HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS

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1 Journal of Applied Analysis and Computation Volume 7 Number August Website: DOI:.948/766 HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS Jingnan Wang and Weihua Jiang Abstract In this paper We investigate Hopf-zero bifurcation with codimension in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore using the Faria and Magalhases normal form method and the center manifold theory we obtain the third order degenerate normal form with two original parameters. Finally through theoretical analysis and numerical simulations we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system and explain the coexistence phenomena of several locally stable states such as the coexistence of multiperiodic orbits as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit. Keywords Predator-prey model with dormancy of predators Hopf-zero bifurcation time delay stability periodic orbit. MSC) 4K8 7G5 7G 9D5.. Introduction In 9 Kuwamura etc [6] proposed a minimum mathematical model of predatorprey systems with dormancy of predators to explain the paradox of enrichment. In the same year Kuwamura etc [7] used the theory of fast-slow systems to demonstrate the mixed-mode oscillations and chaos bifurcated from a coexistence equilibrium. In based on the model in Kuwamura etc [6 p46] taking account of the effects of feedback time delay of the prey growth we proposed a delayed model see Wang etc [9 p54]) with a nondimensionalized change of similar variable transforms see Wang etc [9 p54]) and obtained the following dimensionless the corresponding author. address: wangjingnan@hrbust.edu.cnj. Wang) Department of Mathematics Harbin Institute of Technology Harbin 5 China Department of Applied Mathematics Harbin University of Science and Technology Harbin 58 China The authors were supported by National Natural Science Foundation of China No.7 No.7) the Heilongjiang Provincial Natural Science Foundation No.A8 No.QC4C) and the Foundation of Heilongjiang Educational Committee No. 5468).

2 5 J. Wang & W. Jiang system: xt τ) mxy ẋ = )x N + x ẏ =[ + tanhσx η))] mxy + x + ϑαz d y ż =[ tanhσx η))] mxy + x α + d )z..) where τ is the nondimensionalized change of T which denotes the feedback time delay of the prey to the growth of the species itself; and d is the nondimensionalized change of the mortality rates of the active predatorsee Kuwamura etc [6 p46] and Wang etc [9 p54] for more details about the model). In Wang etc [9] we used time delay as a parameter to discuss the global dynamical behavior such as the stability of the coexistence equilibrium Hopf bifurcation and the chaos of system.). In addition we found some other phenomena: the coexistence of three periodic orbits shown in Figs 4 and 5 in Wang etc [9 p ]; as well as the coexistence of a locally stable coexistence equilibrium and a locally stable periodic orbit When τ =.6 < τ ) =.679 the coexistence equilibrium E from the initial value ) is locally asymptotically stable shown as Fig of Wang etc [9 p547]; at the same time the global Hopf bifurcation graph from E τ ) initial value shown in Fig of Wang etc [9 p547] ). ) shows that there exists a local stable periodic orbit from the additional Since the coexistence of three periodic orbits is an uncommon phenomenon and may lead to chaos of the system the cause of the coexistence of three periodic solutions coexist needs to be further studied. In Wang etc [9] based on theoretical analysis we know that among three periodic orbits one is bifurcated from a coexistence equilibrium; while how the other two periodic orbits are generated remains unknown. In this paper we introduce two bifurcation parameters to find some theoretical explanations for the coexistence of multi-periodic orbits as well as the coexistence of a stable equilibrium and a stable periodic orbit. From some research papers we know that through the analysis of Hopf-zero bifurcation of dynamic systems some complex behaviormulti-periodic phenomenon quasi-periodic phenomenon even chaotic phenomenon) in predator-prey systems with delay [8] neural systems [49] oscillator systems [4] and financial system Ding etc [5] have been revealed. In this paper according to the existence condition and the corresponding characteristic equation of the coexistence equilibrium we choose the mortality rate and time delay τ as two bifurcation parameters to discuss the occurrence condition of Hopf-zero bifurcation of system.). In addition by analyzing the structure of its normal form we discuss whether Hopf-zero bifurcation of the system may lead to the coexistence of the three periodic solutions in Wang etc [9 p547]. The rest of the article is organized as follows. In Sect. we give and prove the existence condition of the coexistence equilibrium. Furthermore in Sect. we obtain the occurrence conditions of Hopf-zero bifurcation from the boundary equilibrium N ). In Sect. 4 we obtain the normal forms of the Hopf-zero bifurcation with the original parameters of the system. In Sect. 5 we perform some bifurcation analysis and numerical simulation which are shown to verify the theoretical predictions. In Sect. 6 biological implications of our results are given.

3 Hopf-zero bifurcation of a delayed predator-prey model Existence conditions of coexistence equilibrium Clearly there is always a trivial equilibrium E = ) and the boundary equilibrium E = N ) in system.). If there exists an the coexistence equilibrium E = x y z ) in system.) then the coexistence equilibrium satisfies the following equations d = mx + x [tanhσx η)) + + ϑα α + d tanhσx η)))]; y = x + x ) N m ; z = m tanhσx η))) x y α + d + x..) Biologically the coexistence equilibrium indicates a mode of persistent coexistence of the predator and prey. In this paper since z denotes the density of predators with dormant state resting eggs) the coexistence equilibrium of system.) implies that x > y > z. In addition from the mathematical point of view the premise of studying equilibrium stability and bifurcation lies in the proof of the equilibrium existence. Thus the proof of the existence of the coexistence equilibrium in Wang etc [9 p54] needs to be added into our discussion. Next we discuss the existence condition of the coexistence equilibrium E = x y z ). According to expression.) let gx) = mx + x [α + d )tanhσx η)) + ) + ϑα tanhσx η))]..) Through analysis we obtain the following results: Proposition.. If < d < gn)/α + d ) then there exists a unique coexistence equilibrium E = x y z ) in system.) where < x < N x y and z are defined in.). Proof. Suppose that there exists a coexistence equilibrium E = x y z ). Then E must satisfy.) if x >. We derive that gx) satisfies If < d < gn)/α + d ). Then g x) > mϑα/ + x) >. gn) > α + d )d >. Since g) = there must be a unique x satisfying < x < N and gx ) = α + d )d. Based on the expression of y z and the inequality < x < N we know that y > and z. Therefore there exists a unique coexistence equilibrium E = x y z ) in system.). The proof is complete.. Occurrence condition of Hopf-zero bifurcation If system.) undergoes Hopf-zero bifurcation at an equilibrium then its corresponding characteristic equation has a zero root except for a pair of imaginary

4 54 J. Wang & W. Jiang roots. In Wang etc [9 p54] the corresponding characteristic equation with the linearization of system.) at E has the following form λ + ã λ + ã λ + ã ) + b λ + b λ)e λτ =.) where ã = α + d + d mx +x [tanhσx η)) + ] mx y +x ) ã = mx y +x ) {σmx [ tanh σx η))] α + d + d ) + m[tanhσx η)) + ]} ã = my +x ) {x σmx [ tanh σx η))]α + d θα) + d α + d )} b = x N b = x N {α + d + d mx +x [tanhσx η)) + ]}. If equation has a zero root λ = then ã =. Since x σmx [ tanh σx η))]α + d θα) + d α + d ) > with the parameter values N =.5 m =. σ = η =.5 d =.4 θ =. α =.8 and d =.5 in Wang etc [9 p546] we know if ã = then y = which is contradictory to y =.94. That is to say if system.) undergoes Hopf-zero bifurcation at an equilibrium then the second coordinate of the equilibrium is zero. Therefore we study the following characteristic equation with the linearizations of system.) at E and E respectively. and λ )λ + d )λ α d ) =.) λ+e λτ )λ +α+d +d mn + N [tanhσn η))+])λ+d α+d ) gn)) =..) Since α > d > equation.) has a negative root and two positive roots for an arbitrary τ thus E is unstable when τ. If d = gn)/α+d ) holds then equation.) has a zero root; and if τ = π + kπ k = ± ± ) holds then equation.) has a pair of pure imaginary roots. By direct calculation we obtain: Proposition.. If τ < π/ and d > gn)/α + d ) hold then the equilibrium E of system.) is locally stable; if d = gn)/α + d ) holds then system.) undergoes a Hopf-zero bifurcation at the equilibrium E when τ = π/ + kπ where k = ± ± and gn) is defined in.) when x = N. Proof. If τ < π/ and d > gn)/α + d ) hold then all the solutions of equation.) have negative real parts so the equilibrium E of system.) is locally stable. By Corollary. in Ruan etc [8 p867] and.) we know that if d = gn)/α + d ) and τ = π/ hold then all the roots of equation.) have negative real parts except a zero and a pair of pure imaginary eigenvalues. Thus we obtain the desired results. The proof is complete. From Proposition. we want to know the dynamic behavior of the boundary equilibrium N ). Then we need to further investigate the properties of Hopfzero bifurcation of singularity N ).

5 Hopf-zero bifurcation of a delayed predator-prey model Normal form of Hopf-zero bifurcation In order to study the properties of Hopf-zero bifurcation of N ) we perform the center manifold reduction and normal form computation for system.). First let ˆx = x N ŷ = y ẑ = z and still denote ˆxŷẑ by x y z. By rescaling the time by t t/τ to normalize the delay and by expanding the functions tanhσx η)) and xy/ + x) in system.) we obtain where ẋ = a τy τxt ) τxt)xt )/N a τxy + a τx y + h.o.t. ẏ =τa b d )y + τϑαz + τa b + Nb )xy τa b b b )x y + h.o.t. ż =τa b )y α + d )z + a b Nb )xy a b + b + b )x y) + h.o.t. a = mn + N ; a = m + N) ; a = m + N) b = + tanhσn η)); b = σ + N) tanh σn η))) b =σ N + N) tanhσn η))tanh σn η)) ). 4.) Next we introduce two bifurcation parameters by d = a [b + b)ϑα α+d ] + µ and τ = π + µ in Eq. 4.) and denote µ = µ µ ). Select C = C[ ] R ) to denote the space of all the continuous functions from [ ] to R. For each X t C define X t θ) = Xt+θ) and X t = as where and F X t µ) = sup <θ< X t θ). Then Eq. 4.) can be written Ẋt) = Lµ)X t + F X t µ) 4.) a π + µ )y π + µ )xt ) Lµ)X t = π + µ ) b )aϑα α+d µ )y + π + µ )ϑαz π + µ )a b )y π + µ )α + d )z N π + µ)xt)xt ) a π + µ)xy + a π + µ)x y π + µ)ab + Nb)xy π + µ)ab b b)x y π + µ)a b Nb)xy π + µ)a b + b + b)x y + h.o.t. Let µ µ ) = ) for ϕ C[ ] R ) L ϕ = dηθ)ϕξ)dξ and select π A + A ) θ = ηθ) = π A θ ) θ = with a A = a b d ϑα ; A =. a b ) α + d )

6 56 J. Wang & W. Jiang Then the linearization equation of 4.) at the trivial equilibrium is and the bilinear form on C C is ψs) ϕθ)) =ψ)ϕ) Ẋt) = L X t θ ψξ θ)dηθ)ϕξ)dξ =ψ)ϕ) + π ψ ξ + )ϕ ξ)dξ ψ s) where ϕθ) = ϕ θ) ϕ θ) ϕ θ)) C Ψs) = ψ s) C. Then the space ψ s) C is decomposed by Λ = { ± π i} into C = P + Q where Q = {ϕ C : ψ ϕ) = for all ψ P } and the bases for P and its adjoint P are e π iθ e π iθ Φθ) = a θ b α+d and D α + d ) D ϑα ) ) Ψs) = D α + d a b )ϑα α+d i e π is D a α + d + i)e π is D a ϑαe π is ) ) D + α + d a b )ϑα α+d i e π is D a α + d i)e π is D a ϑαe π is s where a α + d ) D = α + d ) + ϑαa b ) { D = + π [ α + d a b )ϑα α + d ] [ π + α + d a )] b )ϑα i} α + d D = D. Thus the dual bases satisfy Φ = ΦB and Ψ = BΨ with B = π i π i D α + d ) D ϑα ) ) Ψ) = D α + d ab )ϑα α+d i D a α + d + i) D a ϑα ) ) D + α + d ab )ϑα α+d i D a α + d i) D a ϑα ψ ψ ψ = ψ ψ ψ. ψ ψ ψ

7 Hopf-zero bifurcation of a delayed predator-prey model As in Faria etc [6 p5] we enlarge the phase space C into { } BC := ϕ : [ ] R is a continuous function on [ )and lim ϕθ) exists. θ In BC Eq. 4.) becomes an abstract ODE where u C and A is defined by d dt u = Au + Y F u µ) 4.) A : C BC Au = u + Y [L u u)] and F u µ) = [Lµ) L ]u + F u µ) where Y is the matrix-valued function defined by { I θ = Y θ) = θ <. By the continuous projection G : BC P Gϕ + Y c) = Φ[Ψ ϕ) + Ψ)c] we can decompose the enlarged phase space by Λ = { ± π i} into BC = P KerG. Let u t = Φ xt) + ỹθ) where xt) = x x x ) T. Therefore Eq.4.) is decomposed into the system x = B x + Ψ) F Φ x + ỹθ) µ) 4.4) ỹ = A Q ỹ + I G)Y F Φ x + ỹθ) µ) where ỹθ) Q := Q C KerG A Q is the restriction of A as an operator from Q to the Banach space KerG. By neglecting higher order terms with respect to parameters µ and µ Eq. 4.4) can be written as x = B x +! f x ỹ µ) +! f x ỹ µ) + h.o.t ỹ = A Q ỹ +! f x ỹ µ) +! f x ỹ µ) + h.o.t 4.5) where! f x ỹ µ) =! f x ỹ µ) =! f x ỹ µ) = I G)Y ψ F + ψ F + ψ F ψ F + ψ F + ψ F 4.6) ψ F + ψ F + ψ F ψ F + ψ F + ψ F ψ F + ψ F + ψ F 4.7) ψ F + ψ F + ψ F F F F ;! f x ỹ µ) = I G)Y F F F 4.8)

8 58 J. Wang & W. Jiang with F = a µ x ) + y ) a b 5 x + x + x + y ))x ix + ix + y )) µ x ix + ix + y )) b 6 x + x + x + y )) x ) + y ) a F =b 7 x + x + x + y )) x ) + y ) a F b )a ϑα = µ π ) α + d µ x ) + y ) a ) b + µ ϑα x + y ) + b x + x + x + y )) x ) + y ) α + d a F = b x + x + x + y )) x ) + y ) a F =µ a b ) x ) ) b + y ) µ α + d ) x + y ) a α + d + b x + x + x + y )) x ) + y ) a ) F =b 4 x + x + x + y )) x a + y ) b = πa b + Nb ) b = πa b b b ) b = πa b Nb ) b 4 = πa b + b + b ) b 5 = π N b 6 = πa b 7 = πa. We know that ImM )) c is spanned by the following elements see also Wang etc [ p]): {x e x x e x µ i e µ µ e x x e x µ i e x x e x µ i e } i = with e = ) T e = ) T e = ) T. Thus the normal form of 4.5) on the center manifold of the origin near µ µ ) = has the form ẋ = Bx +! g x µ) + with g x µ) = proj ImM )) cf x µ). To find the third order normal form of the Hopf-zero singularity we need to find the center manifold ỹ = h x) at first hence neglecting the impact of small parameters µ and µ in the third order). Let V C KerG)) be the space of homogeneous polynomials of degree in the variables x = x x x ) T with coefficients in C KerG). Define the operator M : V C KerG)) V C KerG)) M p h) = M p M h) D x p x)b x Bp x) D x h x)b x A Q h x)) where px) V C ) hx) V Q ). From decompositions 7) in Jiang etc [5 p6] the projection associated with the preceding decomposition of V C ) V G) over ImM ) ImM ) is

9 Hopf-zero bifurcation of a delayed predator-prey model denoted by P I = PI P I ). Set U x) = U U ) T = M P If x ) where U /! KerM ) c and U /! = h x) = h x) h x) h x)) T is the unique solution in V Q ) of the equation M U x) = f x ). Thus D x h x)b x ḣ x)θ) + X [ḣ x)) Lh x))] = f / 4.9) where ḣ denotes the derivative of hx)θ) with respect to θ. By a transformation of variables x ỹ) = ˆ x ˆỹ) + U ˆ x)/! the first equation of 4.5) becomes after dropping the hats x = B x +! g x µ) +! f x ) + where f x ) = f x ) +! [Df x y )) y= U x) DU x)g x )]. The value of U x) is found in the Appendix. Let M be the operator defined in V C KerG) with M : V C ) V C ) and M p) x) = D x p x)b x Bp x) where V C ) denotes the linear space of the third order homogeneous polynomials in three variables x x x ) with coefficients in C. Then one may select the decomposition V C ) = ImM ) ImM ) c with the complementary space ImM )) c spanned by the elements {x e x x x e x x e x x e x x e x x e }. The normal form of 4.5) on the center manifold of the origin near µ µ ) = up to the third order is x = B x +! g x µ) +! g x ) + with g x ) = proj ImM )) c f x ). Following Theorem.6 in Faria etc [7 p89] we know that the flow on the center manifold of equation 4.) tangent to P near X t = is governed by the following normal form ẋ = n µ x + b x + ex x x + fx + h.o.t. ẋ = π ix + m + ip )µ x + a + ia )x x +c + ic )x x + d id )x x + h.o.t. 4.) ẋ = π ix + m ip )µ x + a ia )x x +c ic )x x + d + id )x x + h.o.t..

10 6 J. Wang & W. Jiang where n = πψ b = A m + ip = iψ a + ia = A a a e = A + A a π f = A + A a π h ) ) + h ) + h ) + h ) a h ) ) + h ) + ia A A ) a πa c + ic = π { i ψ ψ b 5 ψ b 5 ih ) + ih ) + h ) + h )) + ψ b 6 + ψ b + ψ b )h ) + h ))} d id = ψ b 7 + ψ b ψ b 4 ) + a π {ia A ia A a + A ψ b 5 + h )) ψ b 5 ih ) + h ) + h )) + ψ b 6 + ψ b + ψ b )h ) + h )))}. see Appendix for the detailed calculation process) 5. Bifurcation analysis and numerical simulations To further analyze the bifurcation situation in 4.) we use the coordinate transformation x = Z x = r cos θ+ir sin θ x = r cos θ ir sin θ r >. Then we obtain the normal form of truncated to the third order in the cylindrical coordinates ṙ =m µ r + a rz + c r + d rz Ż =n µ Z + b Z + ezr + fz θ = π + p µ + a Z + c r d Z. 5.) Removing the azimuthal term system 5.) becomes ṙ =rm µ + a Z + c r + d Z ) Ż =Zn µ + b Z + er + fz ). 5.) If a = b = then Eq.5.) has distinct types of unfolding by linear transformation according to the framework by Guckenheimer etc [ p99]; if a and b then we further discuss the bifurcation with parameters µ and µ. In Eq. 5.) M = r Z) = ) is always an equilibrium and the other equilibria are M = M ± = M ± =r Z ) m µ c ) for m µ c b ± b ) 4fn µ f < ) for b ) 4fn µ >

11 Hopf-zero bifurcation of a delayed predator-prey model... 6 where Z = a e b c ± Θ c f d e) r = d Z ) + a Z + m µ c d Z ) + a Z + m µ c Θ = b c a e) 4c f d e)c n µ em µ ) >. < In the normal form 5.) it is quite difficult to estimate the sign of c n m thus we set up some assumptions which are c < n < m > to more conveniently analyze the existence of equilibria M M ± M ±. In addition we briefly obtain the following results: Theorem 5.. If c < n < and m > hold then i) System 5.) undergoes two pitchfork bifurcations and a transcritical bifurcation at the trivial equilibrium M respectively on the curves L = {µ µ ) : µ = b ) 4fn µ } L = {µ µ ) : µ = µ } and L = {µ µ ) : µ = µ < }; ii) System 5.) undergoes a pitchfork bifurcation at the half-trivial equilibrium M and M ± respectively on the curves L = {µ µ ) : µ = c n em µ µ > } and L ± 4 = {µ µ ) : µ = d Z ) a Z m }. Noticing that curve L ± 4 includes two curves L+ 4 and L 4 described as L ± 4 : µ = d n m f µ + d b a f m f b ± b ) 4fn µ ). According to the center manifold theory [ p9] Eq.5.) on the center manifold determines the asymptotic behavior of solutions of the entire equations.) when there exist no unstable manifolds containing the trivial solution. In addition the bifurcation analysis for the three-dimensional system 5.) is based on rotational symmetry. Since system 5.) rotates around the Z-axis the correspondences between -dimensional flows for 5.) and -dimensional flows for 5.) can be established. Thus for 5.) the equilibrium on the Z-axis in 5.) remain a equilibrium while the equilibria outside the Z-axis in 5.) become a periodic orbit. Based on Theorem 5. numerical simulation results in Wang etc [9 p546] and experimental results in [] we give an example and perform numerical simulations to show the relation between the obtained normal form 5.) with the original parameters and the original system.). Example 5.. If we choose N =.5 m =. σ = η =.5 d =.666+µ ϑ =. α =.8 d =.5 and τ = π/ + µ of system.) then system.) undergoes a Hopf-zero bifurcation at the equilibrium.5 ) when µ µ ) = ).

12 6 J. Wang & W. Jiang By direct calculation we obtain n =.578 b =.4 m +p i = i a + a i =.5 +.4i c + c i =.5.9i d d i = i e =.74 f =.58 and ṙ =.45µ r +.5rZ.5r +.99rZ Ż =.578µ Z +.4Z.74Zr.58Z. 5.) Clearly parameters c n and m in Eq.5.) satisfy the conditions of Theorem 5. The d τ) plane according toµ µ ) plane) is divided into eight regions see Fig.) by five lines L L ± 4 : L : d =.779 i.e. µ =. µ ; L : τ = π/ d.666 i.e. µ = µ ; L : d =.666 τ < π/ i.e. µ = µ < ; L : τ =.6645d.666) + π/ τ > π/ i.e. µ =.6645µ µ > ; L ± 4 : τ = d.666) ± d.666) i.e. µ = µ ± µ.. L D.8.6 L D D 8.4 D D 4 τ. D 7.8 M pitchfork bifurcation M transcritical bifurcation D 6.6 L + 4 M pitchfork bifurcation.4 M pitchfork bifurcation L 4 D 5 L M + pitchfork bifurcation..4.6 d Figure. Bifurcation sets around d τ) =.666 π/) with parameters near µ µ ) = ). According to the expressions of M M ± M ± and note that in Fig. an equilibrium with the component r corresponds to the nonconstant periodic solution of system.) we know that L is a pitchfork bifurcation curve of system 5.) and corresponds to a pitchfork bifurcation curve of system.); L is a pitchfork bifurcation curve of system 5.) and corresponds to a Hopf bifurcation curve of system.); L is a transcritical bifurcation curve of system 5.) and corresponds to a fold bifurcation curve of system.); L is a pitchfork bifurcation curve of an equilibrium in system 5.) and corresponds to a pitchfork bifurcation curve of limit cycles in system.); L ± 4 is a pitchfork bifurcation curve of system 5.) corresponds to a Hopf bifurcation curve of system.). According to the expressions of M M ± M ± and the bifurcation curves L L ± 4 we choose eight groups of parameter values µ µ ) =.66.9).66.9) ) ).66.78).4.878).74.8) and.54.8)which belong to the regions D D D D 4 D 5 D 6 D 7 and D 8 to theoretically analyze and numerically simulate the phase portrait with pplane7

13 Hopf-zero bifurcation of a delayed predator-prey model... 6 Z r M M M M M D D D5 D7 M M M M M D D4 D6 D8 Figure. Phase portraits of system 5.) with parameter µ µ ) around ) in D -D 8. program of Matlab and obtain the phase portraits Fig.) of system 5.) with parameterµ µ ) around ) in D -D 8 respectively.in D i i =... 8.) of Fig. the vertical axis represents the Z-axis in Eq. 5.) while the horizontal axis represents the r-axis in Eq. 5.). Therefore the detailed dynamics of system.) in D D 8 near the original parameters d τ) =.666 π/) around µ µ ) = ) without curve L are as follows: According Proposition. system.) has no coexistence equilibrium in D 6 D 7 and D 8. In D 5 µ < µ < µ µ system.) has an unstable equilibrium corresponding to M ) and two stable equilibriums corresponding to M and M + ) which are bifurcated from the equilibrium corresponding to M ) when parameters µ and µ pass through the pitchfork bifurcation curve L. In D 4 µ < µ µ < µ < µ µ a stable periodic orbit corresponding to M + ) in system.) appears when parameters µ and µ pass through the Hopf bifurcation curve L 4 a stable equilibrium corresponding to M + ) becomes unstable and other two equilibriums corresponding to M and M ) remain at the original states: one is stable and the other is unstable. In D µ < µ µ < µ < another stable periodic orbit corresponding to M ) in system.) appears when parameters µ and µ pass through the Hopf bifurcation curve L + 4. System.) has two stable periodic orbits corresponding to M + and M ) and three unstable equilibriums corresponding to M M and M + ). In D µ < < µ <.6645µ an unstable periodic orbit corresponding to M ) in system.) appears when parameters µ and µ pass through the Hopf bifurcation curve L the two stable periodic orbits corresponding to M + and M ) in D remain stable and three

14 z 64 J. Wang & W. Jiang equilibriums corresponding to M M and M + ) remain unstable in system.). In D µ <.6645µ < µ two stable periodic orbits corresponding to M + and M ) in system.) disappear when parameters µ and µ pass through the Hopf bifurcation curve L and the unstable periodic orbit corresponding to M ) in D becomes stable. In fact we can also regard our approach like this : in D a stable periodic orbit corresponding to M ) in system.) appears when parameters µ and µ pass through the Hopf bifurcation curve L. Then the stable periodic orbit corresponding to M ) in D bifurcates two stable periodic orbits corresponding to M + and M ) in D when parameters µ and µ pass through the pitchfork bifurcation of periodic solution ) curve L and the periodic orbit corresponding to M ) becomes unstable. We select three groups of parameter values: µ µ ) = ) in D µ µ ) = ) in D 4 and µ µ ) =.66.78) in D 5 to simulate systems.) see Figs. 4 and 5). In Fig. there are two locally sable periodic orbits corresponding to M + and M respectively. In Fig.4 there is a locally stable equilibrium corresponding to M and a locally stable periodic orbit corresponding to M + in D 4. In Fig.5 there are two locally sable equilibriums corresponding to M + and M respectively. initial value ) initial value ).5 x y z.4. z t t.5 y.5..4 x.6.8 y.5 x.5 Figure. Waveform plots and phase planes of example 5. with the initial values ) and ) when µ µ ) = ). There are two locally stable periodic orbits. x initial value ) initial value.46.6) initial value ) initial value.46.6) y z t 5 t z y.5 x.5 Figure 4. Waveform plots and phase plane of example 5. with the initial values ) and.46.6 ) when µ µ ) = ). There is a locally stable equilibrium and a locally stable periodic orbit.

15 Hopf-zero bifurcation of a delayed predator-prey model x initial value.5.5 ) initial value.5) initial value.5.5) initial value.5) y.5 z 5 5. z.. 5 t 5 t y..5 x.5.5 Figure 5. Waveform plots and phase plane of example 5. with the initial values.5.5 ) and.5 ) when µ µ ) =.66.78). There are two locally stable equilibriums. 6. Conclusion In this paper we take two system variables of.) as two bifurcation parameters. One is the feedback time delay of prey growth leading to the occurrence of Hopf bifurcation to explain the existence of periodic orbits in the system; the other is the mortality rates of the active predator leading to the occurrence of fold bifurcation to explain the coexistence equilibrium and other coexistence of stable states. By applying the equilibrium characteristic equation we deduce that system.) undergoes Hopf-zero bifurcation at the boundary equilibrium under certain conditions. Using the delay τ and the mortality rate d as two parameters we obtain the occurrence condition of the Hopf-zero bifurcation and a corresponding third order normal form. Moreover according to the specific parameter situation in Wang etc [9 p546] we give the bifurcation set and the phase diagram in Section 5 and obtain that the Hopf-zero bifurcation can lead to the coexistence of three periodic orbits as well as the coexistence of a locally stable coexistence equilibrium and a locally stable periodic orbit. In addition we find another bistable state that is the coexistence of two locally stable equilibriums. The different aspects between the bistable state and the monostable state are the different degrees of sensitivity property of the system. That is to say system.) with the bistable state is extra sensitive to the initial value and the variables of time delay τ and mortality rate d. Numerical simulation results show that under the same numerical parameters different initial values have different dynamic behavior. Biologically if the feedback time delay of prey growth is near π/ and the mortality rates of the active predator is near the critical value the states of the predator-prey system are determined sensitively by the different initial densities of system.). By Proposition. we know that when the mortality rates of active predator d is greater than gn)/α + d ) the predator will tend to die that is the corresponding equilibrium N ) is stable; when the mortality rates of active predator d is less than gn)/α + d ) there is the persistent coexistence state of the predator and prey corresponding to the existence of coexistence equilibrium x y z ). In D 4 the persistent coexistence state of the predator and prey in D 5 are destroyed the population of the predator and prey periodically oscillates to maintain persistent coexistence corresponding to the stable periodic orbit because when the parameter value of τ increases and the parameter value of d reduces which means that the predator quickly eats up prey. Due to the lack of food the number of the predator population reduces which leads

16 66 J. Wang & W. Jiang the prey population to slowly recover and this cycle goes on continuously thus the number of the predator population and that of the prey population will periodically change. The immigration of preys or predators can lead to different amplitude population changes of preys or predators which corresponds to the theoretical results in D. The sensitivity properties of the system reflects that the population of predator and prey is affected by the outside condition and the season. That is to say temperature oxygen conditions light and location are the most important factors controlling egg development and dormancy in organisms which induces the complexity of dynamics of the predator-prey system. Acknowledgments The authors wish to express their special gratitude to the editors and the reviewers for the helpful comments given for this paper. Appendix: Derivation of third-order normal form ) Computation of U /! KerM ) c. From 4.5) it follows that p I A a x x A a x x! f ) = A x + iψ b 5 x iψ b 5 x + A x x. A x + iψ b 5 x iψ b 5 x + A x x Therefore according to U /! = M [p I! f )] we have where i U /! = A a x x i A a x x ia π x + ψ b 5 x + ψ b 5 x + i A x x ia x + ψ b 5 x + ψ b 5 x i A x x A =ψ b + ψ b ; A = ψ b ψ b 4 [ A = ψ b 5 ] ψ b 6 + ψ b + ψ b ) a A =A + iψ b 5 ; A = A iψ b 5 [ A = ψ b 5 ] ψ b 6 + ψ b + ψ b ) a A =A + iψ b 5 ; A = A iψ b 5. ) Computation of U /! = h x) = h x) h x) h x)) T V Q ).

17 Hopf-zero bifurcation of a delayed predator-prey model From Eq.4.9) we have! f x ) = I G)Y F = Y Φθ)Ψ))F = e π iθ ψ + e π iθ ψ ψ + e π iθ ψ + e π iθ ψ ψ + e π iθ ψ + e π iθ ψ Y ψ a ψ a ψ b ) α+d ψ b ) α+d b 5 x + e π i x + e π i x )x + x + x ) b 6 x + x + x ) x b x + x + x ) x a ) b x + x + x ) x a ) On the other hand from M U ) = f x ) Eq.4.9) is equivalent to the following equations ḣ x)θ) = A e π iθ + A e π iθ A a ḣ x)θ) πih x)θ) = ib 5 ψ e π iθ + ψ e π iθ ) ḣ x)θ) + πih x)θ) = ib 5 ψ e π iθ + ψ e π iθ ) ḣ x)θ) π ih x)θ) = A e π iθ + A e π iθ A a ḣ x)θ) + π ih x)θ) = A e π iθ + A e π iθ A ḣ x)θ) = ; ḣ x)θ) = A ; ḣ a x)θ) πih x)θ) = ḣ x)θ) + πih x)θ) = ; ḣ x)θ) π ih x)θ) = A a ḣ x)θ) + π ih x)θ) = A ;ḣ x)θ) = ;ḣ x)θ) = b a α+d )a A ḣ x)θ) πih x)θ) = ;ḣ x)θ) + πih x)θ) = ; ḣ x)θ) = ḣ x)θ) π ih x)θ) = b α+d )a A ;ḣ x)θ)+ π ih x)θ) = with boundary conditions: ḣ x)) + πa h x)) + π h x) ) = b 5 + b6 a ḣ x)) + πa h x)) + π h x) ) = b 5 i a ḣ x)) + πa h x)) + π h x) ) = b 5 i ḣ x)) + πa h x)) + π h x) ) = b 5 i) + b6 ḣ x)) + πa h x)) + π h x) ) = b 5 + i) + b6 a ḣ x)) + πa h x)) + π h x) ) = ḣ x)) πb )aϑα α+d ) ḣ x)) πb )aϑα α+d ) ḣ x)) πb )aϑα α+d ) ḣ x)) πb )aϑα α+d ) ḣ x)) πb )aϑα α+d ) ḣ x)) πb )aϑα α+d ) h x)) πϑα h x)) = b a h x)) πϑα h x)) = b a h x)) πϑα h x)) = b h x)) πϑα h x)) = h x)) πϑα h x)) = h x)) πϑα h x)) = ḣ x)) πa b) h x)) + πα+d) h x)) = b a ḣ x)) πa b) h x)) + πα+d) h x)) = b a ḣ x)) πa b) h x)) + πα+d) h x)) = b a ḣ x)) πa b) h x)) + πα+d) h x)) = ḣ x)) πa b) h x)) + πα+d) h x)) = a a a ). b α+d )a A

18 68 J. Wang & W. Jiang ḣ x)) πa b) h x)) + πα+d) h x)) =. Solving these equations noting that Ψs) h q θ)) = q = ) we obtain h x)θ) = π ia e π iθ + π ia e π iθ A a θ + C h x)θ) = π b 5ψ e π iθ π b 5ψ e π iθ + C e πiθ h x)θ) = π b 5ψ e π iθ π b 5ψ e π iθ + C e πiθ h x)θ) = A θe π iθ + i π A e π iθ A πa i + C e π iθ h x)θ) = i π A e π iθ + A e π iθ + A π a i + C e π iθ h x)θ) = ; h x)θ) = A θ; h a x)θ) = ;h x)θ) = h x)θ) = A πa h x)θ) = where C = π [ b 5 + b6 i + C e π iθ ; h x)θ) = A πa i + C e π iθ ; a + π ) A a ]; C = 4 5π i )b 5; C = 4 5π + i )b 5 C = 7A /π)/a; C = 7A /π)/a; C = πa [ b 5 i) + b6 a A + π i)] C = πa [ b 5 + i) + b6 a A π i)]. References [] E. Beninca J. Huisman R. Heerkloss K. D. Johnk P. Branco E. H. Van Nes M. Scheffer and S. P. Ellner Chaos in a long-term experiment with a plankton community Nature [] E. Beninca K. D. Johnk R. Heerkloss and J. Huisman Coupled predator-prey oscillations in a chaotic food web Ecology Letters [] J. Carr Applications of Centre Manifold Theory Springer 98. [4] Y. Ding W. Jiang and P. Yu Hopf-zero bifurcation in a generalized Gopalsamy neural network model Nonlinear Dynamics [5] Y. Ding W. Jiang and H. Wang Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay Chaos Solitons & Fractals [6] T. Faria and L. Magalhaes Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity Journal of Differential Equations [7] T. Faria and L. Magalhaes Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation Journal of Differential Equations [8] S. Guo and W. Jiang Hopf-fold bifurcation and fluctuation phenomena in a delayed rotio-dependent Gause-type predator-prey model International Journal of Bifurcation and Chaos 55. [9] J. Ge and J. Xu An efficient method for studying fold-hopf bifurcation in delayed neural networks International Journal of Bifurcation and Chaos [] S. Guo Y. Chen and J. Wu Two-parameter bifurcations in a network of two neurons with multiple delays Journal of Differential Equations

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