Bifurcation Analysis of a Population Dynamics in a Critical State 1

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1 Bifurcation Analysis of a Population Dynamics in a Critical State 1 Yong-Hui Xia 1, Valery G. Romanovski,3 1. Department of Mathematics, Zhejiang Normal University, Jinhua, China. Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova, SI-, Slovenia 3. Faculty of Nature Science and Mathematics, University of Maribor, Krekova, SI-, Slovenia Abstract: A stage-structured population model in a critical state is studied in this paper. By analyzing the corresponding characteristic equations, the local stability of the equilibria is discussed. Hopf bifurcations occurring at the positive equilibrium under some conditions are demonstrated. Taking time delay τ as bifurcating parameter, the direction and stability of Hopf bifurcation are carried out. The global continuation of periodic solutions bifurcating from the equilibrium is investigated. Finally, an example and numerical simulations are given. Keywords: Bifurcation; Delay; Periodic solutions; Stability; Mathematics Subject Classification: 34K18; 34C3; 9D5 1 Introduction 1.1. Motivations. It is known that Hopf bifurcation of delayed differential systems has been widely studied e.g. see [1] and references cited therein. In particular, many authors successfully applied Hopf bifurcation theorem to study the delayed population dynamical systems. For examples, one can refer to [ 8]. However, most of the works only considered the standard non-critical cases. To illustrate this clearly the characteristic equation should be mentioned. It is known that distribution of the roots of the characteristic equation plays important role in the bifurcation analysis. As for two dimensional system, one can consider the second degree transcendental polynomial equation λ + pλ + r + sλ + qe λτ =, 1.1 where p, q, r, s are real numbers. When τ =, Eq. 1.1 becomes λ + p + sλ + r + q =. 1. Most of the above mentioned works studied the distribution of roots under the standard assumption p + s >, and q + r >. 1 yhxia@zjnu.cn Y. Xia; valery.romanovsky@uni-mb.si V. Romanovski 1

2 In this situation, all roots of Eq. 1. have negative real parts. Thus, it follows from [3], that if transcendental equation 1.1 has no purely imaginary roots, all roots of Eq. 1.1 have negative real parts for all τ. However, to the best of our knowledge, there is no paper considering the transcendental equation 1.1 in the critical case p + s =. In fact, in this situation, Eq. 1. has a pair of purely imaginary roots. So the difficulty arises when determining the distribution of roots of Eq To show how to deal with the critical situation we discuss the bifurcation of a plant-hare dynamical system in critical state. 1.. Model Formulation. It is well known that the classical Lotka-Voterra population models competitive model, predator-prey model have been well investigated. However, in the real world, almost all animals can be classified into juvenile and adult according to the age-structure. Juvenile and adult have different ability of consumption, mating, reproducing, attacking prey, etc. So the stage-structured factor is absolutely necessary for studying the population dynamics. To fit the more realistic environment, stage-structured population growth is introduced in population models. These models assume an average age to maturity which appears as a constant time delay reflecting a delayed birth of immature and a reduced survival of immature to their maturity. For example, some authors have studied a single-species population growth with various stages of life history see [31 34]. Much research has been devoted the models concerning the predator-prey system with stage-structure see [35 49]. Most of the works focused on the predator-prey system with stage structure for the predator. Recently, the authors of [9] considered the influence of a stage structure for prey plant in a plant-herbivore dynamic system. In the model, it is assumed that the plant can produce toxic substance to protect themselves when they are immature, while the plant toxicity can disappear when they grow into the mature plant. In this situation, the herbivores predators choose the mature plants to eat and avoid the immature ones. The model is described as follows: Ḣt = c M b M MtHt d H Ht, Jt = rjt + αmtht rmt τ αb M Mt τht τ d J Jt, Ṁt = rmt τ + αb M Mt τht τ b M MtHt d M Mt βm t, 1.3 where Ht is the population density of herbivore predator species, Jt and Mt denote the densities of the immature and mature plants prey species, respectively; d H denotes the death rate of herbivore; r is the intrinsic growth rate of the immature plant; d J and d M denote the death rate of the immature and mature plant, respectively; τ represents a constant time to maturity; r denotes the rate of immature plant becoming into mature plant; b M denotes the coefficient in herbivore eating immature plant; αb M denotes the rate of mature plant which has been eaten at time t τ becoming into new mature plant; c M denotes the rate of conversing mature plant into new herbivore; β is the intra-specific competition rate of the mature plant. In [9], the parameter β is chosen as β >, by the assumption that mature plants share the nutrients in an closed environment. But in a good situation,

3 there are enough nutrients. In this circumstance, the parameter can be chosen as β = and system 1.3 reduces to Ḣt = c M b M MtHt d H Ht, Jt = rjt + αmtht rmt τ αb M Mt τht τ d J Jt, Ṁt = rmt τ + αb M Mt τht τ b M MtHt d M Mt. System 1.4 is supplemented with the initial conditions of the form Hs = ϕ 1 s, ϕ 1 s >, s [ τ, ], Js = ϕ s, ϕ s >, s [ τ, ], Ms = ϕ 3 s, ϕ 3 s >, s [ τ, ], where ϕ = ϕ 1, ϕ, ϕ 3 C[ τ, ], R 3 +, i = 1,, 3 and C denotes the set of all continuous functions from [ τ, ] into R 3 + = {H, J, M : H >, J >, M > }. Note that the first equation and the third equation of 1.4 can be separated from the whole system. Consider the following subsystem of 1.4. { Ḣt = cm b M MtHt d H Ht, 1.5 Ṁt = rmt τ + αb M Mt τht τ b M MtHt d M Mt. A preliminary result on the positivity of solutions is proved similarly as Lemma 1.1 in [9]. Lemma 1.1. All solutions of system 1.5 are positive Comparison with the previous work. In [9], the authors studied the Hopf bifurcation of system 1.3 under the assumption that β >. In fact, when β =, the results in [9] cannot be valid. Why? Because the distribution of the roots of characteristic equation is completely different. The essential reason is that β > is equivalent to the standard assumption p + s > in Eq. 1., while β = corresponds to the critical case p + s = in Eq. 1.. Moreover, it is very interesting to study the global continuation of periodic solutions bifurcating from the equilibrium in a critical state. The method used in the proof of Lemma 4. is nontrivial and interesting Outline of this work. Model analysis and bifurcation of periodic solution are carried out in Section. Section 3 is devoted to examining the direction and the stability of Hopf bifurcation. An interesting theorem for the global continuation of periodic solution from the equilibrium is given. Finally, an example and numerical simulations are given to show the feasibility of our results. Model analysis and bifurcation of periodic solution In this section we determine the local stability of the equilibria boundary equilibrium and positive equilibrium. 3

4 Letting { xt = Ht, ỹt = c M b M Mt, then system 1.5 can be rewritten as { xt = xtỹt a xt, ỹt = bỹt τ + c xt τỹt τ d xtỹt eỹt,.1. where a = d H, b = r, c = b M α, d = b M, e = d M. From biological point of view, the parameters a, b, c, d, e are all positive. If Ẽ = x, ỹ denotes the equilibrium of system., it must satisfy the algebraic equations { xỹ a x =, bỹ + c xỹ d xỹ eỹ =..3 Clearly, E =, is an equilibrium. If d c and b e, then there are no other equilibria lying on the x-axis or on the y-axis. If d = c and b e, system. has only one equilibrium E. b e d c It is easy to see that there is a unique positive equilibrium, E = b e d c >. So we summarize as follows. Lemma.1. i If d = c and b e, then system. has only one equilibrium E. ii If d c and b e d c >, then system. has a unique positive equilibrium E., a, if d c and To precisely describe the stability of the equilibrium Ẽ, we need the following definition. Definition.. The equilibrium Ẽ of system. is called conditionally stable asymptotically stable on the delays if it is asymptotically stable for some τ j in some intervals, but not necessarily for all delays τ j, 1 j m. The equilibrium Ẽ of system. is called absolutely stable asymptotically stable independent of the delays if it is asymptotically stable for all τ j, 1 j m. To determine the local stability of the equilibria E and E, we need to linearize. around Ẽ = x, ỹ Ẽ may be E or E. To do so, letting xt = xt x and yt = ỹt ỹ,. can be transformed into ẋt = ỹ axt + xyt + xtyt, ẏt = dỹxt d x + eyt + b + c xyt τ + cỹxt τ dxtyt + cxt τyt τ..4 The linear part of system.4 is { ẋt = ỹ axt + xyt, ẏt = dỹxt d x + eyt + b + c xyt τ + cỹxt τ..5 4

5 The corresponding characteristic equation in the unknown λ is λ ỹ a x det dỹ cỹe λτ λ + d x + e b + c xe λτ =, that is, λ + pλ + r + sλ + qe λτ =,.6 where p = d x+e ỹ +a, r = d xỹ d x+eỹ a, s = b+c x, q = [c xỹ b+c xỹ a]. For Ẽ = E, we have the following theorem. Theorem.3. i If b > e, then the equilibrium point E =, is unstable saddle. ii If b < e, then the equilibrium point E =, is asymptotically stable. Proof. For Ẽ = E,.6 reduces to λ + e + aλ + ea bλ + abe λτ =, or λ + aλ + e be λτ =. Obviously, λ 1 = a is a negative root. Let F λ = λ + e be λτ. i If b > e, it is easy to verify that F = e b < and F + = +. Thus, there exists a positive real number λ > such that F λ =. Therefore, E is unstable if b > e. ii Now we consider the case b < e. Note that λ 1 = a is a negative root. To show the asymptotic stability of E in this case, it suffices to prove that all the roots of F λ = have negative real parts. In fact, when τ =, we see that F λ = reduces to λ + e =. That is, λ = e is the unique negative root. So by Rouché s theorem see Dieudonné [3], Theorem , we only need to prove that F λ = does not have any purely imaginary roots. To this end, assume for the contrary that iω is a purely imaginary root of F λ =. Rewrite F λ = in terms of its real and imaginary part as { e b cosωτ =, which implies ω + b sinωτ =, ω = b e <. This is a contradiction. Therefore, if b < e, E is asymptotically stable. For Ẽ = E, replacing x, y with x, y = b e, a in.6, we have d c λ + pλ + r + sλ + qe λτ =,.7 5

6 where p = dx + e = bd ce, r = dx d c y = b eda, s = b + cx d c = bd ce, q = cx d c y = b eca. Note that p + s = in this case. So the system is in a critical state just as we point d c out in the Introduction. When τ =, Eq..7 reduces to λ + p + sλ + q + r =, or λ + q + r =..8 Notice that p + s =, which is different from the standard assumption p + s >. To determine the local stability of the positive equilibrium E more precisely, we proceed in three steps. Step 1. The first step is to find all possible purely imaginary roots of the characteristic equation.7. To this end, we let λ = α + iω, α, ω R, and rewrite.7 in terms of its real and imaginary arts as { α ω + pα + r = e ατ [ sα + q cosωτ sω sinωτ], αω + pω = e ατ.9 [ sω cosωτ + sα + q sinωτ],. When α =,.9 reduces to { ω + r = q cosωτ sω sinωτ, pω = sω cosωτ + q sinωτ..1 It follows by taking the sum of squares that ω 4 s p + rω + r q =..11 Note that for Ẽ = E, p + s =, which implies that s p + r = r <. Hence, the two roots of Eq..11 can be expressed as follows: ω+ = r q, and ω = r + q..1 Thus, if d < c, then r q <, so that Eq..11 has a positive root ω+. And if d > c, then r q >, so that Eq..11 has two positive roots ω±. 6

7 In both cases, the characteristic equation.7 has purely imaginary roots when τ takes certain values. These critical values τ ± j of τ can be determined from system.1 and given by { τ + j = π ω + + jπ ω +, j =, 1,,, { } τ j = 1 ω arccos q p r+q + jπ.13 q +p r+q ω, j =, 1,,. Step. Denote by the root of Eq..7 satisfying λ ± j τ = α± j τ + ω± j τ, j =, 1,,, α ± j τ ± j =, ω± j τ ± j = ω ±..14 This step is to verify that the following transversality conditions hold: d dτ Reλ+ j τ + j >, d dτ Reλ j τ j <..15 To see this, differentiating characteristic equation.7 with respect to τ, we have [λ + p] dλ dτ = sdλ dτ τe λτ [ sλ q] dλ dτ λe λτ [ sλ q]. Noticing that p = s, it follows that { } dλ τe λτ [ sλ q] + λ dτ = λe λτ [ sλ q], which implies Here we have used Eq..7 again. dλ 1 = τe λτ [ sλ q] + λ dτ λe λτ [ sλ q] = e λτ [ sλ q] τ λ = λ + pλ + r τ λ. Thus, in view of.14 and.1, it follows from.16 that { d { dλ 1 } sign dτ }τ=τ Reλ = sign Re + j dτ τ=τ + j { [ = sign Re λ + pλ + r τ ] } λ τ=τ + j { [ = sign Re [iω + + ipω + + r] τ + ]} j iω { + ω+ r } = sign ω+ r + p ω+ { r q r } q = sign = sign{ } = 1. q + p ω+ q + p ω+ 7.16

8 Similarly, Thus, the assertion.15 holds. { d sign dτ }τ=τ Reλ j { q } = sign = 1. q + p ω Step 3. To assure the existence of a positive equilibrium, d c is indispensable. So we shall study the distribution of roots of Eq..7 in two cases d < c or d > c: Case 1: d < c, which implies that q + r <. Setting Gλ = λ + pλ + r + sλ + qe λτ, it follows from Eq..7 that and G = r + q <, G+ = +. Thus, there exists a λ > such that Gλ =. Consequently, the positive equilibrium E is unstable. Case : d > c, which implies that q + r >. Then it follows from Eq..8 that λ = ±i q + r for τ =, which implies Reλ =. On the other hand, it is easy to see from Step that d Reλ >. We claim that Reλτ > for all τ, τ + dτ. If this claim is not true, there exists a δ < δ < τ + < τ such that Reλτ > for τ [, δ and Reλδ =. This implies that λδ is a pure imaginary root of Eq..7 and d Reλδ <. dτ But from Step 1, there is no other ττ except τ ± j j = 1,, such that Eq..7 has a pair of purely imaginary roots. And it follows from Step that τ is the minim number d such that Reλτ dτ j <. This implies δ = τ, which contradicts to the assumption of δ < δ < τ + < τ. Therefore, the real parts of all roots of Eq..7 are positive for all τ, τ +. In view of d Reλτ + dτ > and d Reλτ dτ <, by an analogous argument for the intervals τ +, τ, we see that the real parts of all roots of Eq..7 are negative for all τ τ +, τ. Similarly, we conclude that for the intervals τ 1 +, τ1, τ +, τ,, τ τ + k, τ k, the real parts of all roots of Eq..7 are negative. Similarly, for the intervals τ τ, τ 1 +, τ τ1, τ +,, τ τ k 1, τ + k and τ > τ k, the real parts of the roots of Eq..7 are positive. From the above analysis, it follows that τ ± j are bifurcation values. Thus, we have the following theorem about the distribution of the characteristic roots of Eq..7. Theorem.4. Let ω ± and τ ± j be defined by.1 and.13, respectively. i If d < c and b < e, then Eq..7 has at least one root with positive real part for all τ. 8

9 ii If d > c and b > e, then there is a positive integer k such that there are switches from instability to stability, that is, when τ, τ +, τ, τ + 1, τ 1, τ +,, τ k 1, τ + k and τ > τ k, Eq..7 has a root with positive real part, and when all roots of Eq..7 have negative real parts. τ τ +, τ, τ + 1, τ 1,, τ + k, τ k, Remark.5. In the non-critical case see e.g. [, 3, 1, 11, 6], let ω ± and τ ± j ω ± = 1 s p + r ± 1 [s p + r 4r q ] 1, be defined by and τ ± j = 1 { q ω arccos ± r ps ω } ± ω ± s ω + jπ, j =, 1,,. ± + q ω ± Then there is a positive integer k such that there are switches from stability to instability, that is, when τ [, τ +, τ, τ 1 +, τ 1, τ +,, τ k 1, τ + k, all roots of Eq..7 have negative real parts, and when τ [ τ +, τ, τ + 1, τ 1,, τ + k 1, τ k 1 and τ > τ + k, Eq..7 has at least one root with positive real part. We see that there is a big difference between the critical case and the non-critical one. In particular, the intervals for stability and instability are completely different. In the critical case, there are switches from instability to stability, the intervals, τ +, τ, τ 1 +, τ1, τ +,, τ k 1, τ + k are unstable. On the contrary, in the non-critical case, there are switches from stability to instability, the intervals [, τ +, τ, τ 1 +, τ 1, τ +,, τ k 1, τ + k are locally asymptotically stable. The reason lies in the fact that in the critical case, all roots of Eq..8 are pairs of purely imaginary roots for τ =, while in the non-critical case, all roots of Eq..8 have negative real parts for τ =. Thus for Ẽ = E, we have the following main results on the local stability and bifurcation of the positive equilibrium E. Theorem.6. Let ω ± and τ ± j i If d < c and b < e, then E is unstable. be defined by.1 and.13, respectively. ii If d > c and b > e, then there is a positive integer k such that the equilibrium E switches k times from instability to stability, that is, when τ, τ +, τ 1, τ + 1, τ, τ +,, τ k 1, τ + k 1 and τ > τ k, the positive equilibrium E of. is unstable; when τ τ +, τ 1, τ + 1, τ,, τ + k 1, τ k, the positive equilibrium E of. is asymptotically stable. iii If all the conditions as stated in ii hold, then system. undergoes a Hopf bifurcation at the equilibrium when τ = τ + j j =, 1,, k 1 and τ = τ j j = 1,,, k. 9

10 3 Direction and stability of Hopf bifurcation In the previous section, some sufficient conditions are obtained to guarantee that system. undergoes Hopf bifurcation at the positive equilibrium E when τ = τ + j j =, 1,, k 1 and τ = τ j j = 1,,, k. This section is to derive the explicit formulas determining the direction, stability and period of periodic solutions bifurcating from equilibrium E at these critical values of τ. The method is based on the normal form and the center manifold theory developed by Hassard et al. [1]. Without loss of generality, denote any one of these critical values τ = τ + j j =, 1,, k 1 and τ = τ j j = 1,,, k by τ, at which Eq..7 has a pair of purely imaginary roots iω and system. undergoes a Hopf bifurcation from E. Let u 1 t = xτt x, u t = yτt y and τ = τ + µ, µ R. Then µ = is the Hopf bifurcation value of system. and system. can be rewritten as u 1 t = τ[x u t + u 1 tu t], u t = τ[ dy u 1 t dx + eu t + b + cx u t 1 + cy u 1 t 1 du 1 tu t + cu 1 t 1u t 1] Thus, we can work in the fixed phase space C = C[ 1, ], R, which does not depend on the delay τ. In space C = C[ 1, ], R, system 3.1 is transformed into a FDE as 3.1 ut = L µ u t + fµ, u t, 3. where u = u 1, u T, u t θ = ut + θ C, and L µ are given respectively by x L µ ϕ = τ + µ ϕ dy ϕ 1 dx + eϕ + b + cx ϕ 1 + cy ϕ 1 1 and fµ, ϕ = τ + µ ϕ 1 ϕ dϕ 1 ϕ + cϕ 1 1ϕ 1, 3.3, 3.4 where ϕ = ϕ 1, ϕ C. By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions ηµ, θ in θ [ 1, ] such that L µ ϕ = 1 dηµ, θϕθ, for ϕ C, where bounded variation functions ηµ, θ can be chosen as x ηµ, θ = τ + µ δθ τ + µ dy dx + e cy b + cx δθ + 1, 3.5 where δ is the Dirac function. For ϕ C 1 [ 1, ], R, define Aµϕ = { dϕθ dθ 1, θ [ 1,, dηµ, sϕs, θ =, 1

11 and Rµϕ = Then system 3. is equivalent to {, θ [ 1,, fµ, ϕ, θ =. u t = Aµu t + Rµu t, 3.6 where u = u 1, u T, u t θ = ut + θ, θ [ 1, ]. For ψ C 1 [, 1], R, define { A dψs, s, 1], ψs = ds dη, tϕ t, s =, and a bilinear inner product 1 ψs, ϕθ = ψϕ θ 1 ξ= ψξ θdηθϕξdξ, 3.7 where ηθ = η, θ. Then A and A are adjoint operators. In addition, from Section we know that ±i τω are eigenvalues of A. Thus, they are also eigenvalues of A. We first need to compute the eigenvector of A and A corresponding to i τω and i τω, respectively. To this end, suppose that qθ = 1, α T e iω τθ is the eigenvector of A corresponding to i τω, then Aqθ = iω τqθ. It follows from the definition of A, 3.3 and 3.5 that τ iω dy cy e iω τ x iω + dx + e b + cx e iω τ q =. Thus, we can choose Similarly, let q s = Dβ, 1 T i τω, we can compute q = 1, α T = 1, iω x T. = Dβ, 1e iω τs is the eigenvector of A corresponding to q s = Dβ, 1 T = Dβ, 1e iω τs = D iω + dx + e b + cx e iω τ x, 1e iω τs. In order to assure q s, qθ = 1, we need to determine the value of D. From 3.7, we have q s, qθ = D β, 11, α T θ 1 ξ= D β, 1e iω τξ θ dηθ1, α T e iω τξ dξ, D{ = β + α β }, 1θe iω τθ dηθ1, α T D{ 1 = β + α + τ β }, 1 1, α cy b + cx T e iω τ = D { } β + α + τ[cy + α b + cx ]e iω τ. 11

12 Thus, we can choose D as D = 1 β + ᾱ + τ[cy + ᾱ b + cx ]e iω τ. It is also easy to verify that q s, qθ =. Now we compute the coordinates to describe the center manifold C at µ =. Let u t be the solution of 3.6 when µ =. Define On the center manifold C, we have where zt = q, u t, W t, θ = u t θ Re{ztqθ}. 3.8 W t, θ = W zt, zt, θ, W z, z, θ = W θ z + W 11θz z + W θ z + W 3θ z3 6 +, 3.9 z and z are local coordinates for center manifold C in the direction of q and q. Note that W is real if u t is real. We only consider real solutions. For solution u t C of 3.6, since µ =, we have żt = iω τz + q θ, f, W z, z, θ + Re{zqθ} = iω τz + q f, W z, z, + Re{zq} := iω τz + q f z, z. We rewrite the equation as where gz, z = g z It follows from 3.8 and 3.9 that u t θ = W t, θ + Re{ztqθ} żt = iω τzt + gz, z, + g 11z z + g z + g 1 z z = W θ z + W 11θz z + W θ z + 1, α T e iω τθ z + 1, ᾱ T e iω τθ z +,

13 which together with 3.4 gives gz, z = q f z, z = q f, u t = D β u, 1 τ 1 u du 1 u + cu 1 1u 1 = τ D β u 1 u + τ D du 1 u + cu 1 1u 1 = τ D β [ ] z + z + W 1 z + W 11z z 1 + W 1 z + ] [α z + ᾱ z + W z + W 11z z + W z + + τ D { ] d [z + z + W 1 z + W 11z z 1 + W 1 z + ] [α z + ᾱ z + W z + W 11z z + W z + ] +c [e iω τ z + e iω τ z + W 1 1 z + W 11 1z z 1 + W 1 1 z + ]} [α e iω τ z + ᾱ e iω τ z + W 1 z + W 11 1z z + W 1 z + = [ τ D β α τ Ddα + τ Dcα ] z e iω τ [ + τ D β Re{α } τ DdRe{α } + τ DcRe{α ] } z z + [ τ D β ᾱ τ Ddᾱ + τ Dcᾱ ] z e iω τ + [ τ D β ᾱ W 1 + W + W W11 τ Dd ᾱ W 1 + W + W W11 + τ Dc ] z z ᾱ e iω τ W e iω τ W 1 + α e iω τ W e iω τ W11 1 = τ Dα z β d + ce iω τ + τ DRe{α } β d + c z z + τ Dᾱ z β d + ce iω τ + τ D [ β d ᾱ W 1 + W + W W11 ] z z +c ᾱ e iω τ W e iω τ W 1 + α e iω τ W e iω τ W Comparing the coefficients with 3.1, we have g = τ Dα [ β d + ce iω τ ] g 11 = τ DRe{α }[ β d + c] g = τ Dᾱ [ [ β d + ce iω τ ] g 1 = τ D β d ᾱ W 1 + W + W W11 ] +c ᾱ e iω τ W e iω τ W 1 + α e iω τ W e iω τ W11 1. Since there are W θ and W 11 θ in g 1, we still need to compute them. From 3.6 and 3.8, we have

14 { AW Re{ q Ẇ = u t żq z q = f qθ}, θ [ 1,, AW Re{ q f q} + f, θ =. := AW +Hz, z, θ, 3.14 where Hz, z, θ = H θ z + H 11θz z + H θ z Substituting the corresponding series into 3.14 and comparing the coefficients, we obtain A iω τw = H θ, AW 11 = H 11 θ, From 3.14, it is easy to see that for θ [ 1,, Hz, z, θ = q f qθ q f qθ = gz, zqθ ḡz, z qθ Comparing the coefficients with 3.15 gives that and It follows from 3.16,3.18 and the definition of A that Note that qθ = 1, α T e iω τθ. Hence H θ = g qθ ḡ qθ, 3.18 H 11 θ = g 11 qθ ḡ 11 qθ Ẇ θ = iω τw θ + g qθ + ḡ qθ, W θ = ig ω τ qeiω τθ + iḡ 3ω τ qe iω τθ + E 1 e iω τθ, 3. where E 1 = E 1 1, E 1 R is a constant vector. Similarly, it follows from 3.16 and 3.19 that W 11 θ = ig 11 ω τ qeiω τθ + iḡ 11 ω τ qe iω τθ + E, 3.1 where E = E 1, E R is also a constant vector. In what follows, we shall seek appropriate E 1 and E. From the definition A and 3.16, we obtain dηθw θ = iω τw H θ, 3. 1 and dηθw 11 θ = H 11 θ, where ηθ = η, θ. By 3.14, we have α H = g q ḡ q + τ dα + cα e iω τ,

15 and Note that and H 11 = g 11 q ḡ 11 q + τ Re{α } dre{α } + cre{α e iω τ } iω τi e iω τθ dηθ q =, 1 iω τi Substituting 3. and 3.4 into 3., we obtain which leads to iω dy cy e iω τ 1 iω τi e iω τθ dηθ E 1 = τ 1 e iω τθ dηθ q =. x iω + dx + e b + cx e iω τ Solving Eq. 3.6, we have E 1 1 = A 1 det α dα + cα e iω τ where E 1 = A 1 det A 1 = det iω dy cy e iω τ α dα + cα e iω τ E 1 = τ,. 3.5 α dα + cα e iω τ x iω + dx + e b + cx e iω τ iω α dy cy e iω τ dα + cα e iω τ, x iω + dx + e b + cx e iω τ Similarly, substituting 3.1 and 3.5 into 3.3 gives x Re{α E dy cy dx + e b + cx = } dre{α } + cre{α }., Solving Eq. 3.7, we have E 1 = A det Re{α } x dre{α } + cre{α } dx + e b + cx where E = A det A = det Re{α } dy cy dre{α } + cre{α } x dy cy dx + e b + cx.,, 15

16 Thus, we can determine W θ and W 11 θ from 3. and 3.1. Furthermore, g 1 in 3.13 can be expressed by the parameters and delay. Therefore, we can compute the following values: c 1 = i ω τ [g g 11 g 11 g ] µ = Re{c 1} Re{λ τ}, β = Re{c 1 }, T = Im{c 1} + µ Im{λ τ} 3, + g 1, 3.8 ω τ which determine the quantities of bifurcating periodic solutions at the critical value τ, i.e., µ determines the directions of the Hopf bifurcation: if µ > µ <, then the Hopf bifurcation is supercritical subcritical and the bifurcating periodic solutions exist for τ > ττ < τ; β determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions in the center manifold are stable unstable if β < β > ; and T determines the period of the bifurcating periodic solutions: the period increase decrease if T > T <. Further, it follows from.15 and 3.8 that the following results about the direction of the Hopf bifurcations hold. Theorem 3.1. Assume that d > c and b > e hold. Then the Hopf bifurcations of. at E and τ = τ + j are supercritical respectively subcritical if Rec 1 < respectively Rec 1 >. However, the directions of the Hopf bifurcations. at E and τ = τ j is τ < τ j respectively τ > τ j if Rec 1 < respectively Rec 1 >. Theorem 3.. If d < c, b < e and τ = τ + hold, or d > c, b > e and τ = τ + j, j =, 1,, k and τ = τ j, j =, 1,, k 1, then the bifurcating periodic solution is stable if Rec 1 < and unstable if Rec 1 >. 4 Global existence of periodic solutions This section studies the global continuation of periodic solutions bifurcating from the point E, τ + j of system.. To this end, we need some preliminary results. Recall that system 1.5 can be transformed into. by.1. Moreover, by.1 and 1.4, the initial conditions of. take the form { xs = ϕ1 s, ϕ1 s >, s [ τ, ], ỹs = ϕ 4.1 s, ϕ s >, s [ τ, ], where ϕ = ϕ 1, ϕ C[ τ, ], R +, i = 1,. Thus, we have Lemma 4.1. Solutions of system. with initial conditions 4.1 are positive for all t. 16

17 Proof. Let xt, ỹt be a solution of system. with initial conditions 4.1. Let us consider ỹt for t [, τ]. It follows from the second equation of system. that ỹt = b ϕ t τ + c ϕ 1 t τ ϕ t τ d xtỹt eỹt d xtỹt eỹt, for t [, τ]. By comparison argument, it follows that ỹt ỹ exp{ t [ d xs e]ds} >, for t [, τ]. In a similar way we treat the intervals [τ, τ], [τ, 3τ],, [nτ, n + 1τ], n N, thus, ỹt for t. On the other hand, it follows from the first equation of system. that This completes the proof. xt = x exp{ t [ỹs a]ds} >, for t. Lemma 4.. If d c and b e/d c >, then all the nontrivial p-periodic solutions of system. are uniformly bounded, where p is an arbitrary bounded constant. Proof. Let xt = e v1t, ỹt = e vt, then system. can be rewritten as { v1 t = a + e vt, v t = be v t τ v t + ce v 1t τ+v t τ v t de v1t e. 4. To prove that all nontrivial periodic solutions of system. are uniformly bounded, it suffices to prove that all periodic solutions of system 4. are uniformly bounded. To achieve this goal, let vt = v 1 t, v t be any p-periodic solution of system 4.. Note that v itdt =, i = 1, due to the p-periodicity of vt. Integrating 4. over [, p] gives and [ ] be v t τ v t + ce v 1t τ+v t τ v t dt = d e v t dt = ap, 4.3 e v 1t dt + ep. 4.4 It follows from the first equation of system 4. and 4.3 that v 1 t dt e vt dt + ap = ap. 4.5 Similarly, it follows from the second equation of system 4. and 4.4 that v t [ ] dt be v t τ v t + ce v 1t τ+v t τ v t dt + d e v1t dt + ep d e v1t dt + ep

18 Noticing that ev t v tdt =. Multiplying 4. by e vt on both sides of the second equation of 4. and integrating over [, p], we have b e v t τ dt + c e v 1t τ+v t τ dt = d By the p-periodicity of vt, it is easy to show that e v 1t+v t dt + e e v t dt. 4.7 e v t τ dt = It follows from 4.7 and 4.8 that e v t dt, e v 1t τ+v t τ dt = e v 1t+v t dt. 4.8 b which implies e v t dt + c d c e v 1t+v t dt = d e v 1t e v t dt = b e e v 1t+v t dt + e e v t dt, e v t dt. 4.9 Since vt = v 1 t, v t is p-periodic, there exist ξ i, η i [, p], i = 1, such that It follows from 4.9 and 4.1 that v i ξ i = min v i t, v i η i = max v i t, i = 1,. 4.1 t [,p] t [,p] d ce v 1ξ 1 e v t dt b e Since d c and b e/d c >, it follows that Similarly, we have v 1 ξ 1 ln v 1 η 1 ln Thus, it follows from 4.5 and 4.11 that and v 1 t v 1 ξ 1 + v 1 t v 1 η 1 + From 4.13 and 4.14, we obtain v 1 t max { ln { b e } d c e v t dt. { b e } d c { b e }. 4.1 d c { b e } v 1 t dt ln + ap, 4.13 d c { b e } v 1 t dt ln ap d c + ap, b e } } ap := B d c 18

19 On the other hand, it follows from 4.6 and 4.15 that v t dt d Moreover, from 4.3 and 4.1, it is easy to see that and Thus, it follows from 4.16 and 4.17 that and v t v ξ + v t v η 1 + e v1t dt + ep de B 1 + ep e v 1ξ p ap, or v 1 ξ 1 ln a, 4.17 e v 1η p ap, or v 1 η 1 ln a v t dt ln a + de B 1 + ep, 4.19 v t dt ln a de B 1 + ep. 4. From 4.19 and 4., we obtain { { } v t max ln ln a + de B 1 + ep, ln a de B 1 + ep := B. 4.1 Taking B = max{b 1, B }, we obtain vt B, for any p-periodic solution vt. Although the boundaries of vt depend on the value of p, p is bounded. Therefore, all periodic solutions of 4. are uniformly bounded. Consequently, all periodic solution of. are uniformly bounded. This ends the proof of Lemma 4.. Lemma 4.3. If d c and b e/d c >, system. has no nontrivial τ-periodic solution. Proof. To the contrary, suppose that system. has a τ-periodic solution. Then the system of ordinary differential equations { ẋt = xt[yt a], 4. ẏt = yt[b e d cxt], has a periodic solution. System 4. has the same equilibria as system., which are z =,, z = x, y = b e d c, a. Note that x-axis and y-axis are the invariable manifold of system 4. and the orbits of system 4. do not intersect each other. Thus, there are no solutions crossing the coordinate axes. On the other hand, note the fact that if system 4. has a periodic solution, then there must be an equilibrium in its interior, and that z is located on the origin. Thus, we conclude that the periodic orbit of system 4. must lie in the first quadrant. It is well 19

20 known that the positive equilibrium z is globally asymptotically stable in the first quadrant. Thus, there is no periodic orbit in the first quadrant too. The above discussion means that 4. has no nontrivial periodic solution. It is a contradiction. Therefore, Lemma 4.3 is proved. Below, we follow closely the notations in [5]. For simplification of notations, setting z t = x t, y t, we may rewrite systems. as the following functional differential equation zt = F z t, τ, p, 4.3 where z t θ = zt + θ C[ τ, ], R. From the discussion in Section, we know that system 4.3 has two equilibria z = E, and z = E x, y. If d c and b e/d c >, z is the unique positive equilibrium. In order to apply Theorem 3.3 in [5], define X = C[ τ, ], R, Σ = Cl{z, τ, p X R R + : z is a p-periodic solution of 4.3} N = {z, τ, p : F z t, τ, p = }, and let lz, τ + j, π/ω + denote the connected component of z, τ + j, π/ω + in Σ, τ + j and ω + are defined by.1 and.13, respectively. Theorem 4.4. If d > c and b > e, then for each τ > τ + j at least k 1 periodic solutions., j =, 1,,, k 1, system. has Proof. It suffices to prove that the projection of lz, τ + j, π/ω + onto τ-space is [ τ, for each j 1, where τ τ + j. The characteristic matrix of 4.3 at an equilibrium z = x, ỹ R takes the following form: z,τ,p λ = λi DF z t, τ, pe λ I, where I is an identity matrix. Consequently, the characteristic equation of 4.3 at z is λ + pλ + r + sλ + qe λτ, 4.4 where p = d x+e ỹ +a, r = d xỹ d x+eỹ a, s = b+c x, q = [c xỹ b+c xỹ a]. z, τ, p is called a center if F z, τ, p = and det z,τ, p i π =. A center z, τ, p is said to p be isolated if it is the only center in some neighborhood of z, τ, p. If z = z, Eq. 4.4 reduces to λ + aλ + e be λτ =. 4.5 Obviously, λ 1 = a is a negative root of 4.5. Suppose that iω is a purely imaginary root of F λ =. Rewrite F λ = in terms of its real and imaginary part as { e b cosωτ =, 4.6 ω + b sinωτ =,

21 which implies ω = b e. Therefore, if b < e, then ω <. Consequently, Eq. 4.5 has no any purely imaginary roots. Under the assumption in Theorem 4.4, system 4.3 has no any center of the form z, τ, p. On the other hand, from the discussion about the local Hopf bifurcation in Section, it is easy to verify that z, τ + j, π ω + is also a isolated center. By Theorem.6, there exist ε >, δ > and a smooth curve λ : τ + j δ, τ + j +δ C such that det λτ =, λτ ω + < ε for all τ [τ + j δ, τ + j + δ] and Let λτ + j = iω +, dreλτ dτ τ=τ + j >. Ω ε,π/ω+ = {η, p : < η < ε, p π/ω + < ε}. It is easy to verify that on [τ + j δ, τ + j + δ] Ω ε,π/ω+, det z,τ,pη + πi/p = if and only if η =, τ = τ + j, and p = π/ω +. Therefore, the hypotheses A 1 A 4 in [5] are satisfied. Moreover, if we define H ± z, τ, π/ω + η, p = det z,τ + j ±δ,pη + πi/p, then we have the crossing number of isolated center z, τ + j, π/ω + as follows: γz, τ + j, π/ω + = deg B H z, τ + j, π/ω +, Ω ε,π/ω+ deg B H + z, τ + j, π/ω +, Ω ε,π/ω+ = 1. Thus, we have z, τ, p lz,τ + j,π/ω + γ z, τ, p <, where z, τ, p, in fact, takes the form of z, τ + j, π/ω + j =, 1,,. It follows from Theorem 3.3 in [5] that the connected component lz, τ + j, π/ω + through z, τ + j, π/ω +in Σ is unbounded. From.13, we have τ + j = 1 { hq ω arccos + + mgω } + + jπ, j =, 1,,. ω + g ω+ + h ω + Thus, when j >, we have π/ω + < τ + j. Now we prove that the projection of lz, τ + j, π/ω + onto the τ-space is [ τ,, where τ τ + j. Clearly, it follows from the proof of Lemma 4.3 that system. with τ = has no nontrivial periodic solution. Hence, the projection of lz, τ + j, π/ω + onto the τ-space is away from zero. For a contradiction, we suppose that the projection of lz, τ + j, π/ω + onto the τ-space is 1

22 bounded. This means that the projection of lz, τ + j, π/ω + onto the τ-space is included in interval, τ. Noticing that π/ω + < τ + j and applying Lemma 4.3, we have < p < τ for zt, τ, p belonging to lz, τ + j, π/ω +. This implies that the projection of lz, τ + j, π/ω + onto the p-space is bounded. Then, applying Lemma 4. we get that the connected component lz, τ + j, π/ω + is bounded. This contradiction completes the proof. 5 Example and simulations Example 5.1 Consider the following system: x t = xtỹt 1 xt, ỹ t = ỹt τ + xt τỹt τ xtỹt 1 ỹt, 5.1 Simple computation shows that system 5.1 has a positive equilibrium E =.5,.5. And p = 3/, s = 3/, r = 1/ and q = 1/4. So, it is the critical case in view of p + s =. By the formula.13, we can determine that τ + = 3.676, τ = 4.996,. Thus, the positive equilibrium E switches from instability to stability see Fig. 1 and Fig 3. The bifurcation occurring at critical value τ + takes place when τ crosses τ + to the left see Fig t x plane 3.5 t y plane Figure 1: When < τ =.1 < τ +, the positive equilibrium E is unstable.

23 1 t x plane 1 t y plane x y plane Figure : When < τ =.15 < τ +, the bifurcating periodic solution from E occurs. 1.4 t x plane t y plane Figure 3: When τ = 3.8 > τ +, the positive equilibrium E is asymptotically stable. 3

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