HOPF BIFURCATION FOR TUMOR-IMMUNE COMPETITION SYSTEMS WITH DELAY

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1 Electronic Journal of Differential Equations Vol. 4 (4) No. 7 pp. 3. ISSN: URL: or ftp ejde.math.txstate.edu HOPF BIFURCATION FOR TUMOR-IMMUNE COMPETITION SYSTEMS WITH DELAY PING BI HEYING XIAO Abstract. In this article a immune response system with delay is considered which consists of two-dimensional nonlinear differential equations. The main purpose of this paper is to explore the Hopf bifurcation of a immune response system with delay. The general formula of the direction the estimation formula of period and stability of bifurcated periodic solution are also given. Especially the conditions of the global existence of periodic solutions bifurcating from Hopf bifurcations are given. Numerical simulations are carried out to illustrate the the theoretical analysis and the obtained results.. Introduction Recently there has been much interest in mathematical model describing the interaction between tumor cells and effector cells of the immune system see for example Bi et al [] d Onofrio et al [3 4 5] and Yafia et al [5]. An ideal model can provide insights into the dynamics of interactions of the immune response with the tumor and may play a significant role in understanding of the corresponding cancer and developing effective drug therapy strategies against it. However it is almost impossible to develop realistic models to describe such complex processes. In fact mathematical models for the dynamics of the interaction of the immune components with tumor cells are very idealized. Thus it is feasible to propose simple models which are capable to display some of the essential immunological phenomena. Recently delayed models of tumor and immune response interactions have been studied extensively we refer to Bi et al [ ] Galach [8] Mayer [3] Yafia [6 7 8] and the references cited therein. It would be interesting to consider the nonlinear dynamics of the delayed model. In 994 Kuznetsov et al [] took into account the penetration of TCs by ECs and proposed a model describing the response of effector cells (ECs) to the growth of tumor cells (TCs). They assumed that interactions between ECs and TCs in vitro can be described by the kinetic scheme shown in Figure where E T C T E are the local concentrations of ECs TCs EC-TC complexes inactivated ECs and lethally hit TCs respectively. From the above kinetic scheme and the experimental observations Kuznetsov et al [] claimed that the model can be reduced to two equations which describe Mathematics Subject Classification. 34K8 34K6 9C37. Key words and phrases. Tumor-immune system; Hopf bifurcation; delay. c 3 Texas State University - San Marcos. Submitted October 9 3. Published January 4 4.

2 P. BI H. XIAO EJDE-4/7 the behavior of ECs and TCs only. Taking time scale as that in [8] and [] then Kuznetsov Makalkin and Taylor s model can be written as dx = σ + ζxy δx dy = αy( βy) xy (.) where x denotes the dimensionless density of ECs y stands for the dimensionless density of the population of TCs. All coefficients are positive where ζ > shows the stimulation coefficient of the immune system exceeds the neutralization coefficient of ECs in the process of the formation of EC-TC complexes which is the most biologically meaningful. Figure. Kinetic scheme describing interactions between ECs and TCs Yafia [5] considered the model (.) and studied the linearizing stability of the equilibrium and the existence of the Hopf bifurcation. In [8 6 7] the authors obtained the same results as [5] for the model (.) with delay τ as follows dx = σ + ζx(t τ)y(t τ) δx dy = αy( βy) xy (.) The rest of this paper is organized as follows. In section we study the tumor model with delay (.) and show the properties of the Hopf bifurcated periodic solutions of this system including the direction of Hopf bifurcation the period and stability of bifurcated periodic solutions. The numerical analysis and simulations are also given to illustrate the main results. Section 3 is devoted to the existence of the global Hopf bifurcation. A brief discussion is also given in this section.. Direction and stability of Hopf bifurcation If αδ > σ equations (.) and (.) have two possible nonnegative steady states P ( σ δ ) and P ( α(βδ ζ)+ ζ α(βδ+ζ) αβζ ) where = α (βδ ζ) + 4αβζσ >. Galach[8] and Yafia [6] shows the stability of the equilibria and existence of the Hopf bifurcation for system (.) and (.) the main results are summarized as follows. Lemma. ([8]). If the point P exists and has nonnegative coordinates then it is stable. And there is no nonnegative periodic solution to equation (.).

3 EJDE-4/7 HOPF BIFURCATION 3 Lemma. ([6]). Let αδ > σ. () The equilibrium point P is asymptotically stable for all τ >. () If αβ > ζ then there exists τ l > such that P is asymptotically stable for τ < τ l and unstable for τ > τ l ; (3) If αβ > ζ then there exists ε > such that for each < ɛ < ε system (.) has a family of periodic solutions p l (ε) with period T l = T l (ε) for the parameter values τ = τ(ε) satisfy p l () = P T l () = π τ() = τ l. Where { ω τ l = arccos q(ω r) psω s ω +p if s(r ω ) pq > ω arccos q(ω r) psω s ω +p + π if s(r ω ) pq < ω = (s p + r) + (s p + r) 4(r q ). Let τ j = τ l + jπ j =.... Let P (x y ) be the positive equilibrium where x = α(βδ ζ)+ ζ y = α(βδ+ζ) αβζ. From lemma. we know that if < ζ β < α αδ > σ then (.) undergos Hopf bifurcation at the equilibrium P (x y ) the corresponding purely imaginary roots are λ = ±iτ j ω and the critical values are τ = τ j j =.... As pointed out by Hassard et al [] it is interesting to determine the direction stability and period of these periodic solutions bifurcating from the steady state. 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 Magalhaés [6 7]. Throughout this section we assume that system (.) undergoes Hopf bifurcations at the equilibrium P as the critical parameter τ = τ j j = 3... and the corresponding purely imaginary roots are ±iω. Set z (t) = x(t) x z (t) = y(t) y. Then system (.) can be written as z (t) = α z (t) + α z (t τ) + α 3 z (t τ) + ζz (t τ)z (t τ) z (t) = β z (t) + β z (t) αβz (t) z (t)z (t) (.) where α = δ < α = ζy > α 3 = ζx > β = y < β = α αβy x = αβy <. Normalizing the delay τ in system (.) by the time-scaling t t τ then (.) is transformed into z (t) = τ(α z (t) + α z (t ) + α 3 z (t ) + ζz (t )z (t )) z (t) = τ(β z (t) + β z (t) αβz (t) z (t)z (t)). (.) Let z(t) = (z (t) z (t)) T. We transformed (.) into the FDE ż(t) = N(τ)(z t ) + F (z t τ) (.3) where N(ϕ) : C([ ] R ) R F (ϕ) : C([ ] R ) R ϕ = col(ϕ ϕ ) C([ ] R ) satisfy α ϕ N(τ)(ϕ) = τ () + α ϕ ( ) + α 3 ϕ ( ) β ϕ ( ) + β ϕ () ( ) ζϕ F (ϕ τ) = τ ( )ϕ ( ) αβϕ. () ϕ ()ϕ ()

4 4 P. BI H. XIAO EJDE-4/7 Setting the new parameter γ = τ τ j then (.3) can be written as ż(t) = N(τ j )(z t ) + F (z t γ) (.4) where F (z t γ) = N(γ)(z t ) + F (z t τ j + γ). Let Λ = {iω iω}. Assuming A is the infinitesimal generator of ż(t) = N(τ k )(z t ) satisfying AΦ = ΦB with iω B = (.5) iω and A has a pair of conjugate purely imaginary roots ±iω. Denote P is the invariant space of A associated with Λ then dim P =. We can decompose C := C([ ] R ) as C = P Q by the formal adjoint theory for FDEs in [9]. Considering complex coordinates we still denote C([ ] C ) as C. Let Φ = (Φ Φ ) by the bases for P where Φ = e iωθ v Φ = Φ θ [ ] (.6) where v = (v v ) T is a vector in C that satisfies N(τ j )Φ = iωv. Choose a basis Ψ for the adjoint space P where Ψ = col(ψ Ψ ) Ψ = e iω θu T Ψ = Ψ u u = θ [ ]. (.7) Define (Ψ Φ) = ((Ψ i Φ j )) ij= (ψ ϕ) = ψ()ϕ() θ u ψ(s θ)dη(θ)ϕ(s)ds ϕ P ψ P. (.8) Then (Ψ Φ) can be transformed to the identify matrix I. Thus we can ensure ( ) ( ) v = u = u (.9) iω j (α +α e iω j )τ j τ jα 3e iω j iω j (α +α e iω j )τ j τ jβ with iω j (α + α e iωj )τ j = + v + (α + α 3 v )e iωj. u τ j β Define the enlarged phase space BC as BC := {ϕ : [ ] C ϕ is continuous on [ ) lim ϕ(θ) exists}. θ The projection π : BC P is defined as π(ϕ + X b) = Φ[(Ψ ϕ) + Ψ()b] for each ϕ C and b R thus we have the decomposition BC = P ker π. Let z t = Φx + y x C y ker(π) C := Q we can decompose (.4) as where We write the Taylor expansion ẋ = Bx + Ψ() F (Φx + y γ) dy dx = A Q y + (I π)x F (Φx + y γ) X (θ) = { I θ = ; θ <. Ψ() F (Φx + y γ) = f (x y γ) + 3! f 3 (x y γ) + h.o.t. (I π)x F (Φx + y γ) = f (x y γ) + 3! f 3 (x y γ) + h.o.t. (.) (.)

5 EJDE-4/7 HOPF BIFURCATION 5 where fk f k are homogeneous polynomials in x y γ of degree k k = 3 with coefficients in C and ker π respectively and h.o.t. stands for higher order terms. The normal form method implies a normal form on the center manifold of the origin for (.4) is ẋ = Bx + g (x γ) + 3! g 3(x γ) + h.o.t.. (.) where g (x γ) g 3(x γ) are homogeneous polynomials in x γ and h.o.t. are the higher order terms. We follow the notation in [6]; that is V m+p j (X) = { Σ (ql) =j C (ql) x q γ l : (q l) N m+p C (ql) X } with x = (x x... x m ) γ = (γ γ... γ p ); M j (p h) = (M j p M j h) M j p(x γ) = [B p( γ)](x) = D x p(x γ)bx Bp(x γ) M j h(x γ) = D x h(x γ)bx A Q h(x γ). Since V 3 j satisfies V 3 j (C ) = Im(M j ) ker(m j ) and (.3) ker(m j ) = span { x q γ l e k : (q λ) = λ j λ j Λ j = q N l N (q l) = j } e e is the base of C. Then we can obtain { ker(m x ) = span γ { ( ker(m3 x ) = span x x γ Noting (.) one has i.e. where Therefore ) ( x γ ( x x ) } f (x γ) = Ψ()[N(γ)(Φx) + F (Φx τ j )]; ) } x γ. f A x (x γ) = γ + A x γ + a x + a x x + a x Ā x γ + Āx γ + ā x + ā x x + ā x (.4) A = iω j u T v A = iω j u T v τ j a = τ j [u e iωj ζv v + u ( αβv v v )] a = τ j [ζu (v v + v v ) + u ( αβv v (v v + v v )] a = τ j [u e iωj ζ v v + u ( αβ v v v )]. g (x γ) = Proj ker(m ) f (x γ) = τ j A x γ. (.5) Āx γ Since the terms O( x γ ) are irrelevant to determine the generic Hopf bifurcation we assume { x J = span x } x x then g3(x γ) = Proj J f 3 (x ) + o( x γ )

6 6 P. BI H. XIAO EJDE-4/7 where f 3 (x ) = 3 [(D xf )u (D x u )g ] (x) + 3 [(D yf )u ] (x). Hence we will compute g3(x γ) as follows. Firstly noting f a x (x ) = + a x x + a x ā x + ā x x + ā x u (x ) = ( a x a x x 3 a ) x i 3āx + ā x x ā x one has Proj J [(D x f )u ] (x) = 4 ( ( a a + 3 a ) + a )x x i ( 3 a a + a a )x x A3 x = 4 x Ā 3 x x. (.6) Secondly From (.5) we know g (x ) = then Proj J [(D x u )g ] (x) =. Lastly we will compute Proj J [(D y f )u ] (x) as follows. Let h = u = h x + h x + h γ + h x x + h x γ + h x γ. Noting that g = one has M h(x γ) = f = (I π)x F (Φx γ) = (I π)x [N(γ)(Φx) + F (Φx τ j )]. On the other hand we know M h(x γ) = D x h(x γ)bx A Q h(x γ) = D x h(x γ)bx [ḣ(x γ) + X (L(τ j )(h(x γ)) ḣ(x γ)())]. (.7) If γ = then Let ḣ(x) D x h(x)bx = ΦΨ()F (Φx τ j ) ḣ(x)() L(τ j )(h(x)) = F (Φx τ j ). W (θ) = Φx + y = Φ x + Φ x + y(θ) = e iθ vx + e iθ vx + y(θ) W (θ) = Φx = Φ x + Φ x = e iθ vx + e iθ vx. (.8) From we obtain ( ) u T ζw ( )W ( ) f (x y ) = τ j αβw () W ( )W ( ) ( ) ū T ζw ( )W ( ) αβw () W ( )W ( ) [(D y f )h] (x) ( τ j u T ζ W ( )h ( ) + ζ W ) ( )h ( ) = W ( )h ( ) W ( )h ( ) αβ W ()h () ( τ j ū T ζ W ( )h ( ) + ζ W ) ( )h ( ) W ( )h ( ) W ( )h ( ) αβ W ()h ()

7 EJDE-4/7 HOPF BIFURCATION 7 and where Proj J [(D y f )u A4 x ] (x) = x Ā 4 x x (.9) A 4 = τ j [u ζ(e iωj v h ( ) + e iωj v h ( ) + e iωj v h ( ) ] [ + e iωj v h ( )) + u τ j e iωj v h ( ) e iωj v h ( ) ] e iω k v h ( ) e iωj v h [ ( ) u τ j αβ(v h () + v h ()) ]. To compute A 4 we should get h (θ) h (θ) firstly. From (.8) it follows a ḣ = (Φ Φ ) ā (.) a ḣ () L(τ j )(h ) = τ j b and ( a ) ḣ iω j h = (Φ Φ ) ā a ḣ () L(τ j )(h ) = τ j b (.) where a = [ζ(v v + v v )] b = [ αβv v (v v + v v )] a = [ζv v e iω k ] b = [ αβv e iωj v v ]. Solving the above equations (.) and (.) we obtain where h = [ a iω k Φ ā iω j Φ ] + C h = [ a Φ + ā Φ ] + C e iωkθ iω k 3iω j C a α 3 C = C C b (β + β 3 ) = (α + α ) α 3 β (β + β 3 ) (α + α ) a C β b = (α C + α ) α 3 C = C β (β + β 3 ) τ ja τ j α 3 e iωj C τ j b iω k + τ j β + τ j β 3 e iωj = iω k τ j α τ j α e iωj τ j α 3 e iω k τ j β e iωj iω k + τ j β + τ j β 3 e iωj iω k τ j α τ j α e iω k τ j a C τ j β e iωj τ j b = iω k τ j α τ j α e iωj τ j α 3 e iωj. τ j β e iωj iω j + τ j β + τ j β 3 e iωj

8 8 P. BI H. XIAO EJDE-4/7 Hence g3(x (6A3 + 3A ) = 4 )x x (6Ā3 + 3Ā4)x x. Thus the normal form of the system (.) has the form A x ẋ = Bx + γ + (6A3 + 3A 4 )x x Ā x γ 3! (6Ā3 + 3Ā4)x x + o( x 4 + x γ ). (.) Let x = ξ iξ x = ξ + iξ ξ = ρ cos ω ξ = ρ sin ω. Then system (.) can be written as ρ = r γρ + r ρ 3 + O(γ ρ + (ρ γ) 4 ) ω = ω j Im(A )γ Im(A 3 + A 4)ρ + o( (ρ γ) ) (.3) where r = Re A r = Re(A 3 + A 4). Summarizing we have the following theorem. Theorem.3. The flow on the center manifold of the equilibrium P at γ = is given by (.3). And also we can draw the following conclusion. () The Hopf bifurcation is supercritical if r r < and subcritical if r r > ; () The nontrivial periodic solution is stable if r < and unstable if r > ; (3) The period of the nontrivial solution is P (γ) = π ( Im(A )γ rγ r Im(A 3 + A 4) ) + O(γ 3 ) ω j ω j with P () = π/ω j. To explain the result of Theorem.3 we provide the simulations of Hopf bifurcation at P. In this paper we cite the parameters in [] to illustrate Theorem.3. Assume σ =.8 ζ =.3 δ =.3743 α =.636 β =. then the system (.) has a Tumor-free equilibrium P (.355 ) which is asymptotically stable and a positive equilibrium P ( ) which is locally stable. We only simulate local properties of the positive equilibrium P ( ) here in the following Figure and Figure 3 and the corresponding critical value is τ = Existence of global Hopf bifurcation In section we discussed the direction and stability of the Hopf bifurcated periodic solutions of system (.) at the equilibrium P (x y ) as τ = τ j. However this bifurcated periodic solution exists in a neighborhood of the positive equilibrium so we want to know whether the non-constant periodic solution exist globally. In this section we will study the existence of global bifurcated periodic solution using a global Hopf bifurcation result due to Wu[4]. Throughout this section we follow the notations in [4] and rewrite system (.) as the functional differential equation ż = F (z t τ) (3.)

9 EJDE-4/7 HOPF BIFURCATION y(t) 5 y(t) x(t) x(t) t Figure. (left) Stable equilibrium ( ) when τ = < τ ; (right) oscillation of the solution to (.) with respect to t when τ = < τ y(t) x(t) t Figure 3. (left) Bifurcated periodic solution at the positive equilibrium when τ =. > τ ; (right) oscillation solution x(t) and y(t) of (.) with respect to t when τ =. > τ where z = (x y) T z t (θ) = z(t+θ) C([ τ ] R +). Before stating the main results of this section we will give a known lemma. Let X = C([ τ ] R +) Σ = Cl{(z τ p) X R + R + : zz is a nonconstant periodic solution of (3.)} N = {( z τ p) : F ( z τ p) = }. Lemma 3. ([4]). Only one of the following two results holds () l (Pτ jπ/ ) is unbounded; () l (Pτ jπ/ ) is bounded and ( zτp) l (P γ( z τ p) =. τ j π/ ) N (3.) Assume l (Pτ jπ/ ) is the connected component of (P τ j π/ ) in Σ from lemma. we know that l (Pτ jπ/ ) is nonempty. Next we give two lemmas needed for the proof of the main result.

10 P. BI H. XIAO EJDE-4/7 Lemma 3.. If < ζ/β < min{δ α} αδ > σ then all periodic solutions of (.) are uniformly bounded. Proof. From the second equation of system (.) we have y(t) = y() exp t (α( βy(s)) x(s))ds noting the definition of y(t) and P is a positive equilibrium one knows x() and y() ; that is y(t). It is easy to see dy = αy( βy) xy αy( βy) it follows y β that is y β. We will prove x(t) as follows. Since (x(t) y(t)) is a periodic solution of (.) then x(t) must have the minimum x(η ) η > that is hence σ + ζx(η τ)y(η τ) δx(η ) = δx(η ) ζx(η τ)y(η τ). If x(η τ) then x(t). If x(η τ) < then x(η ) x(η τ) < hence dx = σ + ζx(t τ)y(t τ) δx > ζ β x(η ) δx(t). By the constant variation formula and the comparison theorem it implies that x(t) > ζx(η ) ( exp δt ) + x() exp δt t >. (3.3) βδ noting x() > and < ζ β < min{δ α} it is easy to see that (3.3) does not hold as t = η ; this is a contradiction thus x(t) >. On the other hand we have dx = σ + ζx(t τ)y(t τ) δx σ + ζ x(t τ) δx β Then we prove the bound of the solution x(t) in three cases: () If x(t τ) x(t) for t > since (x(t) y(t)) is a periodic solution of (.) then x(t) must have the maximum x(η) η ( τ) that is σ +ζx(η τ)y(η τ) δx(η) = hence x(η) σ δ + ζ βσ x(η τ) σ δ + ζ ϕ η τ ( τ ). βσ () If x(t τ) < x(t) for t > then dx σ + ζ β x δx from < ζ/β < δ one has x(t) σ δ ζ ; β (3) If there exists t > such that x(t τ) x(t) and t > such that x(t τ) < x(t ). Integrating () and () we know x(t) max{ σ σ δ ζ δ + ζ βσ ϕ }. The proof β is complete.

11 EJDE-4/7 HOPF BIFURCATION Lemma 3.3. If αδ > σ then system (.) has no-nonconstant periodic solution with period τ. Proof. Suppose (.) has no-constant periodic solution with period τ then the corresponding ordinary differential equations dx = σ + ζxy δx (3.4) dy = αy( βy) xy have a non-constant periodic solution. System (3.4) has a positive equilibrium P (x y ) and a boundary equilibrium P ( σ δ ). From lemma. we know that P is stable and (3.4) has no non-constant periodic solution. This is a contradiction. The proof is complete. Theorem 3.4. If < ζ/β < min{δ α} αδ > σ then (.) has at least j periodic solutions for τ τ j j. Proof. The characteristic matrix of (3.) at the equilibrium z = ( z z ) T R is i.e. ( z τ p)(λ) = λid DF ( z τ p)(e λ Id); λ ζ z e ( z τ p)(λ) = λτ δ ζ z e λτ (3.5) z λ α + αβ z + z where Id is the identity matrix DF ( z τ p) is the Fréchet derivative of F with respect to z t evaluated at ( z τ p). On the other hand from the definition of [4] we know that ( z τ p) is called a center of (3.) if ( z τ p) N and det( ( z τ p)( π p i)) = and the center is said to be isolated if it is the only center in the neighborhood of ( z τ p). From (3.5) we know that det( (P τ p)(λ)) = (λ δ)(λ α + σ δ ) = obviously this equation has no purely imaginary roots thus (3.) has no centers with the form (P τ p). From the proof of [ 6 theorem 4.] we know that there exist ɛ > ɛ > and a smooth curve λ(τ) : [τ j ɛ τ j + ɛ ] C such that det( (λ(τ))) = as λ(τ) i < ɛ for all τ [τ j ɛ τ j + ɛ ] furthermore λ(τ j ) = i Re λ(τ) dτ τ=τj > ; thus for all τ > (P τ j π )(j N) is the only center of (3.). Let { Ω ɛπ/ωl = (η p) : < η < ɛ p π } < ɛ. If τ τ j ɛ and (η p) Ω ɛπ/ωl then Define det( (P τ p)(η + π p i)) = η = τ = τ j p = π. H ± (P τ j π/ )(η p) = det( (P τ ± ɛ p)(η + π p i)).

12 P. BI H. XIAO EJDE-4/7 Then the crossing number of isolated center is thus γ(p τ j π/ ) = deg B (H (P τ j π ) Ω ɛπ/ωl ) deg B (H + (P τ j π/ ) Ω ɛπ/ωl ) = ; ( zτp) l (P τ j π/ ) N γ( z τ p) <. (3.6) Noting lemma 3. we know that the connected component l (Pτ jπ/ ) passing through (P τ j π/ ) is unbounded in Σ. From lemma. we know that the projection of l (Pτ jπ/ ) onto z-space is bounded. Noting [6] one has π < τ j. (3.7) From the results of lemma 3.3 we know that (3.) has no non-trivial periodic solutions for τ =. Hence the projection of l (Pτ jπ/ ) onto τ-space is away from zero. If not the projection of l (Pτ jπ/ ) onto τ-space is included in the interval ( τ ) τ > τ j with the help of (3.7) we know ( z τ p) l (Pτ jπ/ ) as p < τ. This implies the projection of l (Pτ jπ/ ) onto p-space is bounded. Thus if we want the connected component l (Pτ jπ/ ) is unbounded the projection of l (Pτ jπ/ ) onto τ-space must be unbounded; i.e. the projection of l (Pτ jπ/ ) onto τ-space must conclude [τ j ) this implies that the system (.) has at least j periodic solutions for all τ τ j (j ). Discussion. We studied the nonlinear dynamics of a Kuznetsov Makalkin and Taylor s model with delay which is analyzed in [6]. In [6] Yafia only give the existence of the Hopf bifurcation. In this paper we give the general formula of the direction the estimation formula of period and stability of bifurcated periodic solution. Especially using a global Hopf bifurcation due to Wu[4] the global existence of periodic solutions bifurcating from Hopf bifurcations is given. we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. Numerical simulations are carried out to illustrate the the theoretical analysis and results. Our results on the existence and stability of the Hopf bifurcated periodic solutions of P describe the equilibrium process. When a global stable periodic orbit exists it can be understood that the tumor and the immune system can coexist for all the time 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 d Onofrio [3] Kuznetsov et al [] Bi et al []. In particular Bi et al. [] have shown that various bifurcations including Hopf bifurcation Bautin bifurcation and Hopf-Hopf bifurcation can occur in such models. Finally we point out that we have studied the dynamical behaviors of P. In fact the dynamical behaviors of P is more rich such as the existence of the Bogdanov- Takens bifurcation and steady-state bifurcation which is studied in []. The two equilibria may coexist correspondingly the system can exhibit more degenerate bifurcations including Hopf-Hopf and resonant higher codimension bifurcations et al. It would be interesting to consider these dynamics of the delayed model.

13 EJDE-4/7 HOPF BIFURCATION 3 Acknowledgments. This research was supported by National Natural Science Foundation of China (No. 7) and Shanghai Leading Academic Discipline Project (No. B47) Project of Key Academic Discipline of East China Normal University. References [] P. Bi S. Ruan; Bifurcations in Delay Differential Equations and Applications to Tumor and Immune System Interaction Models SIAM J. Appl. Dynamical Systems Vol. (4)(3) [] P. Bi H. Xiao Bifurcations of a Tumor-Immune Competition Systems with Delay submitted. [3] A. d Onofrio; A general framework for modeling tumour-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences Phys. D 8 (5) [4] A. d Onofrio; Tumour-immune system interaction: Modeling the tumour-stimulated proliferation of effectors and immunotherapy Math. Models Methods Appl. Sci. 6 (6) [5] A. d Onofrio; Metamodeling tumour-immune system interaction tumour evasion and immunotherapy Math. Comput. Modelling 47 (8) [6] T. Faria L. Magalhães; Normal forms for retarded functional differential equation and applications to Bogdanov-Takens singularity J. Diff. Equ. (995) 4. [7] T. Faria L. Magalhães Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation J. Diff. Equ. (995) 8. [8] M. Galach; Dynamics of the tumor-immune system competition: the effect of time delay Int. J. Appl. Math. Comput. Sci [9] J. K. Hale; Theory of functional differential equations Springer-Verlag[M] New York 977. [] B. Hassard N. Kazarinoff Y. Wan; Theory and Application of Hopf Bifurcation. Cambridge: Cambridge University Press 98. [] V. Kuznetsov I. Makalkin M. Taylor A. Perelson; Nonlinear dynamics of immunogenic tumors: parameter estimationestimation and global bifurcation analysis Bull. Math. Biol. 56 (994) [] D. Liu S. Ruan D. Zhu; Stable periodic oscillations in a two-stage cancer model of tumorimmune interaction Mathematical Biosciences and Engineering (9)5 68. [3] H. Mayer K. Zaenker U. an der Heiden; A basic mathematical model of the immune response Chaos5(995) [4] J. Wu.; Symmetric functional differential equations and neural networks with memory. Trans. Amer. Math. Soc. 35(998) [5] R. Yafia; Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response Nonl. Anal.: Real World Appl. 8(7) [6] R. Yafia; Hopf bifurcation in differential equations with delay for tumor-immune system competition model SIAM J. Appl. Math 67(7) [7] R. Yafia; Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response Discrete Dynamics in Nature and Soc. (6) 9. [8] R. Yafia; Periodic solutions for small and large delays in a tumor-immune system model Electron. J. of Diff. Equ. Conference 4 (6) Ping Bi Department of Mathematics Shanghai Key Laboratory of PMMP East China Normal University 5 Dongchuan RD Shanghai 4 China address: pbi@math.ecnu.edu.cn Heying Xiao Department of Mathematics Shanghai Key Laboratory of PMMP East China Normal University 5 Dongchuan RD Shanghai 4 China address: hawcajok7@63.com