Abstract
The oscillation and instability of systems caused by time delays have been widely studied over the past several decades. In nature, the phenomenon of diffusion is universal. Therefore, it is necessary to investigate the dynamic behavior of reaction-diffusion systems with time delays. In this study, a two-enterprise interaction model with diffusion and delay effects is considered. By analyzing the distribution of the roots of the corresponding characteristic equation, some conditions for the stability of the unique positive equilibrium and the existence of Hopf bifurcation at the steady state are investigated. As the sum of the time delays changes, there are a series of periodic solutions at the trivial steady-state solution of the system. In addition, the direction of Hopf bifurcation and the stability of the periodic solutions are discussed by using the normal form theory and the center manifold reduction of partial functional differential equations. Finally, numerical simulation experiments are conducted to illustrate the validity of the theoretical conclusions.
1. Introduction
As we all know, it is of great significance for an enterprise to consider its developmental progress and growth trend, grasp its position, and comprehensively analyze its interactive relationships with other enterprises in the business ecosystem. In recent years, the dynamic relationships among enterprises described by corresponding mathematical models have been widely studied [1–6]. In [5], Xu proposed the following model for the dynamic competition and cooperation between two enterprises and applied coincidence degree theory to analyze the existence of periodic solutions:where denotes the output of two enterprises, respectively; represents the intrinsic growth rate of two enterprises, respectively; and are the carrying capacities of the natural market, i.e., load capacities of the two enterprises under unrestricted conditions; is the consumption coefficient from the enterprise with output to the one with output ; is the transformation coefficient from the enterprise with output to the one with output ; and represents the initial output of the two enterprises, respectively. It was found that the outputs of the two enterprises reach an equilibrium state and at least have a positive periodic solution when the intrinsic growth rates, natural market loads, competition coefficients, and initial productions of the two enterprises meet certain conditions. Considering that the real dynamic ecological system always depends on information from the historical system, Liao et al. [3, 4] proposed two different competition and cooperation models for two enterprises with a delay (or delays) in succession. They investigated the effect of the delay (or delays) on the stability of the positive equilibrium by choosing (or ) as the bifurcation parameter (or parameters). On the basis of [3], Guerrini [1] further proposed that when the two delays are small, an approximate, simple method can be used to obtain the roots of the characteristic equation. An analysis of the dynamic behavior of a competition and cooperation model for two enterprises with two small delays was also provided. Considering the influence of periodic solutions on the development and operation of enterprises, Xu and Li [6] investigated almost periodic solutions for a competition and cooperation model of two enterprises with time-varying delays and feedback controls. The existence and global stability of the almost periodic solutions were obtained by establishing a suitable Lyapunov functional. In fact, the internal and external effects of time delays were found to be completely different for enterprises. Hence, Li et al. [2] proposed the following differential equations model with four different delays:where , and represent the delay effects within and between enterprises, respectively. Specifically, and denote the gestation periods of the outputs for the two enterprises. and represent the block delay and promoting delay, respectively, imposed by one enterprise on another. One case where in (2) was considered in [4], another case where , was discussed in [3], and some corresponding results were obtained. In [2], Li et al. researched the stability of the positive equilibrium and the existence of Hopf bifurcation, deduced the formula determining the direction of Hopf bifurcation, and achieved the stability of bifurcating periodic solutions by setting and choosing as the parameter. It has shown that time delay has an important impact on the steady-state equilibrium of the interactive relationship between enterprises. The latest research results of some delayed systems can be found in [7–13], which focus on multiple time delays and feedback control, fractional-order delay model, and discrete model with multiple delays.
However, most of these studies only considered the effect of time delays and lost sight of the effect of spatial diffusion. It is well known that diffusion factors are indispensable in the modeling of ecological and biological systems and often utilized to describe the spatial distributions of the densities of some certain substances, such as plants, animals, and other organisms. At present, there are a growing number of dynamic models described by reaction-diffusion equations with time delays in various application areas [14–27]. For example, some scholars [19, 23, 25–27] proposed several different diffusive predator-prey systems with delays to analyze the effect of diffusion on biological systems. Huang et al. [18] researched a plant-pollinator model with diffusion and time delay effects and discussed its bifurcation and temporal periodic patterns. Some scholars [16, 22] established delayed reaction-diffusion systems to investigate the spatiotemporal transmission of bacteria and viruses in epidemiology. As a result of the impact of the growth pole on the economic activities of industrial groups, Hu et al. [17] provided a Kaldor–Kalecki model of the business cycle with diffusion and time delay and described the influence of the diffusion coefficient on the spatial pattern of the system.
Generally, in the development of the regional economy, the enterprises located in the center of the city have the functions of production, information, transportation, trade, service, and decision-making, which act like a “magnetic pole” that attracts production from the surrounding areas and spreads its own production factors to the surrounding areas. Especially when the industrial economic development in the central area is at the mature stage, the diffusion effect continues to increase. Therefore, based on the above argument, we further consider spatial diffusion with zero-flux boundary conditions. A delayed reaction-diffusion system for a two-enterprise interaction should be described by the following model:where denotes the output of two enterprises at time and spatial position , respectively. represents the output of two enterprises in the gestation period at spatial state , respectively. reflects the prevention effect on the output of enterprise caused by its own consumption of limited resources, respectively. As the two enterprises exist in the same environment, the impact of the delay, caused by enterprise 2 consuming the same limited resources in the process of development, on enterprise 1 is expressed by , and the promoting delay by which enterprise 1 supports enterprise 2 is denoted as . In [3], it was suggested that the promoting delay and blocking delay in the gestation period of output for enterprises have important economic significance for their development through changing parameters and . is the Laplacian operator, and denote the diffusion coefficients. represents a bounded domain in with a smooth boundary , and is the outward unit normal vector on . Similar to [2–4], by introducing the terms , we obtain the following model:
For simplicity, we reduce the spatial domain to one dimension . Although there is an abundant literature about the stability and Hopf bifurcation of delayed differential equation models with diffusion, stability analysis of the interactive behaviors among enterprises in the business ecosystem, especially reaction-diffusion financial systems with delays, is very rare. The main contributions of this study are as follows: first, the interaction between two enterprises is discussed from the perspective of ecology, and the factors such as multiple time delay and diffusion affecting the dynamic properties of the system are introduced into the model, which makes the given model more in line with the actual background in economics than the previous models. Second, the complex dynamic properties of the system are discussed, including the stability of the system and the qualitative analysis of the existence of Hopf bifurcation. The parameter calculation expressions of bifurcation direction, stability of bifurcation periodic solution, and period size are determined. Finally, we explain the results from the perspective of application and supplement the relevant conclusions of the existing literature.
The rest of this study is organized as follows. In Section 2, we consider the distribution of the roots of the characteristic equation associated linearized system and explore the stability of the positive steady state and the existence of Hopf bifurcation of system (4) to satisfy some requirements. In Section 3, according to the normal form method and the center manifold reduction developed by Hassard et al. [28], we determine the direction of Hopf bifurcation, give an explicit rule, and analyze the stability of bifurcating periodic solutions. In addition, some numerical simulations are conducted in Section 4 to verify our theoretical predictions. The conclusions are presented in Section 5.
2. Stability and Hopf Bifurcation
In this section, taking as the varying parameter, we analyze the distribution of the roots of the corresponding characteristic equation of (4) at the positive constant steady state and the stability of the positive equilibrium point. Some conditions of Hopf bifurcation of (4) are provided.
According to [3, 4], there is a unique positive constant steady state of system (4), which provides that the following condition holds:
Let ; then, (4) is equivalent to the following system:where . (6) can be written as follows by setting and .
We assume that , and is defined bywith the inner product .
Denote and . For , system (7) can be rewritten as the following abstract differential equation in the phase space :where and , are defined by the following:where
It is easy to see that the linearization of (9) is given byand its characteristic equation iswhere .
According to the properties of the Laplacian operator on the bounded domain [14, 18], the operator on has the eigenvalues and the relative eigenfunctions , , for . is a basis of the phase space . So, any element can be expanded to a Fourier series as follows:where .
By using some simple computations, we have
Hence, (13) is equivalent to
Therefore, all the characteristic roots of (13) are equivalent to the roots of the following equation:
Let , and ; then, (17) can be rewritten as
It is clear that is not a root of (18) for .
When , (18) can be converted into
Suppose and are two roots of (19). For , it is easy to see that
Therefore, if (H1) holds, the unique positive equilibrium point of (4) is asymptotically stable for .
Furthermore, suppose that and is a solution of (18). Then, satisfies the following equation for some :
By separating the imaginary and real parts of (21), we have
It follows from (22) that
Let ; then, (23) becomes
By performing some calculations, one can show that
Note that
Hence, if the following condition holds,then (24) has no positive real roots, and the characteristic equation has no purely imaginary roots for all . Therefore, we have the following conclusion if (H1) and (H2) hold.
Theorem 1. Assume that conditions (H1) and (H2) hold; then, the positive equilibrium point of (7) is locally asymptotically stable for all .
Now, we consider the existence of Hopf bifurcation at the positive equilibrium of (7).
When , we havewhere .
Hence, if , then , and (23) has a unique positive root , whereand . Equation (18) with has a pair of purely imaginary roots . By (22), we can get the values for under this scenario:
Hence, all roots of (18), except , have no zero real parts. According to Theorem 2.7 by Li et al. [2], we know that if (H1) and hold, then the equilibrium remains stable for , and we verify thatwhere is a root of (18), , and .
Apparently, the transversality condition holds in this case, and there is a sign of Hopf bifurcation at and .
Furthermore, we discuss the case when . Let be a root of (18) with . Similarly, we have
By performing some calculations, we can get
It is easy to see that if the following condition holds,then (18) has no purely imaginary roots for . Thus, the following conclusions are true.
Theorem 2. Assume that condition (H1) holds. and be given by (29) and (30), respectively. Then, for system (7), the following conclusions are valid:(i)When (H2) holds, the positive equilibrium is locally asymptotically stable for all (ii)When and (H3) holds, the positive equilibrium is locally asymptotically stable if and unstable if (iii)When , (7) occurs as Hopf bifurcation at
By the same measure, we now make the following assumptions for a certain :
Under condition (H4), equation (23) with has only a positive solution:
Hence, when (H5) and (H6) hold, for , we can deduce thatand for ,
Therefore, it is easy to show that (18) with has no purely imaginary roots, and it follows from (22) that
Lemma 1. Assume that conditions (H1) and (H4)–(H6) hold; then, (18) with has a pair of purely imaginary roots with , and all roots of (18), except , have no zero real parts.
Let be a root of (18) near satisfying and for ; then, the following transversality condition is obtained.
Lemma 2. Assume that conditions (H1) and (H4)–(H6) hold. Then,
Proof. Differentiating both sides of (18) with respect to yieldsIt follows thatAccording to the above analysis,From Lemmas 1 and 2, and the Hopf bifurcation theorem [28], we can have the following result.
Theorem 3. Assume that condition (H1) holds. and are given by (36) and (39), respectively. Then, for system (7), the following results are true:(i)When , the positive equilibrium point is asymptotically stable(ii)When , the positive equilibrium point is unstable(iii)When , (7) occurs as Hopf bifurcation at
3. Direction of Hopf Bifurcation and Stability of Bifurcating Periodic Solution
In the preceding section, we obtained a series of periodic solutions bifurcating from the positive constant steady state of (7) when increases past the critical values . In the present section, by using the normal form method and the center manifold theorem presented in [28–30], we compute the directions, stability, and periods of the bifurcating periodic solution. For brevity, we denote by for some fixed .
Let and . Then, system (7) can be transformed into
Let and . Then, in the phase space , system (44) can be rewritten aswhere , , and are given aswherefor . From the discussion in the last section, we know that (44) has an equilibrium (0,0) and are a pair purely imaginary eigenvalues of characteristic equation of the linear equationand the functional differential equationwhere is a continuous linear operator mapping into .
According to the Riesz representation theorem, for the ordinary functional differential equation,there exists a matrix function of the bounded variation , , such thatwhere . In fact, we can take
Let be the infinitesimal generator corresponding to (50) and be the formal adjoint of under the bilinear inner productfor , and . Then, and are a pair of adjoint operators [31]. Since are a pair of purely imaginary eigenvalues of , they are also a pair of eigenvalues of . Let and be the center spaces of and , respectively, associated with . Thus, is the adjoint space of and [30].
Lemma 3. Let
Then, is a basis of associated with , and is a basis of associated with .
Let and with and .
Through calculation, we can get the following concrete expressions:for and . Also,
Then, by (53) and some calculations, we can further obtain
Then, we defineand structure a new basis for as
Obviously, is a second-order unit matrix. Furthermore, we define and , where , , and . Thus, the center subspace of linear equation (48) is given aswhere and is the complementary subspace of in .
Let be the infinitesimal generator corresponding to (48); then, (44) can be rewritten in the following form:where
Using the decomposition and (60), the solution of (45) on the center manifold can be rewritten aswhere , , and . In fact, on the center manifold, the solution of (45) is given by
Let . By the definitions of and , it is easy to obtain , and we further obtain
Therefore, (64) can be expressed aswhere . By [30], we can deduce that satisfieswhere
Let
From (47) and (64), (66) can be rewritten as
Then,
Therefore, we can have , wherewhere
Note that for .
Let . Comparing the coefficient with that of (70), we can obtain
Next, we compute the terms and that appear in the equation for . Using a method to the one used in [32, 33] and [27], it follows from (69) that
In addition, we can get from [30] thatwhere
Then, from (67) and (70)–(81), we have
Since has only two eigenvalues , (82) has a unique solution in given by
From (81), we can obtain, for , that
Hence, for , we have
From the definition of and (82), we can get
Note that . Therefore,where
By using the definition of and combining (82) and (90), we can obtain the following equation:
Note that
Then, for , we get
By (94), we havewhere
Similarly, we can also havewhere
Substituting (90) and (98) into the expression of , we can compute the following values:
Through the above analysis, we can draw the following conclusions.
Theorem 3. Suppose conditions (H1)–(H6) hold. Then, the parameters , , and determine the properties of Hopf bifurcation, where determines the direction of the Hopf bifurcation, determines the stability of the bifurcating periodic solutions on the center manifold, and determines the period of the bifurcating periodic solutions, i.e.,(i)If , the direction of the Hopf bifurcation is forward (backward)(ii)If , the bifurcating periodic solutions are stable (unstable)(iii)If , the period increases (decreases)
4. Numerical Simulations
Example 1. We choose , and , which satisfy the above conditions. We take the initial conditions asThen, system (4) with the Neumann boundary has a unique positive constant steady state . By performing calculations, we can obtain and . According to Theorem 2, is locally asymptotically stable if and is unstable if . When crosses through the critical point , loses its stability, and Hopf bifurcation occurs. The simulation results are shown in Figures 1–4. Based on formulas in Section 3, we can perform computations and obtain that , , and . Hence, when crosses through , the bifurcation takes place, and a homogeneous, asymptotically stable periodic solution appears near the trivial steady-state solution , both of which are shown in Figures 3 and 4.
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Example 2. In this case, we choose diffusion coefficients , and we keep other parameter values and initial conditions unchanged. We consider Hopf bifurcation of the system with the Dirichlet boundary condition:The results of the simulation are shown in Figures 5–8. Apparently, when passes through the critical value , Hopf bifurcation occurs, and the spatially homogeneous bifurcating periodic solution appears, which suggests that there are periodic oscillations in the interaction between two enterprises.
Figures 9 and 10 show the bifurcation state of the system with and without diffusion for and under the Neumann boundary, respectively. It is found that the diffusion can affect the amplitude of oscillation of the periodic solutions to a certain extent, and the bifurcating periodic solutions are regular and orbitally asymptotically stable.
To illustrate the spatial effect of diffusion on the system, we adopt the method used by references [14, 21] and choose different diffusion coefficients for two different boundary systems under the situations of asymptotically stable periodic solutions. The corresponding spatially dynamic behavior diagrams are shown in Figures 11 and 12.
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5. Conclusions
Since the diffusions of information, technology, and products exist objectively, we deem that the model with reaction-diffusion and delays is more suitable for the interaction between two enterprises, which has no relevant results for system (3) at present. The aim of this study is first to introduce multiple delays and diffusion factors into the traditional two-enterprise interaction model and then to explore the asymptotic stability of the equilibrium and the periodicity of the solution by taking the sum of two time delays as the bifurcation parameter. It is shown that when the time delay passes through a critical value, the system changes from a stable state to an unstable state. The numerical simulations demonstrate that the diffusion effect causes rich, spatially dynamic behaviors.
We acknowledge that there are some economic implications. In the enterprise cluster, two enterprises can achieve a stable state and symbiosis of development via dynamic interaction. Although there exists competition, cooperation is very effective, and the enterprises can achieve long-term codevelopment through appropriate positive interactions. It is noteworthy that in addition to the varying economic environments, competition powers, mark-saturation degrees, and delay effects, the diffusion speed of products or technologies is an important factor in the self-development of enterprises. When enterprises are in a state of cyclical fluctuations, appropriate diffusion effects may change the amplitude of these fluctuations, which is beneficial to the enterprises to a certain extent. Furthermore, it is easy to find that differences in the initial resources or positions of an enterprise have a certain impact on its future self-development. Therefore, it is obvious that with the development of the digital economy and the emergence of industrial clusters, the internal relationships between enterprises are increasingly in line with the ecological model. As a consequence, it is of great practical significance to study the dynamic properties of these systems for understanding the evolution of market structure and taking corresponding regulatory strategies. Benefiting from the latest research studies, we will further consider the dynamic behaviors of discrete-time and fractional-order models in the future.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Chongqing Big Data Engineering Laboratory for Children, Chongqing Electronics Engineering Technology Research Center for Interactive Learning, Chongqing University Innovation Research Group, Chongqing Key Discipline of Electronic Information, Chongqing Natural Science Foundation Project (CSTC2021-msxm1993), the program of Chongqing University of Education (KY202116C), and the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202101614).