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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 639405, 22 pages

http://dx.doi.org/10.1155/2014/639405
Research Article

Dynamical Behavior and Stability Analysis in a Hybrid Epidemiological-Economic Model with Incubation

1Institute of Systems Science, Northeastern University, Shenyang 110004, China

2State Key Laboratory of Integrated Automation of Process Industry, Northeastern University, Shenyang 110004, China

3Changli Institute of Fruit Forestry, Hebei Academy of Agricultural and Forestry Sciences, Changli 066600, China

4Institute of Biotechnology, College of Life and Health Sciences, Northeastern University, Shenyang 110004, China

Received 12 January 2014; Accepted 15 April 2014; Published 12 May 2014

Academic Editor: Weiming Wang

Copyright © 2014 Chao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A hybrid SIR vector disease model with incubation is established, where susceptible host population satisfies the logistic equation and the recovered host individuals are commercially harvested. It is utilized to discuss the transmission mechanism of infectious disease and dynamical effect of commercial harvest on population dynamics. Positivity and permanence of solutions are analytically investigated. By choosing economic interest of commercial harvesting as a parameter, dynamical behavior and local stability of model system without time delay are studied. It reveals that there is a phenomenon of singularity induced bifurcation as well as local stability switch around interior equilibrium when economic interest increases through zero. State feedback controllers are designed to stabilize model system around the desired interior equilibria in the case of zero economic interest and positive economic interest, respectively. By analyzing corresponding characteristic equation of model system with time delay, local stability analysis around interior equilibrium is discussed due to variation of time delay. Hopf bifurcation occurs at the critical value of time delay and corresponding limit cycle is also observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied. Numerical simulations are carried out to show consistency with theoretical analysis.

1. Introduction

In recent decades, plenty of mathematical models describing the population dynamics of infectious disease have been extensively utilized to understand the transmission mechanism of infectious disease within population ecosystem (see [14] and references therein). Much research efforts have been paid to susceptible-infective-recovered (SIR) vector disease model and corresponding model dynamics (see [512] and references therein). Generally, in modelling of communicable disease, the incidence rate (the rate of new infections) is considered to play a vital role in ensuring that the model can provide a reasonable qualitative description of the infectious disease dynamics [3, 4].

In order to discuss the spread of an infectious disease transmitted by a vector (e.g., mosquitoes and rats), Takeuchi et al. [7] formulated a delayed SIR epidemic model with a bilinear incidence rate. Beretta et al. [8] considered the global stability of disease free equilibrium and endemic equilibrium of model system; it was shown that the disease free equilibrium is globally stable for any time delay while the endemic equilibrium is not feasible. By constructing a suitable Lyapunov functional, sufficient conditions were derived to guarantee that if the endemic equilibrium is feasible, it is also globally stable for the delay being sufficiently small. Ruan and Wang [13] studied the global dynamics of an SIR model with vital dynamics and nonlinear incidence rate of saturated mass action and global qualitative and bifurcation analyses are carried out. Ma et al. [14] derived an explicit expression of lower bound of the infective individual of solution of model system, which was proposed as an open problem. They therefore gave an estimation of the length of time delay ensuring global asymptotic stability of the endemic equilibrium. Xu and Ma [15] proposed an SIR epidemic model with nonlinear incidence rate and time delay. By analyzing the corresponding characteristic equations, local stability of an endemic equilibrium and a disease free equilibrium are discussed. An SIR model with distributed delay and a general incidence function is studied in McCluskey [9], and the global dynamics for the SIR epidemiological system is analyzed in Zhou and Cui [10]. Wang et al. [11] considered the asymptotic behavior of the following SIR vector model: where , , and represent the population density of susceptible, infective, and recovered host individuals at time , respectively. It is assumed that the population growth of susceptible host individuals is governed by the logistic growth with a carrying capacity as well as intrinsic birth rate constant . is the average number of constants per infective per unit time and denotes the incubation time, and and stand for the death rate of infective and recovered host individuals, respectively. represents the recovery rate of infective host individuals. The local stability of endemic equilibrium is investigated, and conditions for Hopf bifurcation to occur are derived in [11]. Along with the line of this research, Enatsu et al. [12] analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease and the proposed model with a nonlinear incidence rate satisfying suitable monotonicity conditions.

Nowadays, biological resource within ecosystem is commercially harvested and sold with aim of achieving economic interest [16, 17]. It is well known that harvesting has a strong impact on the dynamic evolution of population and several mathematical models have been established to discuss dynamic effects of harvest effort on population in ecological-epidemiological system, which can be found in [1821] and the references therein. The role of harvesting in a predator-prey-parasite system is discussed in [18]; theoretical results show that, using impulsive harvesting effort as control parameter, it is not only possible to control the cyclic behavior of the system populations leading to the persistence of all species but other desired stable equilibrium including disease free can be obtained. A ratio-dependent eco-epidemiological system is proposed in [19] where prey population is subject to harvesting. Positive invariance, boundedness, stability of equilibria, and permanence of system have been established. In [20], an eco-epidemiological model is studied where prey disease is modeled by a susceptible-infective scheme, and the role of harvesting and switching on the dynamics of disease propagation and/or eradication is discussed. An eco-epidemiological model with distributed time delay and impulsive control strategy is investigated in [21]; local stability and complex dynamical behavior are discussed. Under the system of market economy, harvest effort is usually influenced by variation of economic interest of commercial harvesting [16, 22]. It should be noted that the above mentioned related work [1821] only concentrate on the role of harvest effort on population dynamics, while the dynamic effect of economic interest on commercial harvesting and indirect dynamic effect on ecosystem are not considered. The work done in [12] is an extension of [11] with nonlinear incidence rate, while dynamic effect of harvest effort on population dynamics is not considered.

Recently, some hybrid dynamical models are proposed in [2328], which are utilized to discuss the interaction mechanism of harvested ecosystem from an economic perspective. Compared with the traditional mathematical models (differential equations or difference equations) discussing the population dynamics in ecosystem, the hybrid mathematical models proposed in [2328] are made up of differential equations and algebraic equations, where differential equations concentrate on coexistence and interaction mechanism of population and algebraic equations offer a simpler way to study the effect of harvest effort on ecosystem from an economic perspective. Complex dynamical behavior and stability analysis in prey-predator ecosystems with stage-structured population and gestation delay are considered in [2328]. In general, differential-algebraic models exhibit more complicated dynamics than ordinary differential models. The differential-algebraic models have been applied widely in power systems, aerospace engineering, chemical processes, social management systems, biological systems, network analysis and oil catalysis, and cracking process (see [2931] and references therein). With the help of differential-algebraic model for the power systems and bifurcation theory, complex dynamical behaviors of the power systems, especially the bifurcation phenomena that reveal the instability mechanism of power systems have been extensively studied, which can be found in [3234] and the references therein. Furthermore, some applications of differential-algebraic models in the field of economy, which can be found in [35, 36].

It is well known that the recovered host individuals are naturally immune to vector disease [1], and its potential economic interest can be commercially exploited. Furthermore, harvest effort is usually influenced by variation of economic interest of commercial harvesting [16, 22] under the system of market economy. Consequently, it is necessary to discuss the coexistence and interaction mechanism of population within harvested epidemiological ecosystem as well as dynamical effect of harvest effort due to variation of economic interest. However, as far as knowledge goes, nobody has explicitly proposed a mathematical model to discuss the dynamic effect of commercial harvest on epidemiological system under the system of market economy. The main objective of this paper is to investigate the transmission mechanism of infectious disease and dynamical effect of commercial harvest on population dynamics, especially the complex dynamical behavior and stability switch due to variation of incubation and commercial harvest economic interest. The organisation of the rest section of this paper is as follows. By introducing commercial harvest effort into model system (1), a hybrid epidemiological-economic model is established in Section 2. Positivity and permanence of solutions of model system are discussed in Section 3. In Section 4, qualitative analyses of model system are performed. Conditions for existence of interior equilibrium of model system are studied. Dynamical behavior of model system without incubation around the interior equilibrium is investigated due to variation of economic interest, and state feedback controllers are designed to stabilize model system around the desired interior equilibria. Furthermore, local stability analysis of model system with incubation is analyzed due to variation of time delay; directions of Hopf bifurcation and stability of the bifurcating periodic solutions are also studied. Numerical simulations are made in Section 5, which are utilized to support the theoretical findings obtained in this paper. Finally, this paper ends with a conclusion.

2. Model Formulation

In 1954, Gordon [22] proposed the economic theory of a common-property resource, which studies the effect of harvest effort on ecosystem from an economic perspective. In [22], an algebraic equation is proposed to investigate the economic interest of yield of the harvest effort, which takes form as follows:

Associated with model (1), an algebraic equation, which considers the economic interest of the harvest effort on recovered host individuals in epidemiological system, that is, , is established as follows: where represents the harvest effort on recovered host individuals at time . represents the economic interest of harvest effort on the recovered host individuals. and represent unit price of harvested population and cost of harvest effort, respectively.

Based on (1) and (3), a delayed hybrid model which consists of three differential equations and an algebraic equation can be established as follows: where , , , , and other parameters share the same interpretations mentioned in (1) and (3), and initial conditions for model system (4) are defined in the Banach space: where . It is also assumed that ( ) for a biological reason.

Model system (4) can be expressed in the following form: where

Remark 1. The algebraic equation in model system (6) contains no differentiated variables; hence, the leading matrix in model system (6) has a corresponding zero row.

Remark 2. The model proposed in [11], which composed of differential equations, only discusses the interaction and coexistence mechanism of susceptible, infective, and recovered host individuals. Compared with the model proposed in [11], algebraic equations are incorporated into the model system (4), which focus on the economic interest of harvesting on recovered host individuals. Hence, the established model not only investigates interaction and coexistence mechanism of population in harvested ecosystem but also studies the dynamical behavior due to the variation of economic interest of commercial harvesting and incubation.

3. Positivity and Permanence

Theorem 3. Any solutions of model system (4) with initial conditions are positive.

Proof. For any solutions of model system (4), it is easy to show that is locally Lipschitz and satisfies the condition, , where ( ) have been defined in model system (4).

Due to the lemma in [37] and Theorem A.4 in [38], any solution of the model system (4) with positive initial conditions exists uniquely and each component of the solution remains within the interval for some . Standard and simple arguments show that solutions of model system (4) always exist and stay positive. Hence, this completes the positivity of the solutions of model system (4).

From a viewpoint of biological and economic interest perspective, persistence of solutions of model system (4) in the case of economic interest will be investigated in this section. Some preliminaries are introduced as follows.

Definition 4 (see [39]). Model system (4) is said to be permanent if there exists a compact region such that every solution of model system (4) with initial conditions will eventually enter and remain in region .

Definition 5 (see [39]). Consider a metric space with metric . The distance of a point from a subset of is defined by

It is further assumed that is the closure of an open set , and is nonempty and is the boundary of . Consequently, , . We will also suppose that is a semigroup on satisfying

Let and be the global attractor for .

Lemma 6 (see [39]). Suppose that satisfies (9) and the following conditions hold.(i)There is a such that is compact for ;(ii) is point dissipative in ;(iii) is isolated and has an acyclic covering .Then is uniformly persistent if and only if for each , for .

Lemma 7 (see [40]). Consider the following equation: where and for all ; it derives the following:(i)If , then ,(ii)If , then .

Lemma 8. For any solutions of model system (4), we have

Proof. By using Theorem 3, it follows from the first equation of model system (4) that which derives that .

According to Theorem 3 and the first and second equation of model system (4), it gives that which derives that .

Lemma 9. If , then of solution of model system (4) with initial conditions satisfies where and are independent of corresponding initial values of model system (4).

Proof. Firstly, let denote space of continuous functions mapping into , where :

Denote ,   and .

Next, all conditions in Lemma 6 will be checked. In order to facilitate the proof, we consider the following subsystem of model system (4): where ,   are continuous on and ,   .

By Definition 5 and model system (16), it is easy to see that and are positively invariant, and conditions (i) and (ii) of Lemma 6 clearly hold.

Since model system (16) possesses two constant solutions in : , with the following form:

It follows from simple computation that

Moreover, it follows from the second equation of model system (16) that which reveals that all points in approach to ; that is, . By using the similar analysis mentioned above, it can be also concluded that all points in approach to ; that is, . Based on the above analysis, it shows that invariant sets and are isolated invariant, and is isolated and an acyclic covering. It can be concluded that condition (iii) of Lemma 6 holds.

Finally, we will show that for . Based on the definition of , it is easy to show that . We will show in the following part.

If , then there exists a positive solution to model system (16) with . If , then holds for sufficiently small and there exists a positive constant such that , and for all .

By the second equation of model system (16), it derives that holds for all .

Consider the following equation:

Based on (21) and the comparison principle, it derives that for all .

On the other hand, if , then it follows from Lemma 7 that for all solutions of (21). It can be concluded that , which is a contradiction to . Consequently, it can be derived that .

According to the above analysis, all conditions of Lemma 6 hold. By using Lemma 6, it can be obtained that where and are independent of the corresponding initial values of model system (4).

Theorem 10. If , , and , then all solutions of model system (4) with initial conditions are persistent.

Proof. According to Lemmas 8 and 9, it can be obtained that hold for all , which derive that

When the economic interest , it follows from Theorem 3 and the fourth equation of model system (4) that and . Based on the third equation of model system (4), it can be computed that . According to (24), it derives that provided that .

In the case of , it derives that based on implicit function theory [41]. According to the third equation of model system (4), it can be obtained that which derives that provided that .

It follows from Theorem 3 and the third equation of model system (4) that which derives that

Hence, it gives that , and it can be rewritten as follows:

Based on (23), (24), (25), and (26), it can be concluded that all solutions of model system (4) with initial conditions are persistent in the case of , and it follows from (23), (24), (28), (30) and (31) that all solutions of model system (4) with initial conditions are persistent in the case of .

4. Qualitative Analysis of Model System

Dynamical effects of harvest effort and time delay on population dynamics are discussed in this section. It should be noted that the interior equilibrium biologically interprets that susceptible, infective, and recovered host individuals survive as well as harvest on recovered host individuals exists. Bifurcation phenomenon around the interior equilibria can reveal instability mechanism of model system, which are theoretically relevant to infectious disease control and sustainable yield on recovered host individuals in the real world. Consequently, we will mainly concentrate on dynamical behavior and local stability analysis around interior equilibrium of model system (4) in this paper.

4.1. Model System without Time Delay

In this section, dynamical behavior of model system (4) without time delay is investigated, and local stability analysis around the interior equilibrium is discussed due to variation of economic interest of commercial harvesting. Furthermore, state feedback controllers are designed to stabilize model system around the desired interior equilibria in the case of zero economic interest and positive economic interest, respectively.

4.1.1. Singularity Induced Bifurcation

Theorem 11. Model system (4) without time delay has a singularity induced bifurcation around the interior equilibrium, and is a bifurcation value. Furthermore, local stability switch occurs as increases through 0.

Proof. Based on the economic theory of a common-property resource [22], there is a phenomenon of bioeconomic equilibrium in the case of zero harvest economic interest; that is, . An interior equilibrium can be obtained as follows: , where , , , and .

According to biological interpretation of the interior equilibrium, it follows that ,   , and . In order to guarantee the existence of interior equilibrium, some inequalities are satisfied:

Let be a bifurcation parameter, ,

It can be calculated that By virtue of (32), it can be obtained that

Furthermore, it can be also calculated that

It follows from (32) that

Based on Section IV(A) in [42], can be defined as follows:

By simple computing,

According to (32), it derives that

Based on the above analysis, four items (i–iv) can be obtained as follows.(i)It is easy to show that has a simple zero eigenvalue: and based on (35).(ii)It follows from (37) that is nonsingular around .(iii)By virtue of (40), it can be shown that is nonsingular around ; hence .(iv)It is easy to show and , which follows

It should be noted that the conditions for singularity induced bifurcation, which is introduced in Section III (A) in [42], consist of three conditions, that is, SI1, SI2, and SI3. According to the above items (i)–(iv), SI1, SI2, and SI3 are all satisfied; hence model (4) without time delay has a singularity induced bifurcation around the interior equilibrium and the bifurcation value is .

Along with the line of the above proof, for model (4) without time delay, it follows from simple computing that It follows from (32) that

Inequality (44) satisfies Theorem 3 of [42]. According to Theorem 3 of [42], when increases through , one eigenvalue (denoted by ) of model system (4) without time delay moves from to along the real axis by diverging through infinity; the movement behavior of this eigenvalue influences the stability of model system (4) without time delay.

Since the Jacobian of model system (4) without time delay evaluated around takes the following form: according to the leading matrix in model system (4) and , the characteristic equation of the model system (4) without time delay around is

By virtue of simple computation, the characteristic equation is as follows:

It can be concluded that the rest eigenvalues of model system (4) without time delay (denoted by and ) have negative real parts by using the Routh-Hurwitz criteria [43]. It follows from Theorem  3 in [42] that there is only one eigenvalue diverging to infinity as increases through , and the rest eigenvalues are continuous and nonzero and cannot jump from one half open complex plane to another one as increases through . It has been shown that moves from to along the real axis by diverging through infinity. Therefore, and are continuous and bounded in the half plane as increases through and their movement behaviors have no influence on the stability of model system (4) without time delay around the interior equilibrium .

According to Table 1 and the stability theory, it can be concluded that model system (4) without time delay is stable around as and model system (4) without time delay is unstable around as . Consequently, a stability switch occurs as increases through .

tab1
Table 1: Signs of real parts of eigen values of model (4) without time delay around interior equilibrium .

Remark 12. Some preliminaries of singularity induced bifurcation are introduced below. Parameter dependent differential-algebraic hybrid system of the form where , , and have appropriate dimensions. It has been shown recently that there are generically three types of codimension one local bifurcation associated with the differential-algebraic model (48), namely, saddle-node bifurcation, Hopf bifurcation, and singularity induced bifurcation (see [42]).

The singularity induced bifurcation is firstly introduced and analyzed in [42, 44]. It is a new type of bifurcation and does not occur in usual ordinary differential equation system, which has been characterized for differential-algebraic system, and later improved in [45, 46]. Roughly speaking, the singularity induced bifurcation refers to a stability change of the differential-algebraic hybrid model (48) owing to some eigenvalues of related linearization diverging to infinity when Jacobian is singular.

One of the important consequences of the singularity induced bifurcation is that it leads to an impulse phenomenon, which may result in the collapse of the differential-algebraic system (see [45]). More detailed introductions of the singularity induced bifurcation can be found in [42, 4446].

Remark 13. It follows from Theorem 11 that there is a phenomenon of singularity induced bifurcation around the interior equilibrium when economic interest increases through zero, which can cause local stability switch of model system (4). As stated in Remark 12, the singularity induced bifurcation can result in impulse phenomenon, which may lead to the collapse of the proposed model. In the harvested epidemiological-economic system, the impulse phenomenon is vividly reflected with the outbreak of infectious disease during a short period in the real world. Under this climate, the infected population will be beyond the carrying capacity of environment, which is disastrous for sustainable development of the harvested ecosystem as well as prosperous yield on recovered host individuals.

4.1.2. State Feedback Controller

In order to maintain the sustainable yield on recovered host individuals biological resource as well as economic interest of commercial harvesting at an ideal level, some corresponding control strategies should be taken to eliminate the impulse phenomenon caused by singularity induced bifurcation and stabilize model (4) without time delay. In this subsection, state feedback controllers are designed to stabilize model system (4) without time delay around corresponding interior equilibria in the case of and , respectively.

According to the leading matrix in model system (4) and in (45) (the Jacobian of model system (4) without time delay around the interior equilibrium ), it can be calculated that . By using Theorem 2-2.1 in [47], it is easy to show that the model system (4) without time delay is locally controllable around the interior equilibrium in the case of . Consequently, a state feedback controller can be applied to stabilize the model system (4) without time delay around . By using Theorem 3-1.2 in [47], a state feedback controller ( is a feedback gain and is the component of the interior equilibrium ) can be applied to stabilize model system (4) without time delay around .

Furthermore, the controlled model system (4) without time delay takes the following form:

Theorem 14. When economic interest of harvesting is zero, , if the feedback gain satisfies the following inequality: then singularity induced bifurcation is eliminated and model system (49) is stable around .

Proof. The Jacobian of the model system (49) evaluated at the interior equilibrium takes the form

According to the leading matrix in the model system (4) and , the characteristic equation of model system (49) around is , which can be expressed as follows: where , , and .

By using the Routh-Hurwitz criteria [43], model (49) is locally stable around if and only if satisfies

Consequently, if the feedback gain satisfies the above inequality, then model system (49) is stable around in the case of zero interest of harvesting.

Let denote interior equilibrium of model (4) in the case of positive economic interest of harvesting ( ), where , , , and satisfies the following equation: where , .

Based on Routh-Hurwitz criteria [43], (54) has two positive roots if economic interest satisfies the following inequalities:

As analyzed above, there are two interior equilibria (denoted by and ) when . In this subsection, we only design the controller for the model (4) around the interior equilibrium . Some symmetric results about can be also obtained, and is denoted as for simplicity in the following part.

Theorem 15. When economic interest of harvesting is positive, , if feedback gain of controller satisfies following inequality: then model system (49) is stable around the interior equilibrium .

Proof. The proof is similar to the proof of Theorem 14 of this paper.

Remark 16. It follows from (55) and Theorem 15 that economic interest of commercial harvesting should be regulated within certain interval , which guarantees the existence of interior equilibrium in the case of positive economic interest. After applying the state feedback controller into model system (4) without time delay, model system can be stabilized around the corresponding interior equilibrium, respectively. The elimination of the singularity induced bifurcation means the harvested epidemiological-economic system restores to ecological balance and avoidance of infectious disease outbreak.

4.2. Model System with Time Delay

By analyzing corresponding characteristic equation of model system with time delay, local stability analysis around the interior equilibrium due to variation of time delay is discussed. Conditions for existence of Hopf bifurcation are studied. Furthermore, directions of Hopf bifurcation and stability of periodic solutions are investigated.

4.2.1. Local Stability and Hopf Bifurcation

As analyzed in the above subsection, in the case of time delay and positive economic interest of harvesting where is defined in (55), there are two interior equilibria and for model system (4) with respect to the positive economic interest .

In this subsection, we only investigate dynamical behavior of model system (4) around the interior equilibrium . Some symmetric results about the interior equilibrium can be also obtained, and is denoted as for simplicity. According to Jacobian evaluated at the interior equilibrium and the leading matrix in model system (4), we can obtain the characteristic equation of model system (4) around , which can be expressed as follows: where

Now substituting ( is a positive real number) into (58) and separating the real and imaginary parts, two transcendental equations can be obtained as follows:

By squaring and adding (60) and (61), it can be calculated that where , , .

According to the values of and the Routh-Hurwitz criteria [43], a simple assumption that (58) has at least one positive real root is , which derives that . Hence, under this assumption, (58) will have a pair of purely imaginary roots of the form .

By eliminating from (60) and (61), it can be calculated that the corresponding to is as follows: where .

By using Butler’s lemma [48], model system (4) is locally stable around for . Subsequently, conditions for existence of Hopf bifurcation in [39] are utilized to investigate whether Hopf bifurcation occurs as increases through .

Theorem 17. If , then model system (4) undergoes Hopf bifurcation around the interior equilibrium when ,   . Furthermore, an attracting invariant closed curve bifurcates from interior equilibrium when and .

Proof. As mentioned above, let represent the purely imaginary root of (58). It follows from (58) that , which determines a set of possible values of .

In the following part, we determine the direction of motion of as is varied; namely, we determine

By differentiating (58) with respect to , it can be obtained that

From (62) and the above equation, it can be obtained that

According to the values of given in (58) of this paper, it is easy to show that .

Furthermore, if , then it can be shown that . Hence, it can be concluded that , which derives . Consequently, the transversality condition holds and Hopf bifurcation occurs at , . Furthermore, an attracting invariant closed curve bifurcates from interior equilibrium when and .

Remark 18. signifies that there exists at least one eigenvalue with positive real part for , and the conditions for Hopf bifurcation in [39] are also satisfied yielding the required periodic solution.

4.2.2. Properties of Hopf Bifurcation

By using normal theory and center manifold theorem [49], directions of Hopf bifurcation and stability of the bifurcating periodic solutions are discussed in this section. As analyzed in Section 4.1.2, when economic interest of harvesting ( is defined in (55)), it follows from implicit function theorem [41] and the fourth equation of model system (4) that . Furthermore, model system (4) can be transformed into the following form:

Firstly, some transformations associated with component of interior equilibrium are given as follows: Then is the Hopf bifurcation value of model system (4). Bars of variables are dropped for simplicity of notations; model system (4) is transformed to a functional differential equation in as where is the Banach space of continuous functions mapping the interval into , , for and , are defined as follows, respectively: where , .

It is easy to show that is a continuous linear function mapping into . According to Riesz representation theorem [40], there exists a matrix function of bounded variation for such that where .

In fact, we can choose where denotes the Dirac delta function.

If is any given function in and is the unique solution of the linearized equation of (70) with initial function at zero, then the solution operator is defined by

It follows from Lemma  7.1.1 in [39] that ,   is a strongly continuous semigroup of linear transformation on and the infinitesimal generator of , is as follows: for , the space of functions mapping the interval into which have continuous first derivative and also define then model system (70) is equivalent to

For , the space of functions mapping interval into the three-dimensional row vectors which have continuous first derivative, define and a bilinear inner product where . It follows from the above analysis and are adjoint operators.

By virtue of discussion in Section 4.2.1, are eigenvalues of . Hence, they are also eigenvalues of . In the following, eigenvectors of and are corresponding to and , respectively.

Suppose is the eigenvectors of corresponding to , which derives that . By using the definition of , (71) and (72), it gives that

For , then it can be obtained that