BIFURCATION ANALYSIS IN MODELS OF TUMOR AND IMMUNE SYSTEM INTERACTIONS. Dan Liu. Shigui Ruan. Deming Zhu. (Communicated by Xiaoqiang Zhao)

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1 DISCRETE AND CONTINUOUS doi: /ddsb DYNAMICAL SYSTEMS SERIES B Volume 1, Number 1, Jul 009 pp BIFURCATION ANALYSIS IN MODELS OF TUMOR AND IMMUNE SYSTEM INTERACTIONS Dan Liu Department of Mathematis, East China Normal Universit Shanghai 0006, China Shigui Ruan Department of Mathematis, Universit of Miami Coral Gables, FL 3314, USA Deming Zhu Department of Mathematis, East China Normal Universit Shanghai 0006, China (Communiated b Xiaoqiang Zhao) Abstrat. The purpose of this paper is to present qualitative and bifuration analsis near the degenerate equilibrium in models of interations between lmphote ells and solid tumor and to understand the development of tumor growth. Theoretial analsis shows that these aner models an ehibit Bogdanov-Takens bifuration under suffiientl small perturbation of the sstem parameters whether it is vasularized or not. Periodi osillation behavior and oeistene of the immune sstem and the tumor in the host are found to be influened signifiantl b the hoie of bifuration parameters. It is also onfirmed that bifurations of odimension higher than annot our at this equilibrium in both ases. The analti bifuration diagrams and numerial simulations are given. Some anomalous properties are disovered from omparing the vasularized ase with the avasular ase. 1. Introdution. Caner still remains one of the most dangerous killers of humankind in the 1th entur. Millions of people die from this disease ever ear throughout the world ([9]). The main ause of a remarkabl high inidene of neoplasia liniall derives from immunologial defiien. Investigation ([19]) showed that about ten perent of patients who have spontaneous immunodefiien diseases ma develop aner. Clini and laborator soures also indiate that the immune sstem plas an important role in ontrolling and eliminating tumor ells, and therefore dereasing the observed inidene of aner. This response of immune sstem to the preanerous and anerous is the so-alled immunosurveillane ([17]). More detailed researh about the immune surveillane an be found in [5],[13],[14], and [4]. The interations between the immune sstem and tumor ells are important. Numerous effort and researh have been made to eplore the effets of immune sstem to eliminate and destro tumor ells b stimulating the host s own immune 000 Mathematis Subjet Classifiation. Primar: 34C3, 34C60; Seondar: 37G10. Ke words and phrases. Bogdanov-Takens bifuration, saddle-node, osillation, vasularization, tumor, lmphote. 151

2 15 DAN LIU, SHIGUI RUAN AND DEMING ZHU response to fight aner. But urrent eperimental and lini data reveal that improvement of the immune sstem b immunotherap brings on not suppression but more stimulation of tumor ells growth (see [1], [3]), so the immunotherap is still a restrained treatment modalit in the lini. Nevertheless, the promising future of effetive tumor immunotherap has been lightened b the reent breakthroughs in immunolog suh as the identifiation of immunogeni tumor-assoiated antigens ([5]). In order to qualitativel estimate the funtion of the immune surveillane, a variet of mathematial models of the interation between the immune sstem and solid tumor have been introdued. In 1977, based on some reasonable hpotheses, DeLisi and Resigno [11] proposed the following nonvasularized model dl = λ 1 L + α CL(1 1 L L ) dc = λ C f α CL (1) to desribe immune response to a spherial tumor, where L and C denote respetivel the number of free lmphotes and the total number of tumor ells. C and C f are respetivel the total number of free tumor ells and the number of free ells on a tumor surfae. λ 1, λ and α 1, α are positive onstants. Free ells mean the ells that are not bound b lmphotes. For more detailed eplanation of (1) one an refer to [11]. Model (1) integrates the tumor geometrial harater and renders the interations between the immune sstem and a solid tumor during tumor attak, whih is along the lines of but different from the earlier lassial deterministi model in [6], beause in sstem (1) onl the ells on the surfae of a growing tumor are suseptible to immune attak and destrution. The general diretions of phase portraits for sstem (1) have been studied in [11] eept near the degenerate equilibrium. As an appliation, Arrowsmith and Plae [4] simpl analzed the bifuration at the degenerate equilibrium of (1) in the ase of a usp point b their bifuration theor. However, the did not present the epliit homolini bifuration urve and the orresponding numerial simulations. In the subsequent reviews, Swan etended the mathematial analsis of the model in [6] and studied the field of mathematial modeling in aner researh in [7]. Albert [] set up a mathematial model of the immune sstem with the interation of tumor ells in the presene of a tumor growth modulator b a set of differential equations. In [15], Kuznetsov et al. formulated a model of the totoi T lmphote response to the growth of an immunogeni tumor and studied loal and global bifurations for some realisti values of the parameters. In 1996, Adam [1] proposed a mathematial model desribing ell populations of reative lmphotes and solid tumors b inorporating the effets of vasularization within a tumor or multiell spheroid to model (1), i.e., the vasularized model dl = λ 1 L + α CL(1 1 L L ) ˆβ 1 C /3 dc = λ C f α CL + β C, where ˆβ 1 and β are nonnegative onstants representing the effiien of penetration of the tumor surfae area and volume, respetivel. The appearane of frational eponent in the first equation of the vasularized model is aused b the fat that we take the tumor mass as a spherial form. From (), Adam obtained the possibilit of Hopf bifuration near one of the nondegenerate equilibria and the eistene of a limit le b treating an parameter in the model as a bifuration parameter. ()

3 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 153 There are ver little disussion about the properties of the possible degenerate equilibrium. Following the investigation of the models from [11] and [1], Lin [18] onsidered the eistene of solutions and stabilit of stead states of the immune sstem on both avasular and vasularized ases, determined the regions of unontrolled tumor growth, tumor etintion in finite time and irreversible lmphote deline, and proved the invariane of the sstems in the plane region (0, L ) (0, + ) in both ases. But the trajetories and the dnamial properties near the degenerate equilibrium have not et been onsidered ompletel. In this paper we ontinue to follow the hpotheses in [11] and [1] and fous our attention on the stud of the qualitative properties and bifurations near the degenerate equilibrium of (1) and (). In both models, we stud possible behavior of the trajetories near the degenerate equilibrium b using methods different from [4]. One interesting result of our analsis is that it an ehibit Bogdanov-Takens bifuration of odimension at the degenerate equilibrium for the model of tumor and lmphote interation just like in some predator-pre models (see [], [9], [30]). Thus there ma be a homolini orbit or a limit le bifurated from the degenerate equilibrium when we hoose the partiular values of bifuration parameters in sstem (). The appearane of limit les implies the ourrene of the periodi osillation behavior of these aner models. In other words, the immune sstem and the solid tumor an oeist under some appropriate irumstanes. Also we find that bifurations of odimension 3 or higher annot happen in (1) whether in nonvasularized or vasularized ase, whih rules out man more ompliated ases on the development of the solid tumor and the lmphotes. Our theoretial results maintain the qualitative analsis of DeLisi and Resigno [11] and etend the results of Arrowsmith and Plae [4] for the avasular ase, and the results of Lin [18] for the ases prior to vasularization as well as after vasularization and of Adam [1] for the vasularized ase. Numerial simulations for the nonvasularized model are presented to support the analti onlusions on bifurations.. Bifurations of the nonvasularized model. If the relationship between free and bounded lmphotes is assumed to be equilibrium ontrolled, K is the equilibrium for lmphote and tumor ell interation, and the tumor is spherial, then DeLisi and Resigno [11] derived that C f = C gklc /3 /(1 + KL), C = gc /3 /(1 + KL), where g > 0 is a onstant. Therefore, the following sstem of lmphote and tumor interation is obtained: dl dc = λ 1 L + α 1 ( gc/3 1+KL )L(1 L L ) = λ (C gc/3 KL 1+KL ) α ( gc/3 1+KL )L, (3) Our main goal in this setion is to investigate possible bifurations near the degenerate positive equilibrium of (3). From the biologial point of view, the domain restritions are 0 L L and C 0. Introduing the new variables and parameters = KL, = KC, = KL, α 1 = α 1gK /3, α = gk 1/3 (λ + α K 1 ), we nondimensionalize sstem (3) in a simple epression: d = λ 1 + α1/3 1+ (1 ) = f(, ) d = λ α/3 1+ = g(, ). (4)

4 154 DAN LIU, SHIGUI RUAN AND DEMING ZHU Stead states appear when f(, ) = 0 = g(, ). (5) It is evident to see that (0, 0) is suh a ritial point. Adam [1] has shown that (0, 0) is an unstable equilibrium of (4) b the trajetor analsis. This equilibrium is of little biologial interest beause it means that both lmphote and tumor populations are vanished. So in the following disussion we are not onerned with this trivial equilibrium an more. When and are nonzero, the algebrai equations (5) an be simplified into the following form whih is independent on the variable : λ 1 λ α 1 α = (1 / ) (1 + ) 3 = ψ(, ). (6) Set k 1 = λ1λ α 1α for simpliit, then the absissa of positive equilibrium of sstem (4) is equivalent to the positive solution of the equation ψ(, ) = k 1. In Figure 1, the urve of ψ(, ) of is shown, where m orresponds to the maimum point of ψ(, ) in [0, ]. From that we an find the diret results as below. (a) If k 1 > ψ(, m ), sstem (4) has no interior equilibrium. (b) If k 1 = ψ(, m ), sstem (4) has a unique interior equilibrium S( m, m ). () If 0 < k 1 < ψ(, m ), sstem (4) has two different interior equilibria S 1 ( 1, 1 ) and S (, ) satisfing 1 < m <. ψ(, ) k 1 ψ(, ) ψ( 1, ) 0 m Figure 1. The urve ψ(, ) at different values of with 1 <. In ase (a), (0, 0) is the onl equilibrium whih is unstable and Adam [1] onluded that the trajetor of sstem (4) approahes unontrollable tumor growth for an initial nonzero value of (, ). For the ase (b), there is another equilibrium besides the origin. DeLisi and Resigno [11] gave a global analsis of trajetories near this positive equilibrium. But the did not disuss the propert of sstem (4) at the point in detail. The two different equilibria in ase () were studied suessivel b Adam [1] in 1996 and Lin [18] in 004. The proved respetivel that the equilibrium S (, ) is an unstable saddle point while S 1 ( 1, 1 ) ma be a enter, fous or node and either stable or unstable. We are onerned with properties of S( m, m ) in ase (b) in the following. Sine these fied points are far awa from the origin, we an make the equivalent transformation u = 1/3 whih does not hange their qualitative properties. Let us redenote respetivel u, λ /3, and α /3

5 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 155 as, λ, and α, then sstem (4) beomes d = λ 1 + α1(1 ) 1+ = F(, ) d = λ α 1+ = G(, ). (7) B simple alulations, we know that m = 3+ and m = αm λ (1+ m). Substituting m into equation (6), the following lemma an be proved. Lemma.1. The parameter set Σ SN = {(λ 1, λ, α 1, α, ) λ 1λ α 1 α is the saddle-node bifuration surfae of sstem (7). = 4 7(1 + ),, λ i, α i > 0, i = 1, } (8) When the parameters pass through the surfae Σ SN from one side to the other side, the number of the interior equilibria hanges from zero to two. And there is onl one suh point on that surfae. Now we take a mathematial analsis for sstem (7) near the point S( m, m ). To stud the propert at S( m, m ), it is neessar to make some tehnial transformations and use the anonial form of sstem (7) about this equilibrium. For the sake of simpliit, let 1 = m, 1 = m so we an translate the interior equilibrium into the origin and epand sstem (7) in a power series about the origin, then we have d 1 = a 1 + b 1 + p p p 1 + P 1( 1, 1 ) d 1 = 1 + d 1 + q q q 1 + Q 1( 1, 1 ), where P 1 ( 1, 1 ) and Q 1 ( 1, 1 ) are C funtions of ( 1, 1 ) with at least the third order and the oeffiients of first and seond order term are the derivatives of F and G suh that J(, ) (m, m) = P(, ) (m, m) = 1 Q(, ) (m, m) = 1 ( F F G G ( F F ( G G ) ) ( m, m) F F )( m, m) G G ( m, m) ( ) a b =, d ( ) p11 p = 1, p 1 p = ( ) q11 q 1. q 1 q As a diret observation, we obtain that q 1 = q = 0 independent of the value of ( m, m ). Moreover, the determinant DetJ and the trae TrJ at S( m, m ) an be determined. It is eas to find that DetJ( m, m ) = 0 and TrJ( m, m ) = (3+) 3(1+) λ 1 + λ after substitution several times, whih means that S( m, m ) is a degenerate equilibrium. We divide into two ases in order to investigate the propert of this equilibrium..1. Case A: λ (3+) 3(1+ λ ) 1. This ondition implies that the Jaobian matri J of the linear part of sstem (7) at the nonhperboli equilibrium ( m, m ) is similar to the Jordan normal form ( a + d (9) (10) ). B a linear oordinate and time hange

6 156 DAN LIU, SHIGUI RUAN AND DEMING ZHU X = MX 1, τ = (a + d)t, sstem (9) is hanged into d dτ = (dw 1 bw 4 ) + (dw bw 4 ) + (dw 3 bw 4 ) + P (, ) = p(, ) d dτ = + (aw 1 + bw 4 ) + (aw + bw 4 ) + (aw 3 + bw 4 ) + Q (, ) = q(, ), (11) ( ) d b where X i = ( i, i ) T, i = 1,, M =, and w a b j for j varing from 1 to 4 are defined as w 1 = b p 11 abp 1+a p b (a+d), w 3 = (b p 11 abp 1+bdp 1 adp ) b (a+d), w 3 3 = b p 11+adp 1+d p b (a+d), w 3 4 = b q 11 b (a+d), and T denotes the transposition of a matri. 3 We now determine the phase portraits of sstem (9) near (0,0). Appling the theor in [3], we first onsider the equation q(, ) = 0. B the impliit funtion theorem, this equation has a solution = ϕ( ) in a small neighborhood of the origin, where ϕ( ) = (aw 1 + bw 4 ) + (aw 1 + bw 4 )(aw + bw 4 ) 3 + O( 4 ) is an analti funtion suh that ϕ(0) = ϕ (0) = 0. Define a funtion ψ( ) b ψ( ) = p(, ). Here it needs to be mentioned that the funtion ψ( ) annot vanish identiall. That is beause ( m, m ) is an isolated equilibrium of sstem (7) and so is the equilibrium (0,0) for (9). If ψ( ) = 0, it would dedue from the definitions of ϕ and ψ that all points of the urve = ϕ( ) are stead states of sstem (11), whih ontradits with the isolation of the equilibrium (0,0). Therefore we ma epand the funtion ψ( ) as the form of the power series: ψ( ) =(dw 1 bw 4 ) (dw bw 4 )(aw 1 + bw 4 ) 3 + [(dw bw 4 )(aw 1 + bw 4 )(aw + bw 4 ) + (dw 3 bw 4 )(aw 1 + bw 4 ) ] 4 + O( 5 ), (1) 9(1+ where dw 1 bw 4 = )(3+ ) 3 λ 1λ ((3+ )λ 1 3(1+ )λ ) whih is atuall reasonable under the 3 ondition of Case A. From the results of Andronov et al. [3], we obtain the possible topologial struture of the equilibrium state (0,0) of sstem (9) in the following onlusion. Theorem.. If Case A is valid, then (0,0) is a saddle-node of sstem (9) whih onsists of two hperboli setors and one paraboli setor. The orresponding phase portraits in the neighborhood of the origin are analzed and drawn in Figure (a) and (b). We selet the parameters α 1 = , α = , g = 9., L = , K = 10 8 in the immune sstem (1). When λ 1 = 0.01, λ = 0.01, one gets that λ > (3+) 3(1+ λ ) 1, and the trajetories near the interior equilibrium ( m, m ) = (1.9976, ) b numerial simulations are shown in the left piture (a) of Figure 3, where ( m, m ) is unstable, so almost all trajetories will head to (, ) and the population of aner ells will be unontrollable as t inreases to infinite. However, when λ 1 = 0.01, λ = , we find λ < (3+) 3(1+ λ ) 1 and there are

7 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 157 trajetories originating from some regions tend to the same interior equilibrium ( m, m ) and the other regions bring unontrolled growth of aner as t goes to infinite, see Figure 3(b). In suh a ase, ( m, m ) is a semi-stable equilibrium. If the initial values are hosen suitabl, aner an oeist with the immune sstem beause trajetories originating from suh regions will lose to this equilibrium as t inreases (a) (b) Figure. The outline of trajetories for sstem (.5) near ( m, m ) where the origin denotes the equilibrium ( m, m ) in the plane of (, ). (a) orresponds to the ase λ > (3+) 3(1+ λ ) 1 and (b) orresponds to λ < (3+) 3(1+ λ ) (a) (b) Figure 3. The phase portraits near the degenerate equilibrium ( m, m ) when λ (3+) 3(1+ ) λ 1... Case B: λ = (3+) 3(1+ λ ) 1. In this ase, the Jaobian matri J of the linear part of sstem (7) at the equilibrium S( m, m ) is similar to the Jordan blok form ( ). Appling the bifuration theor in [10], [0] and [16] and taking the

8 158 DAN LIU, SHIGUI RUAN AND DEMING ZHU affine transformation = 1, = 1 +d 1, we an rewrite sstem (9) as follows: d = + d q 11 d q q 11 + P (, ) d = [ d (p 11 + dq 11 ) dp 1 + p ] + [ d (p 11 + dq 11 ) + p 1 ] + p11+dq11 + Q (, ), (13) where P (, ) and Q (, ) are power series in (, ) with powers at least 3. Performing the net C hange of variables of sstem (13) in a small neighborhood of the origin: 3 = p11 dq11 1 q 11 3 = + d q 11 p11+dq11, (14) we eliminate the term, then sstem (13) is C equivalent to d 3 = 3 + P 3 ( 3, 3 ) d 3 = [ d (p 11 + dq 11 ) dp 1 + p ] 3 + ( d p 11 + p 1 ) Q 3 ( 3, 3 ), (15) where P 3 ( 3, 3 ) and Q 3 ( 3, 3 ) are C funtions in ( 3, 3 ) at least of the third order. In order to use the results from [10] to make sure if the origin of sstem (15) is a usp point, we make the transformation 4 = 3, 4 = 3 + P 3 ( 3, 3 ), (16) whih brings sstem (15) to the anonial normal form d 4 = 4 = [ d d 4 (p 11 + dq 11 ) dp 1 + p ] 4 + ( d p 11 + p 1 ) Q 4 ( 4, 4 ), (17) where Q 4 ( 4, 4 ) is a C funtion in ( 4, 4 ) at least of the third order. Mathematia works out that d 1 = d (p 11 + dq 11 ) dp 1 + p = 9(1+)λ3 4α > 0, d = d p 11 + p 1 = 3(1+)(9+)λ (3+ )α < 0, whih means d 1 d 0 for an positive values of λ, α 1, α, and. Thus we have the following theorem b the qualitative theor of ordinar differential equations and the theor of differential manifolds. Theorem.3. For an (λ 1, λ, α 1, α, ) Σ SN, S( m, m ) is a usp-tpe equilibrium of odimension (i.e. a Bogdanov-Takens bifuration point) under the ondition of Case B. The above theorem implies that sstem (7) annot ehibit bifurations of odimension greater than at the degenerate equilibrium. We will prove that Bogdanov- Takens bifuration ours in sstem (7) under a small parameter perturbation b hoosing suitable bifuration parameters in the net setion. Under the hpothesis of Case B, we take λ 1 and λ as bifuration parameters to stud bifuration analsis of the versal unfolding for the odimension- usp point b the results in [10] and [16]. Denote the new parameter famil as (λ 1 µ 1, λ +µ, α 1, α, ) after perturbing the parameter famil (λ 1, λ, α 1, α, ), then the perturbed sstem is written as (18) d = ( λ 1 + µ 1 ) + α1(1 ) 1+ d = (λ + µ ) α 1+. (19)

9 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 159 In order to simplif (19) into normal form as (17), we first make two affine translations 1 = m, 1 = m and = 1, = 1 + d 1, where d = d + µ. For the sake of simplifiation, we redenote d as d in the following disussion. Choose the C hange of oordinates same as (14) and (16) in a small neighborhood of (0, 0), then sstem (19) is equivalent to d 4 = 4 d 4 = [1 + l 1 (µ)]( m µ 1 + d m µ ) l (µ) m µ + [l 1 (µ) m (µ) m µ ] 4 +[l 11 (µ) + l (µ)] 4 + m 1 (µ) 4 + m (µ) R 1 ( 4, 4, µ), (0) where µ = (µ 1, µ ), m i (µ) = d i + O( µ ), d i is epressed as (18), l ij (µ), m 1j (µ) = O( µ ) are C funtions of µ and have the following epressions l 11 (µ) = ( q 11 mµ 1 + p 11 mµ ), l 1 (µ) = 1 q 11 m µ, l 1 (µ) = d(µ 1 + µ ) + b + d + d q 11 m µ p 11 + dq 11 ( m µ 1 + d m µ ), l (µ) = µ 1 + µ p 11 + dq 11 m µ, m 11 (µ) = p 11 dq 11 l 11 d q 11l 1 1 q 11[ d(µ 1 + µ ) + b + d ], m 1 (µ) = 1 q 11[l 11 (µ 1 + µ )] + p 11 + dq 11 l 1, here p ij and q ij are defined as in (10), R 1 ( 4, 4, µ) = O( µ 3 ) + O( µ ( 4, 4 ) ) + O( µ ( 4, 4 ) )+O( ( 4, 4 ) 3 ) is the power series of ( 4, 4, µ) with at least degree 3 and the oeffiients depend on the perturbing parameters µ, i, j = 1,. For sstem (0), one an appl the Malgrange Preparation theorem to simplif the seond equation in a normal form (see [10], Chapter 3, pp.194). Here we break this method and make a diret linear transformation 5 = 4 + l1(µ) m(µ)mµ m 1(µ), 5 = 4 depending on the parameters. Note that for suffiientl small µ 1 and µ, m i (µ) = d i + O(µ) 0 for i = 1,. Thus resaling 5, 5, t b 6 = m m 1 5, 6 = 5, τ = m1 m t and putting the epressions of l ij, m ij, m, m into the new equations, we obtain that m 3 m 1 d 6 = 6 d 6 = 4(9 + ) 4 81(1 + )(3 + ) 4 λ {(3 + ) λ [(3 + )µ 1 3(1 + )µ ] + (1 + )(9 + ) µ 1 + ( )µ 1 µ + 4 (1 + )µ } + { (9 + )[ ( )µ 1 + 6( 3 + )µ ] 9(1 + )(3 + ) λ + O( µ )} R 3 ( 6, 6, µ) =ν 1 (µ) + ν (µ) R 3 ( 6, 6, µ), where R 3 has the same properties as R 1. Sine ( ) ν1 16(9 + ) 5 Det = 81(1 + )(3 + ) 4 λ > 0 ν 1 µ 1 µ ν ν µ 1 µ (µ 1=0, µ =0) (1)

10 160 DAN LIU, SHIGUI RUAN AND DEMING ZHU for an values of the parameters, λ > 0, whih implies that the loal parameter representation transformation ν 1 = ν 1 (µ), ν = ν (µ) is nonsingular. Therefore, based on the results in [7], [8], [1] and [8], the following onlusion is valid. Theorem.4. Sstem (1) is a universal unfolding of the usp point of odimension. There is a neighborhood Ω of (µ 1, µ ) = (0, 0) in R suh that sstem (19) undergoes Bogdanov-Takens bifuration inside Ω. I SN + II III H HL µ SN O µ 1 IV Figure 4. Bifuration diagram at ( m, m ) after perturbing (λ 1, λ ). The loal bifuration urves in this small neighborhood Ω of the origin onsist of Table 1. The phase portraits of nonvasularized model. 6 O O O I SN + O 3 O II 6 O H III O 3 O HL O IV SN O

11 BIFURCATIONS IN TUMOR-IMMUNITY MODELS (a) (b) () (d) Figure 5. The phase portraits for different (µ1, µ ) when λ = in the avasular ase. (3+ ) 3(1+ ) λ1 SN = {(µ1, µ ) : ν1 (µ1, µ ) = 0} orresponds to the saddle-node bifuration urve on the plane of (µ1, µ ). Along this urve sstem (1) has a unique equilibrium with a zero eigenvalue. Crossing SN from the top down implies the appearane of two equilibria, the right one is a saddle p and the left one is a stable fous. H = {(µ1, µ ) : ν (µ1, µ ) = ν1 (µ1, µ )} orresponds to the Hopf bifuration urve on the plane of (µ1, µ ). There will our a stable periodi orbit when (µ1, µ ) Ω goes through H from II to III and the left equilibrium turns into an unstable fous from a stable fous. 49 HL = {(µ1, µ ) : ν1 (µ1, µ ) = 5 ν (µ1, µ ) +O(ν (µ1, µ )5/ ), ν (µ1, µ ) > 0} orresponds to the homolini loop bifuration urve. When (µ1, µ ) HL, there is an inner stable homolini orbit of sstem (19). But the homolini orbit will be broken one (µ1, µ ) traverses HL from III to IV. The bifuration diagram of sstem (19) for (µ1, µ ) Ω is displaed in Figure 4, where the regions I-IV are shaped b the above three bifuration urves. For the perturbed sstem (19), the orresponding phase portraits belonging to eah bifuration region are listed in Table 1. In addition, we draw the trajetories on the phase plane (, ) b numerial simulations shown in Figure 5 when (µ1, µ ) takes partiular values in eah bifuration region of Ω, whih is onsistent with the analti results in Table 1. To simulate the stable singular orbits, we hoose the value of the original pa rameters as follows: α1 = , g = 9., L = , K = 10 8, α =

12 16 DAN LIU, SHIGUI RUAN AND DEMING ZHU , λ 1 = 0.01, λ = , whih followed b the values of the new parameters λ 1 = 0.01, λ = , α 1 = , α = , = 500 in sstem (7) and the unique interior equilibrium ( m, m ) = (1.9976, ). In Figure 5, (a) orresponds to the trajetories of unperturbed sstem (4) near the usp point B. The unontrollable tumor ell population will eventuall leads to death of patient. When (µ 1, µ ) = ( , ) lies in the region II, the orresponding diagram of phase portrait is shown in Figure 5(b), where two interior equilibria bifurate from the saddle-node, the left one is a stable fous and the right one is a saddle. There is a region D in the first quadrant suh that an orbits originating from D will approahes to one of the equilibria. In other words, the growth of tumor ell population is under ontrol. Figure 5() orresponds to the trajetories of sstem (19) near the stead states when (µ 1, µ ) = ( , ) lies in the region III, where a stable limit le enirling the left unstable fous ours. When (µ 1, µ ) = ( , ) lies on the urve HL, there is an inner stable homolini loop and the orresponding phase portrait is drawn in (d). 3. Bifurations of the vasularized model. The qualitative features of the aner model with neovasularization was studied b Adam [1] in Considering the vasularization to model (3), the new sstem is written as dl dc = λ 1 L + α 1 ( gc/3 1+KL )L(1 L L ) ˆβ 1 C /3 = λ (C gc/3 KL 1+KL ) α ( gc/3 1+KL )L + β C. For sstem (), Adam disussed the Hopf bifuration near the positive nondegenerate equilibrium when there is not vasularization, i.e. both ˆβ 1 and β are 0. B a fresh look at the theor of immunosurveillane, Lin [18] onsidered the eistene, stabilit and behavior in the rather simple deterministi model. This setion presents the qualitative analsis near the degenerate interior equilibrium for sstem () if it eists. We ontinue to use the notation in setion although there ma be some differenes between the vasularized and nonvasularized ases. Based on the results of the nondimensionalization in [1], sstem () an be rewritten as d = λ 1 + α1/3 1+ (1 ) β 1 /3 = f(, ) d = (λ + β ) α/3 1+ = g(, ) b hanges of variables and parameters = KL, = KC, = KL, α 1 = α 1gK /3, α = gk 1/3 (λ + α K 1 ), β 1 = ˆβ 1 K 1/3. Furthermore, the interior equilibria satisf the equation of the -loation: λ 1 (λ + β ) α 1 α () (3) = [(1 / ) k (1 + )] (1 + ) 3 = ψ(, k, ), (4) where k = β 1 /α 1 is a nonnegative onstant not more than 1/ in terms of the parameters range in [18]. We need to point out that ψ(, k, ) in (4) depends on k and ma be zero or negative for some values of (, k, ) in their respetivel reasonable range, while this annot happen for the nonvasularized ase. The equilibrium being far awa from the origin guarantees that sstem () is equivalent to d = λ 1 + [ α1(1 ) 1+ β 1 ] = F(, ) d = (λ + β ) α 1+ = G(, ) (5)

13 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 163 in terms of the transformation 3, (λ + β )/3 λ + β and α /3 α. ψ(, k, ) k 1 ψ(, k, ) 0 m ψ(, k 1, ) Figure 6. The urve ψ(, k, ) at different positive values of k when > 4k /(1 k ) with k 1 < k. Sine the solutions of (4) an be regarded as the intersetion of the horizontal line = λ1(λ+β) α 1α with the urve of = ψ(, k, ). The degenerate equilibrium of sstem (5) ours on the etremum point of ψ(, k, ) for (see Figure 6). The possible absissa of this point is whih is meaningful onl when both m = k (1 + 3/ k ) 1 + 3/ k, (6) 3k 1 k +k and ψ(, k, m ) > 0 hold. An one of < 3k 1 k +k and ψ(, k, m ) 0 being valid will result in the noneistene of degenerate equilibria for sstem (5). Owing to (1 / ) k (1 + ) = 1 [ (1 k ) ] + (1 k ) k, 4 we find that the eistene of m requires > 4k (1 k ) > 3k 1 k + k. Otherwise, there are also no interior equilibria for sstem (5). The Jaobian matri J, Haissein matries P and Q of sstem (5) at the unique interior equilibrium ( m, m ) are epressed in the same wa as the nonvasularized ase eept ˆb = b m β 1, ˆd = d + β and ˆp = p β 1, where b, d and p are defined as in α m (λ +β )(1+ m) (10), m =. For the sake of onveniene, we drop the hat to take the uniform smbols in the vasularized ase just like (10). Obviousl, Det(J( m, m )) = α 1(λ + β ) m (1 + m) 3 ψ (, k, m ) = 0. That is wh we all ( m, m ) a degenerate equilibrium. On the other hand, to have the double zero eigenvalues for the matri of the linear part of (5) at the degenerate equilibrium ( m, m ), namel, to have Tr(J( m, m ))

14 164 DAN LIU, SHIGUI RUAN AND DEMING ZHU = 0 make sense, we should have m (1 + 1 ) k (1 + m ) > 0, whih means that 1 m > 1/[ (1 + 1 ) 1]. k Using the epression of m, we know that the above inequalit is equivalent to 1 + k < whih an be simplified into > dedues the following lemmas: 1 k (1 + 3 k ) + k (1 + 1 ), 4k (1 k ). Therefore, assoiating with 3k 1 k +k Lemma 3.1. Sstem (5) has possible interior equilibria onl if > 4k (1 k ). If the interior equilibrium eists, then its absissa satisfies the equation λ1(λ+β) α 1α = ψ(, k, ). If the degenerate equilibrium ( m, m ) for whih the matri of the linear part of (5) has double zero eigenvalues, then its absissa satisfies the equation α 1 α = (λ +β ) 3 (1+) 4 [ (1+ 1 ) k(1+) ]. Lemma 3.. The parameter surfae Σ SN = {(λ 1, λ, α 1, α, β 1, β, ) λ 1(λ + β ) λ i, α i, β i > 0, i = 1, }. α 1 α = m(1 m / ) k m (1 + m ) (1 + m ) 3, orresponds to the saddle-node bifuration of sstem (5), where k = β 1 /α 1 and > 4k /(1 k ). Net we restrit on the ondition > 4k (1 k ) to onsider the degenerate interior equilibrium ( m, m ) of (5) where the matri of the linear part has double zero eigenvalues. B a hange of oordinates 1 = m, 1 = m, we simplif and epand (5) in a power series about the origin: d 1 d 1 = d 1 d 1 + p p p 1 + P 1 ( 1, 1 ) = 1 + d 1 + q Q 1( 1, 1 ). Iterating three more hanges of oordinates in (13), (15) and (17), sstem (7) is transformed into the form of (17), where the onl differene lies in d 1 = d (p 11 + dq 11 ) dp 1 + p = (λ+β) (1+ m)[(+ 3 d = d p 11 + p 1 = α1α m {[1+ k (1 k + 3 (7) k)m (1+k)] α [ m (1+, 1 ) k(1+m) ] )]m+(3+ k )} (λ +β )(1+ m) 3. (8) Aording to Lemma 3.1 and Lemma 3., we have the following theorem for the vasularized ase. Theorem 3.3. For an (λ 1, λ, α 1, α, β 1, β, ) Σ SN with > 4k (1 k ), the possible interior equilibrium ( m, m ) is a nondegenerate Bogdanov-Takens bifuration point of odimension for sstem (5) when α 1 α = where m is defined as in (6). (λ +β ) 3 (1+ m) 4 m[ m (1+ 1 ) k(1+m) ],

15 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 165 Proof. From (6), we have ( + 3 k ) m (1 + k ) = (+3/ k) 1 k (1+3/ k )+1 k (1+3/ k ) 1+3/ k. Sine > 4k /(1 k ) > 3k /(1 k + k) and 0 k 1/, the numerator of the above epression is alwas positive, whih implies that d 1 > 0 for an positive onstants λ, β, α 1, and α. Now we need to show that d is nonzero. Suppose otherwise d = 0, it would follow from (8) that m = (3+ k )/[1+ k (1 k + 3 )]. Putting the above epression into (6), we an obtain b using Mathematia that = 9 + 1k 8k + (3 k ) k (1 k ) < 4k (1 k ), whih ontradits with the hpothesis > 4k (1 k ). Thus, d 1 d annot be vanished. As a result, ( m, m ) is a nondegenerate usptpe point of odimension and sstem (5) will ehibit nondegenerate Bogdanov- Takens bifuration at this equilibrium. The proof is omplete Figure 7. The phase portrait near the usp-tpe equilibrium ( m, m ) of odimension for the vasularized ase. In fat, we ma hoose appropriate parameters in sstem (5) as bifuration parameters to have the Bogdanov-Takens bifuration our near the degenerate equilibrium just like the proedure in setion. It turns out to be muh more ompliated to alulate the bifuration equations and urves beause of the eistene of the parameters of neovasularization. Nevertheless, we are luk to find that there are similar results at the degenerate equilibrium between the nonvasularized and vasularized models, so the disussion of the usp-tpe bifuration of odimension for the seond ase is omitted in this setion. In order to larif the similarit to the avasular ase, we selet the reasonable values of the original parameters in sstem () as: α 1 = , g = 9., L = , K = 10 8, α = , λ 1 = , λ = 0.005, ˆβ1 = 0.03, β = , whih produe the values of the new parameters λ 1 = , λ = , α 1 =

16 166 DAN LIU, SHIGUI RUAN AND DEMING ZHU , α = , = 500, β 1 = , β = in sstem (5) and the interior equilibrium ( m, m ) = ( , ). Under these parameter values, the phase portrait b numerial simulations near the unique interior equilibrium for sstem (5) is depited in Figure 7. Based on Theorem 3.3, one an make the following assertion immediatel. Remark 1. An bifurations of odimension greater than two annot take plae near the usp-tpe equilibrium for perturbed vasularized aner model (5). 4. Disussion. The qualitative analsis and some bifuration results near the degenerate equilibrium have been given for the aner models (1) and () in this paper. B appling the transformation and bifuration theor in [10] and [9], we have disovered that the degenerate equilibrium is a nondegenerate usp of odimension two when the parameters take some ritial values whether the aner model suffers the neovasularization or not. We have also shown that the sstem in avasular ase ould ehibit Bogdanov-Takens bifuration in the small neighborhood of the ritial values of parameters. It is valuable to find out that an bifurations with odimension greater than two annot appear in the aner model, whih avoids more omple dnamial behavior. In ontrast with previous papers, our results sustain the qualitative analsis of Delisi and Resigno [11] about the phase portraits in the (, )-plane and improve the qualitative studies of Adam [1] and Lin [18] near the degenerate equilibrium. More importantl, we present more detailed and learer results on the dnamis of these models than Adam [1] and Lin [18], who found that tumor ell population was unontrolled and trajetories all tended to (, ) if k 1 = ψ( m ). As a matter of fat, the degenerate equilibrium is proved to be a odimension- usp aording to the realisti ranges of these parameters meeting an atual biologial situation in [18]. Therefore, b hoosing partiular values of bifuration parameters (µ 1, µ ) inside Ω, limit les or homolini orbits ma appear in aner models. From the biologial point of view, the speial hoie of parameters an lead to the ourrene of the periodi osillation behavior or oeistene of immune sstem and tumor ells. The amplitude and the loation of the equilibria in the phase plane determine the influene of those osillations. When the amplitude of the orresponding osillations is suffiientl small suh that the host an put up with the maimum levels of solid tumor and lmphote ells, then both health and arinogeni tissue an survive. On the ontrar, the survival of the host ma fail sine the solid tumor reahes a high level when the amplitude is too large ([11]). In the avasular ase, we have obtained the interesting numerial results about the eistene of a stable limit le and a homolini orbit in Figure 5(d) and (e), respetivel. When the periodi or homolini orbit eists, it an be seen as safe in the interior of these losed orbits beause the trajetories originating from there will never go beond them. So the aner ells and the immune sstem an oeist for a long term although the aner is not eliminated eventuall. We an interpret this situation biologiall that while the immune sstem fights with aner in the host, there is a balane between them beause of the periodi hanges in internal tissues and the eternal irumstanes suh that the oeist in a bounded region. For the vasularized aner model, the ourrene of Bogdanov-Takens bifuration was predited though we did not provide the proof. Intuitivel, the presene of neovasularization enhanes the possibilit of tumor survival, however, the predition of the eistene of a limit le or a homolini orbit of sstem () bifurated

17 BIFURCATIONS IN TUMOR-IMMUNITY MODELS 167 from the degenerate equilibrium whih is similar to the avasular ase has been made. Therefore, the qualitative dnamial feature near the interior degenerate equilibrium does not alter even after the aner model inorporates the terms with respet to the neovasularization of the tumor. Periodi osillation behavior is still able to our after vasularization. In fat, omparison between numerial simulations in Figure 5(a) and Figure 7 has ehibited the same dnamial behavior. Suh similar anomalous properties like that have also been notied in [1] and [11]. Aknowledgments. We would like to thank the anonmous referee for his/her helpful omments and valuable suggestions. The researh of D. Liu was supported b the State Sholarship Fund of China Sholarship Counil when she was visiting the Universit of Miami, the kind hospitalit and assistane of the Department of Mathematis at the Universit of Miami is also gratefull aknowledged. The researh of S. Ruan was partiall supported b NSF grant DMS The researh of D. Zhu was supported b NSFC of P. R. China grant no REFERENCES [1] J. A. Adam, Effets of vasularization on lmphote/tumor ell dnamis: Qualitative features, Math. Comp. Modelling, 3 (1996), [] A. Albert, M. Freedman and A. S. Perelson, Tumor and the immune sstem: The effets of tumor growth modulator, Math. Biosi., 50 (1980), [3] A. A. Andronov, E. A. Leontovih, I. I. Gordon and A. G. Maier, Qualitative Theor of Seond-Order Dnamial Sstems, John Wile and Sons, New York, [4] D. K. Arrowsmith and C. M. Plae, Bifurations at a usp singularit with appliations, Ata Appliandae Mathematiae, (1984), [5] R. W. Baldwin, Immune surveillane revisited, Nature, 70 (1977), 559. [6] G. I. Bell, Predator pre equations stimulating and immune response, Math. Biosi., 16 (1973), [7] R. Bogdanov, Bifurations of a limit le for a famil of vetor fields on the plane, Seleta Math. Soviet., 1 (1981), [8] R. Bogdanov, Versal deformations of a singular point on the plane in the ase of zero eigenvalues, Seleta Math. Soviet., 1 (1981), [9] P. Bole, A. d Onofrio, P. Maisonneuve, G. Severi, C. Robertson, M. Tubiana and U. Veronesi, Measuring progress against aner in Europe: Has the 15% deline targeted for 000 ome about? Ann. Onol., 14 (003), [10] S.-N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifuration of Planar Vetor Fields, Cambridge Universit Press, Cambridge, [11] C. DeLisi and A. Resigno, Immune surveillane and neoplasia-i: A minimal mathematial model, Bull. Math. Biol., 39 (1977), [1] J. Gukenheimer and P. Holmes, Nonlinear Osillations, Dnamial Sstems, and Bifurations of Vetor Fields, Springer-Verlag, New York, [13] R. B. Herberman and H. T. Holden, Natural ell-mediated immunit, Adv. Caner Res., 7 (1978), [14] G. Klein, Immunologial surveillane against neoplasia, Harve Let. Ser., 69 (1975), [15] V. A. Kuznetsov, I. A. Makalkin, M. A. Talor and A. S. Perelson, Nonlinear dnamis of immunogeni tumors: Parameters estimation and global bifuration analsis, Bull. Math. Biol., 56 (1994), [16] Y. A. Kuznetsov, Elements of Applied Bifuration Theor, Appl. Math. Si., 11, Springer- Verlag, New York, [17] R. Lefever and R. Gara, Loal disription of immune tumor rejetion, in Biomathematis and Cell Kinetis, North-Holland Biomedial Press, (1978), [18] A. H. Lin, A model of tumor and lmphote interations, Disrete Continuous Dnam. Sstems - B, 4 (004), [19] C. J. M. Melief and R. S. Shwartz, Immunoompetene and Malignan in Caner: A Comprehensive Treatise (eds. F. F. Beker), I, Plenum, New York, 1975,

18 168 DAN LIU, SHIGUI RUAN AND DEMING ZHU [0] L. Perko, Differential Equations and Dnamial Sstems, Appl. Math., 7, Springer, New York, 000. [1] R. T. Prehn, Review/ommentar. The dose-response urve in tumor immunit, Int. J.Immunopharm., 5 (1983), [] S. Ruan and D. Xiao, Global analsis in a predator-pre sstem with nonmonotoni funtional response, SIAM J. Appl. Math., 61 (001), [3] D. Sampson, T. G. Peter, S. D. Lewis, J. Metzig and B. E. Murtz, Dose dependene of immunopotentiation and tumor regression indued b levamisole, Caner Res., 37 (1977), [4] W. T. Shearer and M. P. Fink, Immune surveillane sstem: Its failure and ativation, Prog. Hematol., 10 (1977), [5] M. J. Smth, D. I. Godfre and J. A. Trapani, A fresh look at tumor immunosurveillane and immunotherap, Nat. Immunol., 4 (001), [6] G. W. Swan, Immunologial surveillane and neoplasti development, Rok Mountain J. Math., 9 (1979), [7] G. W. Swan, Some Current Mathematial Topis in Caner Researh, Universit Mirofilms International, MI, [8] F. Takens, Fored osillations and bifurations, in Appliations of Global Analsis I, Comm. Math. Inst. Rijksuniv. Utreht, 3 (1974), [9] D. Xiao and S. Ruan, Bogdanov-Takens bifurations in predator-pre sstems with onstant rate harvesting, Fields Instit. Commun., 1 (1999), [30] H. Zhu, S. A. Campbell and G. S. K. Wolkowiz, Bifuration analsis of a predator-pre sstem with nonmonotoni funtional response, SIAM J. Appl. Math., 63 (00), Reeived Jul 008; revised September address: liudan enu@163.om address: ruan@math.miami.edu address: dmzhu@math.enu.edu.n