Abstract

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

1. Introduction

It is of great significance to study the dynamical behaviors of chemical reaction models to understand the reaction mechanism and evolution law of the reaction process of reactants. Experiments show that the description of the internal reaction mechanism of the chemical reaction system by delay reaction-diffusion equation is more realistic. Therefore, many scholars have conducted in-depth studies on the equation and obtained a large number of practical conclusions. In 1990, De Kepper et al. discovered Turing patterns in the CIMA reaction experiment, which also verified Turing’s theoretical work over 40 years ago. In 1991 and 1992, Lengyel and Epstein in [1, 2] proposed the following mathematical model to depict the experimental process of the CIMA:where and are the concentrations of two chemical reactants, respectively, and are positive constants representing the quantities related to the feed concentrations, the ratio of diffusion coefficients and the rescaling parameter are denoted as and , respectively, is a bounded open domain in with smooth boundary , and is the Laplace operator.

System (1) has been extensively studied by many scholars (see [3–7]). In [5], the researchers mainly analyzed some basic properties of system (1). It can be found through the study that when the parameter (size of the reactor), (effective diffusion rate), and (initial concentration of the reactants) are relatively small, the system has no nonconstant steady state. In addition, when the initial concentration of the reactant is within an appropriate range, the system has nonconstant steady states for a large diffusion rate, . In 2005, on the basis of [3], Jang et al. studied the global bifurcation problem of the nonconstant positive steady-state solutions in one-dimensional case and considered its limiting behavior by using the shadow system method. In [6], Yi et al. discussed the conditions of homogeneous equilibrium solution and periodic solution and gave the detailed Hopf bifurcation analysis for the ODE and PDE models. In 2009, Yi et al. [7] proved the global asymptotical behavior of the constant positive equilibrium solution and convergence of all solutions of the system by constructing the Lyapunov function. In [4], Jin et al. proved the bifurcations of steady-state solutions and spatially nonhomogeneous periodic solutions of (1).

Obviously, in the initial study of the model, most scholars did not consider the influence of time-delay factors. It is well known that the ordinary differential equation reflects that the development of things only depends on the current state, while the delay differential equation is used to describe the development system which depends on both the current state and the past state. In practical problems, if the delay factor is not considered, the properties and states of the system may change, even lead to wrong conclusions. Usually, a long delay will destroy the stability of the equilibrium point of the system. Therefore, it is more realistic to consider the delay factor in the system (see [8–15]). Based on this fact, Celik et al. [16] considered a coupled delayed-Lengyel–Epstein model as follows:

By choosing as the varying parameter and analyzing the related characteristic equations, Celik et al. [16] investigated the stability of the constant equilibrium point and the existence of Hopf bifurcation of system (2) and determined the necessary conditions for the parameters.

In [17, 18], Merdan and Kayan proposed the following delayed models:and

They investigated the existence of Hopf bifurcation of the systems and the properties of Hopf bifurcation.

Recently, Zhang and He [19] assumed that the delay only occurred in the self-decomposition of the activator and further studied the delayed differential equation model as the following form:

They analyzed the influence of the change of delay on the dynamical behaviors and found that the equilibrium of system (5) would eventually become unstable after passing through several stable switches and Hopf bifurcations at some certain critical values of .

For a long time, most scholars are more concerned about biological mathematics. Therefore, in a period of time, biological mathematics has made a rapid development, and some practical conclusions have been obtained. However, there are many chemical reaction functional differential equations in nature, so it is of great practical significance to study the existence, stability, and Hopf bifurcation of periodic solutions of these functional differential equations in the field of chemistry. Of course, researchers did not realize the influence of time delay on the properties of periodic solutions of differential equations in the early stages. With the maturity of the biological mathematics theory, researchers began to realize the importance of time delay in chemical reaction systems. Apparently, these models (2)–(5) discussed above are based on the hypothesis that the delay effect is a single activator or inhibitor alone. In the field of chemistry, scholars are concerned about the range of time delay, when the system produces Hopf bifurcation and the equilibrium point is stable, which means that the parameters of the model should be selected in an appropriate range to save energy to the greatest extent and control environmental pollution for the model of the system. Many models represent the dynamic systems of chemical reactions through ordinary differential equations with no delay or differential equations with the same delay. However, the effects of time delay in the activator and inhibitor are completely different. Based on the fact, a more reasonable mathematical model to describe CIMA reaction should be depicted by the Lengyel–Epstein model with two delays as follows:where are the time lags of the activator and the inhibitor.

We know that the most widely studied delay differential equations are the cases with a single delay or multiple identical delays. When the delay is taken as a parameter, the unique positive equilibrium of the system often loses stability at the critical value after Hopf bifurcation. However, there are multiple stable intervals and unstable intervals in the system when it has many different time delays because of their different critical values. We pay more attention to the changes happened to the dynamic properties of the system in this case. If we fix one of the time delays to its stable interval and take the other as a dynamic parameter, we can study the rich dynamic properties of the system. Since the time lags between catalysts and inhibitors are different in actual chemical reactions, it would be interesting and realistic to consider the two different delays in the system. To the best knowledge of the authors, the Lengyel–Epstein model with two different delays is seldom studied by scholars. By choosing different delays as the bifurcation parameter, the main purpose of this article is to analyze the effect of the two different delays on the dynamical behaviors of system (6).

This article is organized as follows. In Section 2, we investigate the stability of the positive equilibrium and the existence of local Hopf bifurcation of system (6) in three cases according to different symbols of . In Section 3, by choosing as a parameter and fixing in its stable interval, the direction and stability of bifurcating periodic solutions are obtained by using the normal form method and the center manifold reduction for delayed differential equations developed by Hassard et al. [20]. In Section 4, some numerical examples and simulations results are included to illustrate the validity of the main results.

2. Stability and Hopf Bifurcation

In this section, we use the method in [21] to discuss the system’s positivity and uniform boundedness.

Lemma 1. Let . Then, the solutions of system (6) are nonnegative, for all .

Proof. From the first equation, we havethat is,Therefore, one can show that , for all .
Similarly, we can apply the same method to prove , for all . In fact, from the second equation, we can obtainthat is,Therefore, one can show that , for all .
For the boundedness of all solutions of system (6), according to [5], we have the following result.

Lemma 2. Let . Then, all solutions of system (6) are uniformly bounded, that is,

Now, we will analyze the stability of the positive equilibrium point and the existence of local Hopf bifurcation of system (6) in detail in three cases , and .

Let ; it is easy to get that system (6) has a unique positive equilibrium point . Then, the linearized system of (6) is given as follows:where

The associated characteristic equation of the linear system (12) is given by the following form:

That is,

Case 1. .
Then, the characteristic equation (15) can be simplified toBy the Routh–Hurwitz criterion, we can see that all roots of (16) have always negative real parts if the condition,holds.
Hence, the positive equilibrium of system (6) is locally asymptotically stable if holds.

Case 2. .
Then, the associated characteristic equation (15) can be transformed toLet be a root of the characteristic equation (18); then, we haveSeparating the real and imaginary parts of equation (19), we obtainIt follows from (20) thatLet . Then, (21) becomesNotice that andHence, one can see that if the conditionholds, then equation (22) has no positive solution. Therefore, all solutions of equation (18) have negative real parts when , and condition holds.

Theorem 1. Assume that and holds; then, the positive equilibrium of system (6) is asymptotically stable, for all .

Now, we suppose that the following condition is given:

Then, one can obtain that is a unique positive root of equation (22) and is a pair of purely imaginary roots of the characteristic equation (18). Here,

By substituting into (20), one can solve a series of as

Let be a root of (18) near such that and . Differentiating both sides of equation (18) with respect to , the following can be performed:

It follows that

Furthermore, we can know that

According to the above discussion and Corollary 2.4 of [3], we can present the following results.

Theorem 2. Assume that and (H3) hold, can be defined by (27), and can be given by (26). Then,(i)The positive equilibrium of system (6) is asymptotically stable, for all .(ii)The positive equilibrium of system (6) is unstable for (iii)System (6) undergoes Hopf bifurcation at the positive equilibrium for

Case 3. .
In this case, we consider the characteristic equation (15) by choosing as a parameter and fixing in its stable interval . Without loss of generality, we analyze system (6) under condition .
Multiplying both sides of equation (15) by , then one can get the following equation:Let be a root of the characteristic equation (15); then, equation (31) becomes the following form:Separating the real and imaginary parts of equation (32), we obtainIt follows from (33) thatAccording to (34), we can obtainwhereNow, we give the following assumption for equation (35).
: assume that equation (35) has at least one positive root.
If condition holds, then equation (35) has a positive root such that equation (15) has a pair of purely imaginary roots . Thus, we haveDifferentiating both sides of equation (15) with respect to , the following equation can be easily obtained:It follows thatLet , and we can know thatwhereNext, we assume that the following condition holds:Therefore, from the above discussions and Hopf bifurcation Theorem 2, the following results can be directly deduced.