Hopf bifurcation and Turing instability in the reaction diffusion Holling Tanner predator prey model

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1 IMA Journal of Applied Mathematics , doi:0.093/imamat/hxr050 Advance Access pulication on Novemer 7, 20 Hopf ifurcation and Turing instaility in the reaction diffusion Holling Tanner predator prey model XIN LI AND WEIHUA JIANG Department of Mathematics, Harin Institute of Technology, Harin, Heilongjiang 5000, People s Repulic of China Corresponding author: jiangwh@hit.edu.cn AND JUNPING SHI Department of Mathematics, College of William and Mary, Williamsurg, VA , USA [Received on 0 January 20; revised on 9 July 20; accepted on 6 August 20] The reaction diffusion Holling Tanner predator prey model with Neumann oundary condition is considered. We perform a detailed staility and Hopf ifurcation analysis and derive conditions for determining the direction of ifurcation and the staility of the ifurcating periodic solution. For partial differential equation PDE, we consider the Turing instaility of the equilirium solutions and the ifurcating periodic solutions. Through oth theoretical analysis and numerical simulations, we show the istaility of a stale equilirium solution and a stale periodic solution for ordinary differential equation and the phenomenon that a periodic solution ecomes Turing unstale for PDE. Keywords: Hopf ifurcation; Turing instaility; reaction diffusion model; prey predator system; Holling type-ii functional response.. Introduction We consider the following Holling Tanner prey predator model May, 973; Yi et al., 2008 dh dτ = v H H khp K H + D, dp dτ = sp cp, H where v and s are the intrinsic growth rates of the prey H and predator P populations, respectively. The per capita growth rate of the prey is of the common logistic form, and K is the carrying capacity, the maximum numer of the prey allowed y the limited resource. The predation function is represented y a Holling type-ii function which is extensively used in inverterate ecology. The constant k is the maximum of the predation rate when the predator will not or cannot kill more prey even when the latter is availale. The constant D refers to some value of the prey population eyond which the predators attacking capaility egins to saturate. The predator population also grows in logistic form, and c is the numer of preys required to support one predator at equilirium, when P equals H/c Tanner, 975. We consider the two populations in a spatial domain r 0,π, and we use xr, t and yr, t to denote the population densities of the prey and the predator, respectively. The dispersal of species in. c The Authors 20. Pulished y Oxford University Press on ehalf of the Institute of Mathematics and its Applications. All rights reserved.

2 288 X. LI ET AL. the spatial domain is assumed to e random, so that Fick s law holds, and it leads to the well-known diffusion equations as follows: x τ = d 2 x r 2, y τ = d 2 y 2 r 2, where d i i =, 2 are the diffusion coefficients. With the addition of diffusion and non-dimensionalization, the system. ecomes x 2 x = d + x βx my, r 0,π, t > 0, t r 2 x +.2 y 2 y = d 2 t r 2 + sy y, r 0,π, t > 0, x where t = vτ, x = H/D, y = cp/d, m = k/vc and β = D/K. We assume the no-flux oundary conditions so the ecosystem is a closed one: x0, t r = xπ, t r = y0, t r = yπ, t r The system. was first studied y Tanner 975, and he showed that under certain conditions the relative sizes of growth rates could determine the staility of the system and stated a hypothesis for the conclusion that either a stale prey population possesses strong self-limitation or the growth rate of the prey species is less than that of its predator. He tested his hypothesis y estimating the intrinsic growth rates for certain prey species and their predators. In his ook, Murray 2002 studied the staility of the positive equilirium and the existence of the limit cycles of system.. Hsu & Huang 995 dealt with the question of gloal staility of the positive equilirium in a class of predator prey systems including system. with certain conditions on the parameters, applying Dulac s criterion and Lyapunov functions construction. In Hsu & Huang 998, they proved the uniqueness of limit cycle when the unique positive equilirium is unstale. In Hsu & Huang 999, they showed that for some parameter range the Hopf ifurcation of system. is sucritical, i.e., near the Hopf ifurcation point, there exists a small-amplitude repelling periodic orit enclosing a stale equilirium and there are multiple limit cycles. Gasull et al. 997 gave a negative answer to the question: does the asymptotic staility of the positive equilirium of the system. imply the gloal staility? They computed the Poincaré Lyapunov constants in case a weak focus occurs, and in this way they were ale to construct an example with two limit cycles. For the partial differential equation PDE model, Du & Hsu 2004 considered a diffusive Leslie Gower predator prey model, which was a special case of system.2, ut in a heterogeneous environment. They showed that positive steady-state solutions with certain prescried spatial patterns can e otained if the coefficient functions are chosen suitaly and oserved some essential differences in the ehaviour of their model from that of the Lotka Volterra model that seemed to arise only in the heterogeneous case. Peng & Wang 2005 studied system.2, and they otained the existence and nonexistence of positive non-constant steady states. In Peng & Wang 2007, they otained some results for the gloal staility of the unique positive equilirium of system.2. Recently, Chen et al. 200 considered a diffusive Leslie Gower predator prey model with delay, and they proved the gloal staility of constant equilirium even with delay effect, which improved an earlier result of Du & Hsu For the studies of Hopf ifurcation for reaction diffusion R-D system, Yi et al. 2009a derived an explicit algorithm for determining the direction of Hopf ifurcation and staility of the ifurcating = 0.

3 HOPF BIFURCATION AND TURING INSTABILITY 289 periodic solutions for an R-D system consisting of two equations with Neumann oundary condition. In particular, they have shown the existence of multiple spatially non-homogeneous periodic orits while the system parameters are all spatially homogeneous. Earlier, Yi et al considered a Lengyel Epstein R-D system of the chlorite iodide malonic acid reaction, and they derived the precise conditions on the parameters so that the spatially homogeneous equilirium solution and the spatially homogeneous periodic solution ecome Turing unstale. They also studied the gloal asymptotical ehaviour of the Lengyel Epstein R-D system in Yi et al Similar work along this line has een done recently for Gierer Meinhardt model Liu et al., 200; Ruan, 998, Sel kov model Han & Bao, 2009 and a iomolecular model with autocatalysis and saturation law Yi et al., 200. See also Shi 2009 for a recent survey on astract ifurcation theorems and applications to spatiotemporal models from ecology and iochemistry. In this article, we analyse the staility and Hopf ifurcation of the positive equilirium in oth ordinary differential equation ODE and PDE models and derive conditions for determining the direction of ifurcation and the staility of the ifurcating periodic solution. For PDE, we also derive the precise conditions on the parameters so that the spatially homogeneous equilirium solution and the spatially homogeneous periodic solution ecome Turing unstale. By oth theoretical analysis and numerical simulations, we show the coexistence of a stale equilirium point, an unstale limit cycle and a stale limit cycle for ODE and a Turing unstale periodic solution is attracted y a stale non-constant steady state for PDE. The latter result confirms the results otained in Peng & Wang The rest of this article is organized as follows: In Section 2, we investigate the asymptotical ehaviour of the equilirium and occurrence of Hopf ifurcation of the local system the ODE model. In Section 3, we consider the diffusion-driven instaility of the equilirium solution. In Section 4, we analyse the staility of the ifurcating periodic solution, which is a spatially homogeneous periodic solution of the R-D system, through the Hopf ifurcation when the spatial domain is a ounded interval. In Section 5, we carry out some numerical simulations to illustrate the analytical results. In Section 6, we end our investigation with concluding remarks, and we summarize our result in a ifurcation diagram. 2. Analysis of the local system For system.2, the local system is an ODE in the form of dx = x βx mxy := f x, y, dt x + dy = sy y := gx, y, dt x x0 >0, y0 >0. The non-dimensional form which we use here is different from the ones used in previous papers. By using this form, we can deal with the staility of the equilirium solutions in an easier way. System 2. has two non-trivial equilirium points, a oundary equilirium point E /β, 0 and a positive equilirium point E 2 x, y, where x = y = 2β R 2 + 4β R, R = β + m. By simple calculations, we know that the oundary equilirium E is a saddle point with the positive x-axis as its stale manifold. We are interested in studying the properties of the positive equilirium 2.

4 290 X. LI ET AL. E 2 x, y. Before considering the staility of E 2, we present the following result, which states that system 2. is as well ehaved as one intuits from the iological prolem. A corresponding result for a different non-dimensional form is presented in Hsu & Huang 995 without proof. LEMMA 2. The solutions of system 2. are positive and eventually ounded, i.e., there exists T 0 such that xt </β, yt </β for t T. Proof. The phase portrait of 2. is shown in Fig.. The nullclines of the systems are: C : y = m βx+x, on which dx/dt = 0; and C 2 : y = x, on which dy/dt = 0. The first quadrant is divided into four parts D, D 2, D 3 and D 4 y C and C 2. The intersection of C and C 2 is the positive equilirium point x, y. Set L ={x, y: x = /β, 0 y /β} and L 2 ={x, y:0 x /β, y = /β}. Denote D as the rectangular region whose oundary consists of L, L 2, x-axis and y-axis. It is clear that D is an invariant set and attracts any trajectory starting in the first quadrant. Hence, the solutions are eventually ounded. Next we prove the positivity of the solutions y showing that trajectories starting from the first quadrant cannot reach the y-axis. To this end we only need to prove that trajectories cannot arrive the y-axis in D 2. FIG..Phase portrait of 2. in xy-plane.

5 HOPF BIFURCATION AND TURING INSTABILITY 29 For a given point x 0, y 0 D 2, denote T as the time of the trajectory running from x 0, y 0 to C and T 2 N as the time of the trajectory running from x 0, y 0 to the line x = x 0 /N, N N and N 2. We estimate the time T and T 2. T It is clear that T is finite. While since T 2 N = y 0 sy y dy x = x 0N y0 x 0 x βx mxy x+ my 0 ln lim N + there exists an N 0 N such that my 0 ln my 0 ln sy y x 0 dy = s ln dx x 0N βx0 + Nmy 0 βx 0 + my 0 x 0 y 0. x 0 x 0 xβx + my 0 dx, βx0 + Nmy 0 =+, βx 0 + my 0 βx0 + N 0 my 0 βx 0 + my 0 > s x 0 ln y 0, x 0 hence, T 2 N 0 >T. This shows that the time of the trajectory running to the y-axis is far longer than that to C, that is the trajectory runs into D 3 efore it reaches the y-axis. From the properties of the vector field shown in Fig., the trajectories cannot reach the y-axis in D 3, so any trajectories starting in the first quadrant cannot reach the y-axis. From the aove discussion, we know that there is no homoclinic or heteroclinic orit in the domain D. Then the conclusion is proved. Now we study the staility of E 2. The Jacoian matrix of system 2. at x, y is s0 Js :=, s s where s 0 = 2βx my mx + x, = 2 + x. The characteristic equation corresponding to E 2 is λ 2 s 0 sλ ss 0 + = Note that s 0 + = + βx 2 + x < 0, where we use the fact that y = m βx + x. So all roots of 2.2 have negative real parts if and only if s > s 0. Now we consider the following two cases: s 0 > 0 and s 0 0. Note that s 0 = x + x m R 2 + 4β H β < and m >,sos 0 > 0 if and only if + β2 2 β,

6 292 X. LI ET AL. s 0 0 if and only if H + β2 β, or β<and m 2 β. When H holds, we always have s > 0 s 0,soE 2 is locally asymptotically stale. Furthermore, we state the gloal staility of E 2 in the following result. THEOREM 2. Suppose that β, m, s > 0.. If H holds, then x, y is gloally asymptotically stale; 2. If H holds and s > s 0, then x, y is locally asymptotically stale; 3. If H holds and s < s 0, then x, y is unstale and there exists at least one periodic orit for 2.. Proof. Part 2 from the analysis given aove. Part follows from Theorem 2.2 in Hsu & Huang 995. Part 3 is from Lemma 2. and the Poincaré Bendixson Theorem. Suppose H holds, we have x x βx + x y < 0 for x > 0, x x. m Next we analyse the Hopf ifurcation occurring at x, y. Since the equilirium point x, y is gloally asymptotically stale when H holds, we always assume that β and m are fixed so that H holds in the following, and we use s as the ifurcation parameter. When s is near s 0, the characteristic equation 2.2 has a pair of complex roots λs = αs ± iωs, where αs = 2 s 0 s, ωs = 2 4s s0 + s 2 and αs 0 = 0, α s 0 = /2 < 0. By the Poincaré Andronov Hopf Bifurcation Theorem, we know that system 2. undergoes a Hopf ifurcation at x, y when s = s 0. However, the detailed property of the Hopf ifurcation needs further analysis of the normal form. To that end we transform the equilirium x, y to the origin y the transformation x = x x and ỹ = y y. For convenience, we still denote x and ỹ y x and y, respectively. Thus, the local system ecomes Here dx dt dy dt = Js f x, y, s = A 20 x 2 + A xy + A 30 x 3 + A 2 x 2 y + O x 4, x 3 y, x f x, y, s y + f 2 x, y, s. 2.3 f 2 x, y, s = B 20 x 2 + B xy + B 02 y 2 + B 30 x 3 + B 2 x 2 y + B 2 xy 2 + O x 4, x 3 y, x 2 y 2, where A 20 := βx β + x 2 m + x 2, A := + x 2, A 30 := βx + x 3, A m 2 := + x 3,

7 HOPF BIFURCATION AND TURING INSTABILITY 293 Set matrix B 20 := s x, B := 2s x, B 02 := s x, B 30 := N T := M, 0 s x 2, B 2 := 2s x 2, B 2 := s x 2. then αs ωs T JsT = Λs :=, ωs αs where M = s ω and N = s 0+s 2ω. Via the transformation x, y = T u,v, system 2. ecomes where F F 2 du dt dv dt = Λs u F +, 2.4 v f Nu + v, Mu := T = M f 2Nu + v, Mu f 2 Nu + v, Mu f N. M f 2Nu + v, Mu Rewrite 2.4 in the following polar coordinates form: then the Taylor expansion of 2.4 at s = s 0 yields F 2 ṙ = αsr + asr 3 +, θ = ωs + csr 2 +, ṙ = α s 0 s s 0 r + as 0 r 3 + Os s 0 2 r,s s 0 r 3, r 5, θ = ωs 0 + ω s 0 s s 0 + cs 0 r 2 + Os s 0 2,s s 0 r 2, r 4. In order to determine the staility of the periodic solution, we need to calculate the sign of the coefficient as 0, which is given y as 0 := 6 [F uuu + F uvv + F2 uuv + F2 vvv ] + 6ωs 0 [F uv F uu + F vv F2 uv F2 uu + F2 vv F uu F2 uu + F vv F2 vv ],

8 294 X. LI ET AL. where all partial derivatives are evaluated at the ifurcation point x, y, s = 0, 0, s 0. The explicit calculation of the coefficient as 0 can e found in Guckenheimer & Holmes 983, Hassard et al. 980, Marsden & McCracken 976 and Wiggins 990. Thus, we can calculate that as 0 = 8 [3A A 2 N A 30 2B 2 ] 8ω 0 [ 2A 20 + A A 20 + A N A 20 + A A 20 B 20 N 0 2B 20 A 20 B 20 N 0 ], where N 0 := N s=s0 = s 0 s 0 + and ω 0 := ωs 0 = s 0 s 0 +. Clearly, we can regard as 0 as a function aout β and m. The expression of as 0 y β and m is cumersome to present here, thus we omit it. But as 0 = 0 defines a curve in the βm-plane. With the symolic mathematical software + β2 Matla, we plot the curve as 0 = 0intheβm-plane, together with the curve m = in Fig β Now from Poincaré Andronov Hopf Bifurcation Theorem, α s s=s0 = /2 < 0 and the aove calculation of as 0, we summarize our results. THEOREM 2.2 Suppose that H holds, and let s 0 = R = β + m. R 2 + 4β R 2β + R 2 + 4β R m R 2 + 4β where. The coexistence equilirium x, y of system 2. is locally asymptotically stale when s > s 0 and is unstale when s < s System 2. undergoes a Hopf ifurcation at x, y when s = s 0. When as 0 <0, the direction of the Hopf ifurcation is supercritical and the ifurcating periodic solutions are oritally asymptotically stale; when as 0 >0, the direction of the Hopf ifurcation is sucritical and the ifurcating periodic solutions are unstale. The phenomenon of ifurcation for model. shown y Theorem 2.2 coincide with that in Gasull et al. 997 and Hsu & Huang 999. We have the following result directly from Theorem 2.2 and Poincaré Bendixson Theorem. COROLLARY 2. Under the assumption in Theorem 2.2, when as 0 >0 and s s 0, s 0 + ɛ, there exist at least two periodic orits for system 2.. In Fig. 2, a sucritical Hopf ifurcation occurs if β, m is aove the curve P 2, while a supercritical Hopf ifurcation occurs if β, m falls etween the curve P and P 2.Forβ, m elow P, the gloal staility of x, y always holds. A corresponding result for a different non-dimensional form is presented in Gasull et al Turing instaility of coexistence equilirium In 953, Turing 952 showed that, a system of coupled R-D equations can e used to descrie patterns and forms in iological systems. Turing s theory shows that the interplay of chemical reaction and diffusion may cause the stale equilirium of the local system to ecome unstale for the diffusive

9 HOPF BIFURCATION AND TURING INSTABILITY 295 FIG. 2.Graph of as 0 = 0 on the condition H. The curve P = {β, m m = +β2 2 β },, 0 <β< the curve P 2 ={β, m as 0 = 0, 0 <β<}, the region Q = {β, m m > +β2 2 β },, 0 <β< which is divided y P 2 into two regions Q where as 0 <0and Q 2 where as 0 >0. system and lead to the spontaneous formulation of a spatially periodic stationary structure. This kind of instaility is called Turing instaility or diffusion-driven instaility. In this part, we derive conditions for the Turing instaility for the spatially homogeneous equilirium solution of the R-D Holling Tanner model. Here we consider the special case with the no-flux oundary condition in a one-dimensional interval 0,π: x t y t = d 2 x r mxy + x βx, r 0,π, t > 0, 2 x +, r 0,π, t > 0, = d 2 2 y r 2 + sy y x x r 0, t = x r π, t = 0, t > 0, y r 0, t = y r π, t = 0, t > 0. 3.

10 296 X. LI ET AL. The operator φ φ on 0,πwith oundary condition φ 0 = φ π = 0 has eigenvalues and normalized eigenfunctions μ 0 = 0, μ k = k 2, k =, 2, 3,..., φ 0 r = 2 π, φ kr = coskr, k =, 2, 3,... π The linearized system of 3. at x, y has the form: xt x Δx x = Ls := D + Js, 3.2 y Δy y y t where Js is the Jacoian matrix defined in Section 2 and d 0 D := 0 d 2. Ls is a linear operator with domain D L = X C := X ix ={x + ix 2 : x, x 2 X}, where X := {x, y H 2 [0,π] H 2 [0,π]: x 0 = x π = y 0 = y π = 0} is a real-valued Soolev space. Consider the following characteristic equation of the operator Ls: φ φ Ls ψ = μ ψ. Let φr, ψr e an eigenfunction of Ls corresponding to the eigenvalue μ, and let φ ak = coskr, ψ k=0 where a k and k are coefficients. We otain that ak D k 2 cos kr + Js then k=0 k Js k 2 D ak k k=0 = μ k ak ak k k cos kr = μ k=0 ak k, k = 0,, 2,... cos kr, Denote J k := Js k 2 D = s0 k 2 d s s k 2 d 2, k = 0,, 2,...

11 HOPF BIFURCATION AND TURING INSTABILITY 297 It is clear that the eigenvalues of Ls are given y the eigenvalues of J k for k = 0,, 2,... The characteristic equation of J k is where μ 2 T k μ + D k = 0, k = 0,, 2,..., 3.3 T k := tr J k = s 0 s k 2 d + d 2, D k := det J k = d d 2 k 4 + d s d 2 s 0 k 2 ss 0 +. By analysing the distriution of the roots of 3.3, we can otain the following conclusion. THEOREM 3. Suppose that H holds and s > s 0, such that x, y is a locally asymptotically stale equilirium for system 2.. Then x, y is a locally asymptotically stale equilirium solution of system 3. if and only if one of following is statisfied H 2 d s 0 ; H 3 d d 2s 0 ; s { H 4 d < min s 0, d } 2s 0 s and x, y is an unstale equilirium solution of 3. if { H 5 d < min s 0, d } 2s 0 s and s > d 2k 2 s 0 d k 2 s0 d k 2, for all k satisfying k <, s 0 + d and s < d 2K 2 s 0 d K 2 s0 d K 2, for some K N satisfying K <. s 0 + d Thus, the equilirium x, y is Turing unstale if s elongs to the interval: { } I K = s: s 0 < s < d 2K 2 s 0 d K 2 d K 2. s 0 + That is, if s I K, then x, y is locally asymptotically stale with respect to the ODE dynamics 2., and it is unstale with respect to the PDE 3.. Proof. First, it is clear that, T k+ < T k for k 0 from the definition of T k, and T 0 < 0. So T k < 0, for all k 0. Hence, the signs of the real parts of roots of 3.3 are determined y the signs of D k, respectively. We regard D k as a quadratic function aout k 2 denoted y Dk 2, that is, Dk 2 := d d 2 k 4 + d s d 2 s 0 k 2 ss 0 +, k N. The symmetry axis of the graph of k 2, Dk 2 is ls = d 2 s 0 d s/2d d 2. H 2 implies that d k 2 s 0 0 for all k, that means D k = d d 2 k 4 +d s d 2 s 0 k 2 ss 0 + > 0 for all k 0. H 3 implies that ls <0, then we can conclude that D k > 0 for all k 0 since D 0 > 0. Clearly, H 4 implies that D k > 0 for all k 0. So all roots of 3.3 will have negative real parts under any one of assumptions H 2, H 3 and H 4. When H 5 holds, DK 2 <0, 3.3 has at least one root with positive real part. Hence, x, y is an unstale equilirium solution of system 3.. The interval I K is non-trivial only for a finite numer of eigenmodes K and apparently I K = if K 2 s 0 /d. To guarantee that I K is non-empty for K 2 < s 0 /d, one can select a large enough

12 298 X. LI ET AL. d 2 once d is fixed. When these conditions are met and certain transversality and simplicity conditions are satisfied, a pitchfork ifurcation for the non-constant equilirium solutions occurs at s = s K = d 2 K 2 s 0 d K 2 d K 2 s 0 + see Yi et al., 2009, so for decreasing s, the constant equilirium x, y loses staility to a non-constant equilirium efore the Hopf ifurcation at s = s 0 < s K. The first such ifurcation point is d 2 K 2 s 0 d K 2 s = max K N d K 2 s 0 +. When s > s, x, y is locally asymptotically stale for the PDE system 3., and it is unstale if s s. 4. Staility of spatially homogeneous periodic orits The PDE 3. possesses any periodic solution of 2. as a spatially homogeneous periodic solution, including the ones from Hopf ifurcation in Theorem 2.. We can also perform a Hopf ifurcation analysis Crandall & Rainowitz, 977; Hassard et al., 980 for 3. at the same ifurcation point in 2., and ifurcating spatially homogeneous periodic solutions exist near s = s 0. But the staility of these periodic solutions with respect to 3. could e different from that for 2.. If φt is an unstale periodic solution of 2., then it is clearly also unstale for 3.; while if φt is a stale periodic solution of 2., it could e unstale for 3. ecause of diffusion. We use the normal form method and centre manifold theorem in Hassard et al. 980 to study the direction of the Hopf ifurcation. Let L e the conjugate operator of L defined as 3.2 in Section 3: L x xrr s := D + J x s, 4. y y rr y where J s := J s, with domain D L = X C. Let a0 q = := s 0 + iω 0 0, q a = 0 0 := 2πω 0 ω0 + s 0 i. i It is easy to see that L a, = a, L for any a D L, D L, and Ls 0 q = iω 0 q, L s 0 q = iω 0 q, q, q =, q, q =0. Here a, = π 0 ā dr denote the inner product in L 2 [0,π] L 2 [0,π]. According to Hassard et al. 980, we decompose X = X C X S, with X C :={zq + z q: z C}, X S :={w X: q,w =0}. For any x, y X, there exists z C and w = w,w 2 X S such that x, y = zq + z q + w,w 2, z = q,x, y. Thus, x = z + z + w, s0 + iω 0 y = z + z s0 + iω 0 + w

13 HOPF BIFURCATION AND TURING INSTABILITY 299 In z,wcoordinates, system 3. ecomes dz dt = iω 0z + q, f, dw dt = Lw + [ f q, f q q, f q], with f = f, f 2, where f and f 2 are defined as 2.3. Straightforward calculations show that q, f = 2ω 0 [ω 0 f s 0 f + f 2 i], q, f = [ω 0 f + s 0 f + f 2 i], 2ω 0 q, f q = 2ω 0 q, f q = 2ω 0 ω 0 f s 0 f + f 2 i [ω 0 f s 0 f + f 2 i] s 0 + iω 0 ω 0 f + s 0 f + f 2 i [ω 0 f + s 0 f + f 2 i] s 0 iω 0 Hz, z,w:= f q, f q q, f q = 0. 0 Write w = w 20 /2z 2 + w z z + w 20 /2 z 2 + O z 3 for the equation of the centre manifold; we can otain 2iω 0 Lω 20 = 0, Lω = 0 and ω 02 = w 20. This implies that ω 20 = ω 02 = ω = 0. Thus, the equation on the centre manifold in z, z coordinates ecomes where and,, 4.3 dz dt = iω 0z + 2 g 20z 2 + g z z + 2 g 02 z g 2z 2 z + O z 4, 4.4 g 20 = q,c 0, d 0, g = q,e 0, f 0, g 2 = q,g 0, h 0, 4.5 c 0 := f xx a f xya f yy 2 0 = 2A A 0, d 0 := g xx a g xya g yy 2 0 = 2B B 0 + 2B , e 0 := f xx a f xy a 0 0 +ā f yy 0 2 = 2A 20 + A 0 + 0, f 0 := g xx a g xy a 0 0 +ā g yy 0 2 = 2B 20 + B B ,

14 300 X. LI ET AL. g 0 : = f xxx a 0 2 a 0 + f xxy 2 a a f xyy a 0 + 0ā0 2 + f yyy = 6A A , h 0 : = g xxx a 0 2 a 0 + g xxy 2 a a g xyy a 0 + 0ā0 2 + g yyy = 6B B B Here all the partial derivatives are evaluated at the point s, x, y = s 0, 0, 0. Then we otain [ g 20 = A 20 2s ] [ 0 + B 20 i A 20 + A s 0 + 2s s ] B 20, ω 0 [ ] g = A 20 s 0 A i s 0 A 20 s2 0 ω 0 A + s 0 + B 20, g 2 = 3A 30 2 [A 2s 0 + s 0 + B 2 ] + i ω 0 According to Hassard et al. 980, c s 0 = i 2ω 0 [ s0 2s 0 A 2 B 2 3A B 2 s 0 3B 30 ]. g 20 g 2 g 2 3 g g 2, 4.6 then { i Rec s 0 = Re g 20 g 2 g 2 3 2ω g } 0 2 g 2 = Reg 20 Img + Img 20 Reg + 2ω 0 2 Reg 2 = 2ω0 2 2s 0 A s 0 2s0 2 A 20 A s2 0 A + 2s 0 + A 20 B 20 s 0 s 0 + A B 20 2s B A 30 s 0 A 2 s 0 + Comparing the result aove with the expression of as 0, we find that Rec s 0 = 4s 0+ as 0. Then Rec s 0 <0if and only if as 0 <0since 4s 0+ > 0. We summarize our analysis results in the following manner. THEOREM 4. Suppose that β, m, s satisfy the same conditions as Theorem 2.2, then system 3. undergoes a Hopf ifurcation at x, y when s = s 0.. The direction of Hopf ifurcation of system 3. is the same as that of system 2.; 2. When as 0 < 0, the direction of Hopf ifurcation is supercritical, and the ifurcating periodic solutions are asymptotically stale on the centre manifold. Furthermore, they are oritally asymptotically stale for system 3. if and only if one of H 2, H 3 and H 4 holds; they are unstale if H 5 holds; B

15 HOPF BIFURCATION AND TURING INSTABILITY When as 0 >0, the direction of Hopf ifurcation is sucritical, and the ifurcating periodic solutions are unstale on the centre manifold; thus they are unstale for system Numerical simulations In this section, we present some numerical simulations to illustrate our theoretical analysis. The ODE model 2. has three parameters: β, m, s. We choose parameters: β = 0.2, m = β = 0.2, m = Under 5., we have the equilirium point x, y.0895,.0895, and the critical point s and as < 0. By Theorem 2.2, the equilirium is asymptotically stale when s > s 0 ; a Hopf ifurcation occurs at s = s 0, the ifurcating periodic solutions occur when s < s 0 and the ifurcating periodic solutions are asymptotically stale Fig. 3. Under 5.2, the equilirium point x, y , , the critical point s and as > 0. By Theorem 2.2, the equilirium is asymptotically stale when s > s 0 ; the ifurcating periodic solutions occur when s > s 0 and the ifurcating periodic solutions are unstale. These are shown in Fig. 4. The PDE model has five parameters: β, m, s, d, d 2. We choose three sets of parameters as follows: β = 0.2, m =.5, d = 0.008, d 2 =, s = 0.25, 5.3 β = 0.2, m =.5, d =, d 2 =, s = 0.3, 5.4 β = 0.2, m =.5, d =, d 2 =, s = 0.5, 5.5 β = 0.2, m =.5, d = 0.008, d 2 =, s = FIG. 3.Phase portraits of 2. with parameters in 5.. Left: s = 0.2 > s 0, the positive equilirium is asymptotically stale initial condition is x 0, y 0 =.5,.5; right: s = 0.5 < s 0, the ifurcating periodic orit is stale initial condition x 0, y 0 =.,. and x 0, y 0 = 3,.8.

16 302 X. LI ET AL. FIG. 5. Numerical simulations of the Turing instaility of the equilirium solution of system 3. under 5.3. The solution appears to converge to a non-homogeneous steady state initial condition x0, y0 = 0. cos r +.08, 0. cos r Left: component x; right: component y. Under 5.3, s and s = 0.25 T, i.e., H5 holds for K =, 2, 3, 4. By Theorem 3., the homogeneous equilirium solution x, y.0895,.0895 of system 3. is unstale. Figure 5 shows the Turing instaility of the equilirium solution. Under 5.4, s , Rec s < 0 and H2 holds. By Theorem 3., the homogeneous equilirium solution x, y of system 3. is locally asymptotically stale see Fig. 6. Under 5.5, s , Rec s < 0 and H2 holds; the choice of s satisfies s < s0. By Theorem 4., Hopf ifurcation occurs at s = s0, the ifurcating periodic orits exist for s < s0, which are oritally asymptotically stale. This is shown in Fig. 7. Under 5.6, s , Rec s < 0 and H5 holds for K = 3 in Theorem 3.; the choice of s satisfies s < s0. By Theorems 4. and 3., the ifurcating periodic orits exist for s < s0, which are Turing unstale. In Fig. 8, we can see the solution from x0, y0 = 0. cos r + FIG. 4. Phase portraits of 2. with parameters in 5.2. Left: s = 0.62 > s0, the positive equilirium is asymptotically stale, and the ifurcating periodic solutions are unstale; right: s = 0.5 < s0, the positive equilirium is unstale, there exists a stale periodic orit initial condition x0, y0 = 0.34, 0.34, x0, y0 = 0.5, 0.5 and x0, y0 = 0.9, 0.9 in the left panel and x0, y0 = 0.3, 0.3 in the right panel.

17 HOPF BIFURCATION AND TURING INSTABILITY 303 FIG. 7. Numerical simulations of the stale periodic solution of system 3. under 5.5 and s = 0.5 < s0. The solution appears to converge to a homogeneous periodic orit initial condition x0, y0 = 0.5 sin r +.08, 0.5 cos r Left: component x; right: component y..08, 0. cos r +.08 is attracted y a positive non-constant stale equilirium. This verifies the results previously proved in Peng & Wang Conclusions A rigorous investigation of the dynamics of an R-D Holling Tanner model suject to Neumann oundary condition is attempted, and the main purpose of this article is to identify the parameter ranges of staility and instaility of spatially homogeneous equilirium solutions and ifurcating periodic orits. We summarize our investigation on the ifurcation diagram of the parameters m and s see Fig. 9. For the ODE system 2., a curve L 2 : s = s0 separates the stale region aove L 2 and the unstale region elow L 2 ; a Hopf ifurcation occurs when the parameter crosses L 2 transversally, and at least one periodic orit exists for parameter values elow L 2. A vertical line L a : Rec s0 = as0 = 0 separates the region of supercritical Hopf ifurcations on the left of L a and sucritical ones on the FIG. 6. Numerical simulations of the stale equilirium solution of system 3. under 5.4. The solution appears to converge to a homogeneous steady state initial condition x0, y0 = 0.5 sin r +.08, 0.5 cos r Left: component x; right: component y.

18 304 X. LI ET AL. FIG. 8. Numerical simulations of the Turing instaility of the ifurcating periodic orit and the existence of positive nonhomogeneous steady states of system 3. under 5.6. The solution appears to converge to a non-homogeneous steady state initial condition x 0, y 0 = 0. cos r +.08, 0. cos r Left: component x; right: component y. FIG. 9.Bifurcation diagram in m, s parameter space. Here we choose β = 0.2 and d 2 /d = 2. L is the curve s 0 + s 2 + 4s = 0 where the Jacoian Js has repeated eigenvalues, and L divides the first quadrant into three parts: M stale/unstale focus, M 2 stale node and M 3 stale node. The ODE Hopf ifurcation curve L 2 is s = s 0, aove elow which x, y is a stale unstale focus for the ODE system. The curve L 3 is s = d 2 s 0 /d, and etween L 2 and L 3, Turing instaility could occur for x, y for the PDE system. The vertical line L H is s 0 = 0m = 0.9 here, and on the left side of L H x, y is gloally asymptotically stale with respect to the ODE system. The vertical L a is Rec s 0 = as 0 = 0m = 3.06 here, and on the left right side of L a the Hopf ifurcation at s = s 0 is supercritical sucritical.

19 HOPF BIFURCATION AND TURING INSTABILITY 305 right of L a. Hence, on the right side of L a, 2. may have two periodic orits for s s 0, s 0 + ɛ. A further vertical line L H : s 0 = 0 identifies a parameter region on the left side of L H where the coexistence equilirium x, y is gloally asymptotically stale with respect to the ODE dynamics 2.. On the right side of L H and aove L 2, x, y is locally asymptotically stale for 2.. Finally, a curve L : s 0 + s 2 + 4s = 0 separates the region where x, y is a focus or node. The equilirium solutions and periodic solutions of the ODE system 2. are spatially homogeneous solutions of the R-D system 3.. Hence, the ODE dynamics of 2. is a sudynamics of the one of 3.. The direction of Hopf ifurcation of system 3. st s = s 0 is also same as that of system 2.. But the staility of the solutions can change ecause of the effect of diffusion. If the parameter value m, s is chosen aove curve L 3, then the staility of the coexistence equilirium x, y with respect to the PDE 3. is the same as that of 2., and there will e no Turing instaility of the spatially homogeneous equilirium. But such a diffusion-induced instaility can occur for parameter values chosen etween the curves L 2 and L 3. The Turing ifurcations of spatially non-homogeneous equilirium solutions that occurred etween L 2 and L 3 and further Hopf ifurcations of spatially non-homogeneous periodic orits that occurred elow L 2 will e of interest for further investigation. Funding National Natural Science Foundation of China; Heilongjiang Provincial Natural Science Foundation A200806; Program of Excellent Team and Science Research Foundation in Harin Institute of Technology. REFERENCES CHEN, S., SHI, J.& WEI, J. 200 Gloal staility and Hopf ifurcation in a delayed diffusive Leslie-Gower predator-prey system sumitted for pulication. CRANDALL, M. G.& RABINOWITZ, P. H. 977 The Hopf ifurcation theorem in infinite dimensions. Arch. Ration. Mech. Anal., 67, DU, Y.& HSU, S. B A diffusive predator-prey model in heterogeneous environment. J. Differ. Equ., 203, GASULL, A., KOOIJ, R. E.& TORRGROSA, J. 997 Limit cycles in the Holling-Tanner model. Pul. Mat., 4, GUCKENHEIMER, J.& HOLMES, P. J. 983 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. New York: Springer. HAN, W.& BAO, Z Hopf ifurcation analysis of a reaction-diffusion Sel kov system. J. Math. Anal. Appl., 356, HASSARD, B. D., KAZARINOFF, N.& WAN, Y.-H. 980 Theory and Applications of Hopf Bifurcation. Camridge: Camridge University Press. HSU, S.B.&HUANG, T. W. 995 Gloal staility for a class of predator-prey system. SIAM J. Appl. Math., 55, HSU, S.B.&HUANG, T. W. 998 Uniqueness of limit cycles for a predator prey system of Holling and Leslie type. Can. Appl. Math. Quart., 6, 9 7. HSU, S. B.& HUANG, T. W. 999 Hopf ifurcation analysis for a predator-prey system of Holling and Leslie type. Taiwanese J. Math., 3, LIU, J., YI, F.& WEI, J. 200 Multiple ifurcation analysis and spatiotemporal patterns in a -D Gierer- Meinhardt model of morphogenesis. Int. J. Bifurcation Chaos, 20, MARSDEN, J.E.&MCCRACKEN, M. 976 The Hopf Bifurcation and its Applications. New York: Springer.

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