Bifurcations of Tumor-Immune Competition Systems with Delay

and Applied Analysis 3 2.1. Tumor-Free Point. The characteristic equation of system (8) at the tumor-free equilibrium P 0 is Δ (λ) = (λ + δ) (λ − α + σ δ e −λτ ) = 0. (9) Then we have the following results. Lemma 1. (I) If α = σ/δ, then (1) Equation (9) has a simple zero root, and all other roots have negative real parts as 0 ≤ τ < δ/σ; (2) Equation (9) has a double zero root, and all other roots have negative real parts as τ = δ/σ; (3) Equation (9) has at least one root with positive real parts as τ > δ/σ. (II) If α < σ/δ, then (1) all roots of (9) have negative real parts as 0 ≤ τ < τ 0 ; (2) Equation (9) has a pair of conjugate purely imaginary roots ±iω + , and all other roots have negative real parts as τ = τ 0 ; (3) Equation (9) has at least one root with positive real parts as τ > τ 0 . (III) Equation (9) has a negative root−δ, and all other roots have positive real parts as α > σ/δ. Proof. λ = 0 is a root of (9) if and only if α = σ/δ. If τ = 0, (9) has two roots λ 1 = −δ and λ 2 = α − σ/δ. Then there are three cases: (1) λ 2 = 0 as α = σ/δ; (2) λ 2 < 0 as α < σ/δ; (3) λ 2 > 0 as α > σ/δ. We will consider the case τ > 0 as follows. If α = σ/δ, τ = δ/σ, then Δ 󸀠 (λ) = 2λ − α + σ δ e −λτ − τλ σ δ e −λτ + δ − στe −λτ ; (10) hence, Δ󸀠(λ)| λ=0 = 0, and Δ󸀠󸀠(λ)| λ=0 = δ 2 /σ > 0. Thus λ = 0 is the double zero root of (9). If (9) has purely imaginary roots, then the roots must be the solution of Δ 0 (λ) = λ − α + σ δ e −λτ = 0. (11) Assume that λ = iω(ω > 0) is the root of (11); that is,


Introduction
In this century, cancer remains one of the most dangerous killers of humankind; every year millions of people suffer from cancer and die from this disease throughout the world; see Boyle et al. [1].Recently, there has been much interest in mathematical modeling of immune response with the intruder (see, e.g., Liu et al. [2,3], Yafia [4], d'Onofrio et al. [5,6], and the references cited therein).In fact, mathematical models are feasible to propose simple models which are capable of displaying some of the essential immunological phenomena.The delayed models of tumor and immune response interactions have been studied extensively; we refer to Bi and Ruan [7], Yafia [8], Mayer et al. [9], Yafia [10], and the references cited therein, which have shown that various bifurcations can occur in such models.It is interesting to consider the nonlinear dynamics of the delayed tumorimmune model.
In 1994, Kuznetsov et al. [11] took into account the penetration of tumor cells (TCs) by effector cells (ECs) and proposed a model describing the response of ECs to the growth of TCs.They assumed that interactions between ECs and TCs in vitro can be described by the kinetic scheme shown in Figure 1, where , , ,  * , and  * are the local concentrations of ECs, TCs, EC-TC complexes, inactivated ECs, and lethally hit TCs, respectively.Then the Kuznetsov and Taylor model is as follows: d d =  +  (, ) −  1  −  1  + ( −1 +  3 ) , where  is the normal rate of the flow of adult ECs into the tumor site, (, ) describes the accumulation of effector cells in the tumor cells localization due to the presence of the tumor,  1 ,  2 , and  3 are the coefficients of the processes of destruction and migration for E, EC, and TC, respectively,  is the coefficient of the maximal growth of tumor, and  is the environment capacity.Kuznetsov et al. [11] claimed that experimental observations motivate the approximation / ≈ 0; therefore, it is reasonable to assume that  ≈  with  =  1 /( 1 + 2 + 3 ).Kuznetsov et al. [11] also suggested that the function  is in the Michaelis-Menten form (, ) = (, ) = (/( + ))(,  > 0).In 2003, Gałach [12] suggested that the function  should be in a Lotka-Volterra form (, ) = (, ) =  1 ; then the model ( 1) can be reduced to where  denotes the dimensionless density of ECs,  stands for the dimensionless density of the population of TCs,  1 =  2 ,  2 =  3 , and all coefficients are positive.Set  =  0   ,  =  0   ,  = (1/ 2  0 )  ,  0 > 0,  0 > 0. Replace  with   and  with   .Then (2) can be written as where  = / 2  2 0 ,  =  1  0 / 2  0 ,  =  1  0 / 2  0 ,  =  1 / 2  0 ,  = / 2  0 , and  =  0 .
Mayer et al. [9] and Asachenkov et al. [13] pointed out that the delays should be taken into account to describe the times necessary for molecule production, proliferation, differentiation of cells, transport, and so forth.In fact, the immune system needs time to develop a suitable response after the invasion of tumor cells; the binding of EC and TC also needs time.Therefore, we introduce time delays into the model of immune response.Integrating models [9][10][11], we will consider the model as follows: where  =  − ; if the stimulation coefficient of the immune system exceeds the neutralization coefficient of ECs in the process of the formation of EC-TC complexes, then  > 0.
In this paper, we will consider the dynamical behaviors of model ( 4) with  1 =  2 = .The rest of this paper is organized as follows.In Section 2, the linear analysis of the model is carried out and local stability of the equilibria and the conditions of Hopf bifurcation are given.Section 3 is devoted to the analysis of Hopf, steady-state bifurcations, and B-T bifurcation.Numerical results and simulations are carried out to illustrate the main results.A brief discussion and more numerical simulations are given in Section 4.

Local Analysis
In this section, we will study the local stability of the equilibria and the Hopf bifurcations of system It is easy to obtain that system (5) have three equilibria  0 (/, 0),  1 ( 1 ,  1 ), and  2 ( 2 ,  2 ), where It is easy to see that  1 < 0.
Because the number of tumor cells or effect cells is positive, we only consider the dynamical behaviors of the equilibria  0 (tumor-free point) and  2 in the rest of the paper.
Abstract and Applied Analysis 3 2.1.Tumor-Free Point.The characteristic equation of system (8) at the tumor-free equilibrium  0 is Then we have the following results.
Lemma 1. (I) If  = /, then (1) Equation ( 9) has a simple zero root, and all other roots have negative real parts as 0 ≤  < /; (2) Equation (9) has a double zero root, and all other roots have negative real parts as  = /; (3) Equation ( 9) has at least one root with positive real parts as  > /.
(III) Equation (9) has a negative root −, and all other roots have positive real parts as  > /.
Differentiating both sides of ( 11) with respect to , we have Using Rouché theorem, we know that the conclusions of (I)( 1) and (I)(2) are true. Thus Hence the conclusion (I)( then (II)(3) is proved.Then all the proof is finished.
Thus the following results can be obtained by Lemma 1.

Positive Equilibrium.
If  −  > 0, then the positive equilibrium  2 exists.The characteristic equation of system (8) at the point  2 is where thus (19) has no zero root.If  = 0, then (19) can be written as It is easy to see and then all roots of (22) have negative real parts.
If  > 0, we assume that ( 19) has a pair of purely imaginary roots  =  ( > 0); thus and hence Noting  +  > 0,  < 0, then we have  −  < 0 and  2 −  2 = ( + )( − ) < 0. That is to say, (25) has only one positive root and the corresponding critical value is We can also give the following transversal condition: Then all results of this theorem have been proven.
From Lemma 3, the following theorem can be obtained directly.

Direction and Stability of the Bifurcations
3.1.Hopf Bifurcation.In the previous section, we know that system (5) undergoes Hopf bifurcation at the tumor-free equilibrium  0 and positive equilibrium  2 under certain conditions.In this section, we will study the stability and direction of the Hopf bifurcated periodic solution by using the center manifold reduction and normal form theory of retarded functional differential equations due to the ideals of Faria and Magalhães [15,16].Throughout this section, we always assume that system (5) undergoes Hopf bifurcations at the equilibrium  ( 0 or  2 ) as the critical parameter  =   and the corresponding purely imaginary roots are ±  .
Normalizing the delay  in system ( 7) by the time-scaling  → /, then ( 7) is transformed into This scaling is irrelevant for the study of the stability of the equilibrium but will be crucial for the Hopf bifurcation analysis.
Assume that  is the infinitesimal generator of ż () = (  )(  ) satisfying Φ = Φ with and  has a pair of conjugate purely imaginary roots ±  .Denote that  is the invariant space of  associated with Λ; then dim  = 2.We can decompose  := ([−1, 0], R 2 ) to  = ⨁ by the formal adjoint theory for FDEs by Hale [17].Considering complex coordinates, we still denote  as be the bases of , where ) is a vector in  2 and (  )Φ 1 =   V. Choose a basis Ψ for the adjoint space  * , such that (Ψ, Φ) =  2 , where (⋅, ⋅) is the bilinear form on  * ×  associated with the adjoint equation.Thus, Ψ = col(Ψ 1 , Ψ 2 ) with and Take the enlarged phase space , defined as The projection  :  →  is defined as where We write the Taylor expansion as follows: where  1  and  2  are homogeneous polynomials in , , and  of degree ,  = 2, 3, with coefficients in C 2 and Ker , h.o.t.stands for higher order terms.The normal form method implies a normal form on the center manifold of the origin for (32) which is where  1 2 (, 0, ) and  1 3 (, 0, ) are homogeneous polynomials in  and , respectively.
Firstly, knowing that On the other hand, we know that Let Abstract and Applied Analysis 7 we obtain where In order to obtain  4 , we need to compute ℎ 110 (), ℎ 200 ().From (53), it follows that where . Solving (59), we can obtain where ) , Hence Thus, the normal form of system (42) has the form where Summarizing all above, we have the following theorem.
Theorem 5.The flow on the center manifold of the equilibrium  at  = 0 is given by (64).Also the following results hold: (  (2) the bifurcated periodic solution is stable if  2 < 0 and unstable if  2 > 0; (3) the period of the bifurcated periodic solution is In the following, we will give some simulations to illustrate the results of Theorems 4 and 5 for model (4).We cite the parameters in [11], that is,  = 0.1181,  = 0.0031,  = 0.3743,  = 1.636, and  = 0.002.Then (4) has a tumor-free equilibrium (0.3155, 0), which is unstable, and a positive equilibrium (1.33435, 92.1911), which is locally asymptotically stable.We only simulate local properties of the stable equilibrium (1.33435, 92.1911) here in Figures 2(a Remark 6.From Figures 2(c) and 2(d), we can see that the amplitude vibration for () is much bigger than that of (); also both () and () with respect to  are not so smooth.Then the Hopf bifurcated periodic solution on ((), ()) plan is not given here.At the same time, we can see that the dynamical behaviors of the system have been changed although  is small.

Steady-State Bifurcation.
From Section 2, we know that system (5) undergoes a steady-state bifurcation at the tumorfree equilibrium  0 as  = /, 0 <  < /.In this section, we will discuss the properties of the steady-state bifurcation by using the center manifold reduction and normal form theories of retarded functional differential equations.
Assuming that  is the infinitesimal generator of ż () = (/)(  ), then  has a simple zero root.Set Λ = {0} and we denote by  the invariant space of  associated with Λ; then dim  = 1.We can decompose  := ([−1, 0], R 2 ) to  =  ⨁  by the formal adjoint theory for FDEs by Hale [17].Let  = span(Φ) be the bases for , where Φ Choose a basis Ψ for the adjoint space  * , where Ψ = ( 1 ,  2 ), which is a vector in R 2 * satisfying Ψ(/) = − Ψ(0).Thus we can obtain According to the method of Faria and a similar computation in the last section, we can obtain Noting one has Thus, the normal form of system ( 5) is Then the following two results are obvious.

Discussion
We have studied the nonlinear dynamics of Kuznetsov, Makalkin, and Taylor's model with delay, which is a twodimensional model of tumor cells and immune system.We first provided linear analysis of the model with delays at the possible equilibria, namely, the tumor-free and positive equilibria, and discussed the existence of Hopf bifurcation at the equilibria.We investigated the Hopf bifurcation, Bogdanov-Takens bifurcation, and steady-state bifurcation in the model.Numerical simulations were presented to illustrate the theoretical analysis and results.
Our analysis on the existence and stability of the tumorfree equilibrium corresponds to this elimination process and on the existence and stability of the positive equilibrium corresponds to coexistence of the immune system and the tumor system.Our results on the existence and stability of the Hopf bifurcated periodic solutions of  2 describe the equilibrium process.When a stable periodic orbit exists, it can be understood that the tumor and the immune system can coexist although the cancer is not eliminated.The conditions for the parameters provide theories basis to control the development or progression of the tumors.The phenomena have been observed in some models as d'Onofrio [5], Kuznetsov et al. [11], and Bi and Xiao [14].In particular, Bi and Ruan [7] have shown that various bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation, can occur in such models.Our results on the existence and stability of the bifurcated (Hopf, Bogdanov-Takens, and steady-state) periodic solutions describe rich dynamical behaviors of  0 , which show that the elimination process is so complex and difficult to control.
Finally, we should point out that we have studied the local dynamical behaviors of  0 and  2 .As the example in our paper showed these two equilibria may coexist.Correspondingly, the system can exhibit more degenerate bifurcations including Hopf-Hopf and resonant higher codimension bifurcations.It would be interesting to consider these dynamics of the delayed model.

Figure 1 :
Figure 1: Kinetic scheme describing interactions between ECs and TCs.