 and λ 3 both have negative real parts.
erefore, according to the Hurwitz discriminant [17,18], it can be known that the non-negative equilibrium point E 2 is locally asymptotically stable.e proof is completed.
e non-negative equilibrium E 0 is unstable, indicating that the commodity will inevitably be born.In a local area, if the condition βe 1 − ke 2 < 0 holds, then the equilibrium state of the self-sufficient economy is E 1 .If the conversion rate of supply-predation price is less than its consumption rate (δ − e 3 < 0), the state of price without market can also be stabilized in local areas.

□
Theorem 5.If assumptions H 2 − H 4 hold, then the only positive equilibrium E 3 of system ( 1) is locally asymptotically stable.
Proof.Because the Jacobi matrix of system (1) at E 3 is the expression of |J(D * , P * , S * ) − λI| � 0 is where According to Routh-Hurwitz theorem [18], it is known that all roots of the equation |J(D * , P * , S * ) − λI| � 0 have negative real parts.erefore, the only positive equilibrium E 3 is locally asymptotic stable.e proof is completed.eorem 5 gives the conditions H 2 − H 4 that the only positive equilibrium is locally asymptotically stable.ey are also the conditions for the existence of the positive equilibrium.□ Lemma 2 (see [21]).Consider a continuously differentiable three-dimensional differential system dD dt � F(D, P, S), defined in some open simply connected subset U ⊂ R 3 .Assume that there exists a continuously differentiable B(D, P, S): vanishing only on a set of zero Lebesgue measure.en, system (23) has no periodic orbits contained in U.

is function B is usually called a Dulac function of system (23).
Lemma 2 is called the Bendixson-Dulac theorem.Theorem 6.If k − β ≥ 0, then system (1) does not have the limit cycle in R 3 + .
Proof.In system (1), let and take the Dulac function [21] B(D, P, S) � ω + P DPS . (26) According to Lemma 2, it can be known that system (1) does not have the limit cycle in R 3  + . is ends the proof.
According to the conclusions of eorems 4-6, the following theorem can be obtained.□ Theorem 7. Let k − β ≥ 0, and there are the following conclusions.

Numerical Simulation of System (1).
To observe the global asymptotic stability of the equilibriums of system (1), it will take the initial values If parameters are taken, it is easy to verify k − β ≥ 0, βe 1 − ke 2 < 0. According to conclusion (1) of eorem 7, we can get that the equilibrium E 1 � (0.4, 0, 0) is globally asymptotically stable.If the parameter group (29) is selected, the numerical simulation of system (1) is shown in Figure 1.
If the supply S � 0 and the price P � 0, the demand for self-sufficiency D � 0.4 can be sustained for a long time.
If parameters are taken, it is easy to verify that the above parameters meet the conditions for the existence of the positive equilibrium and k − β > 0. By calculating, the positive equilibrium E 3 � (1.33, 0.44, 1.66).e numerical simulation of system ( 1) is shown in Figures 3 and 4. Figures 3 and 4 show that with the passage or extension of time, although the values of demand D(T), price P(T), and supply S(T) will fluctuate continuously, the amplitude will gradually decrease and tend to the positive equilibrium E 3 .

Analysis of the Commodity Dynamic Relationship When Production Technology Is Improved
With the development of technology and the improvement of production technology, the production cost of goods will inevitably be reduced, and the supply of goods will be temporarily increased, which will lead to the decline of price, and the reduction of price will also promote the temporary increase of demand.In practice, the decline of price is a gradual process.At the same time, the changes of demand and supply in this process also occur gradually.However, to simplify the problem, this article assumes that production technology cyclically increases, and the demand, price, and supply have changed immediately.Assuming that at t � kτ(k � 1, 2, . ..), after the production technology is improved, the increase in demand is d 0 , the decrease in price is p 0 , and the increase in supply is s 0 .e specific differential dynamic system is as follows.
Proof.Still choose the function where m 1 � min e 2 , e 3  .Also, ω − 1 ≥ 0, so dV dt Because the function β(e 1 + m 1 )D − kβD 2 is bounded, let one of its upper bounds be m 2 > 0, and it has Solving the above differential inequality, we can obtain According to the last three expressions of system (32) and the construction method of the function V(D, P, S), we can get In summary, Solve this system of differential inequalities where kτ < t ≤ (k + 1)τ(k � 1, 2, . ..).Take the limit erefore, V(t) is uniformly bounded.Furthermore, all positive solutions of system (32) are uniformly bounded.e proof is completed.
Consider the following impulsive differential equation: According to the conclusion in [23], if the initial value is U(0 + ) � u 0 /1 − e − e 3 τ , the impulsive differential equation (43) has a global asymptotic stability positive periodic solution at is, for any solution of (43), there are According to the first and second equations of system (32), we can get Using the same method as above, it can be proved that there exist T D > 0, T P > 0, m D , m P > 0, so that if t > T D , t > T P , we can get e proof is completed.e conclusions of eorems 8 and 9 show that no matter how science and technology improve, with the passage of time, the demand, price, and supply of goods will have upper and lower limits.□ Theorem 10.If ω − 1 ≥ 0, then system (32) has the positive periodic solution.
According to the supremum and infimum principle [24] and the continuity of function D(t), we can get Define the mapping According to Brouwer's fixed point theorem [25], there is at least one point t 0 (∈ [T, +∞)) such that D(t 0 ) � D(t 0 + τ).erefore, D(t) has at least one positive periodic solution with τ as the period.
In the same way: P(t) and S(t) all have at least one positive periodic solution with τ as the period.
In summary, it can be obtained if ω − 1 ≥ 0, system (32) has a positive periodic solution.e proof is completed.
eorem 10 shows that demand, price, and supply will fluctuate periodically when production technology is improved periodically.□
It can be seen from Figure 5 that although the demand will increase instantaneously when the productivity increases, it will return to the normal state immediately.When t ∈ [0, 300], the integral curve of system (32) gradually changes from irregular to regular.When t ∈ [300, +∞), D(T), P(T), S(T) appear in the form of cyclic periodic solutions.It is verified that the solution of system (32) is periodic.In addition, when t ∈ [0, +∞), D(t), P(t), and S(t) are in a band-shaped region, which verifies that the solution of the system (32) is uniformly bounded and persistence.

Conclusion
e cobweb model is a traditional model for analyzing economic dynamic changes, which is used to describe the change trend of the interaction of supply, demand, and price.
e existing research results are mainly as follows.
(2) Most of the research results use the external expression relationship among supply, demand, and price to build a cobweb model, which lacks the influence of internal driving force.(3) A few research results (such as [9]) have constructed a continuous cobweb model, but the focus is on the convergence rate, and the qualitative analysis of the model needs to be more in-depth.
In this paper, a non-linear continuous cobweb economic differential dynamics model is constructed by using the "predator-prey" theory.e specific research results are as follows.
(1) Without the influence of other factors, the relevant conclusions of model (1) show that under certain conditions, the two economic states of self- sufficiency and high price but little demand can also be sustained for a long time.
(2) In the state of imbalance, although the supply, price, and demand will continue to fluctuate around the positive equilibrium, the amplitude decreases over time and approaches equilibrium.(3) Model (32) reveals that when the production technology is cyclically improved, although the supply, price, and demand will fluctuate irregularly in the initial period, all three will change periodically.
is paper mainly studies the dynamic change behavior of the cobweb economic model from the perspective of differential dynamics and gives some properties of the model.
ese conclusions can provide valuable help for people in the production, sales, or purchase of goods.

Figure 1 :
Figure 1: Plane phase diagram of the global asymptotic stability at E 1 .

Figure 2 :Figure 3 : 3 Figure 4 :
Figure 2: Plane phase diagram of the global asymptotic stability at E 2 .

Table 1 :
Economic meaning of parameters.