Hopf bifurcation analysis for a model of plant virus propagation with two delays

Transcription

1 Li et al. Advances in Difference Equations 08 08:59 R E S E A R C H Open Access Hopf bifurcation analysis for a model of plant virus propagation with two delays Qinglian Li Yunxian Dai * XingweiGuo and Xingyong Zhang * Correspondence: dyxian976@sina.com Department of Applied Mathematics Kunming University of Science and Technology Kunming People s Republic of China Full list of author information is available at the end of the article Abstract In this paper we consider a model of plant virus propagation with two delays and Holling type II functional response. The stability of the positive equilibrium and the existence of Hopf bifurcation are analyzed by choosing τ and τ as bifurcation parameters respectively. Using the center manifold theory and normal form method we discuss conditions for determining the stability and the bifurcation direction of the bifurcating periodic solution. Finally we carry out numerical simulations to illustrate the theoretical analysis. Keywords: Delay differential equation; Virus propagation; Hopf bifurcation; Holling type II Introduction As we know plants play a vital role in the everyday life of all organisms on earth. Sometimes however plants become infected with a virus which can have a devastating effect on the ecosystem that depends on it. An insect-vector can cause the transmission of the virus from plant to plant. The propagation characteristics and epidemiology of plant viruses were studied in [ ]. In [3] the transmission pathways of plant viruses were analyzed in detail from the perspective of plants and media; the authors established a model of plant infections disease and analyzed the dynamics of the model. Although there are many models that describe the interaction between vectors and humans there are not as many that describe the relationship between plants and vectors. Recently Jackson and Chen-Charpentier [4] have proposed a plant virus propagation model with the functional response Holling type II of the following form: ds βy = μk + di μs dt +αy S di dt = βy S d + μ + γ I +αy dr dt = γ I μr dx = β IX dt +α I mx dy = β IX dt +α I my. St It Rt Xt and Y t denote the susceptible plants infected plants recovered plants susceptible insect vectors and infected insect vectors respectively. The total The Authors 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Li et al. Advances in Difference Equations 08 08:59 Page of number of plants will be denoted by the fixed positive constant K K = S + I + Randthetotal number of insects will be denoted by N = X + Y. The parameters β β α α μ m γ and d are positive real numbers; β is the infection rate of plants due to vectors β is the infection rate of vectors due to plants α is the saturation constant of plants due to vectors α is the saturation constant of vectors due to plants μ is the natural death rate of plants m is the natural death rate of vectors γ istherecoveryrateofplants is the replenishing rate of vectors birth and/or immigration and d is the death rate of infected plants due to the disease. In the model. the authors make the following assumptions [4]: The susceptible plants can be infected only by the infected insect vectors. This model does not consider that infection can be transmitted from plant to plant. The interaction between the insects and the plants is modeled using Holing type II since insects can only bite a limited number of plants. The total number of plants is denoted by the fixed positive constant K becausein one area one can always keep the total number fixed by adding a new plant when a plant has died. The new plant shares the same characteristics of the plant it replaced before it was infected. 3 The replenishment rate of insect vectors is a positive constant and all of the new born vectors are susceptible. 4 A susceptible vector can be infected only by an infected plant host and after it is infected it will hold the virus for the rest of its life. Further there is no vertical transmission of the virus and vectors cannot transmit the virus to another vector. Notice that adding dx dt and dy dt yields dn dt t. So equation. can be reduced to the following equations: ds dt di dt = βy dy = β I dt = μk S βy +αy S + di +αy S ωi +α I m Y my = mn N = X + Y andn m as. ω = d + μ + γ. In [4] the authors analyzed the stability of equilibria with the basic reproduction number using the generation matrix approach. Considering time for the virus to enter the plant cells and to spread in the plant and time for the virus to infect the insect the authors proposed the model with two discrete delays: ds dt = μk S βy t τ +αy t τ St τ +di di dt = βy t τ +αy t τ St τ ωi dy = β It τ dt +α It τ m Y t τ my.3 τ is the time it takes a plant to become infected after contagion and τ is the time it takes a vector to become infected after contagion. Since the delay differential equations are extensively used in the practical life it is very important to study the stability of differential equations with delays. Recently a great deal of scholars have achieved very good research results in terms of multidelay differential equations [5 5].

3 Li et al. Advances in Difference Equations 08 08:59 Page 3 of In our paper we continue the work of Jackson and Chen-Charpentier [4]. Viewing delays as bifurcation parameters we discuss the stability of equilibrium and the existence of Hopf bifurcation of system.3infourcases:τ =0τ =0;τ >0τ =0;3τ = τ = τ > 0; 4 τ 0 τ 0 τ >0τ τ.whenτ τ we study the properties of Hopf bifurcation by using the normal form theory and center manifold theorem. This paper is organized as follows. In Sect. westudythestabilityofpositiveequilibrium and the existence of local Hopf bifurcation of system.3. In Sect. 3 the direction and stability of Hopf bifurcation are determined by using normal form theory and central manifold theorem. Some numerical simulations are carried out to support our results in Sect. 4. Finally a conclusion is given in Sect. 5. Stability and existence of Hopf bifurcation System.3 has a unique positive equilibrium ES I Y provided that the following conditions are satisfied: H mω + Kμβ + α m>dm αβ K μ + mmω + Kμβ + α m dm>0 β βk μ > m μω αmμω + βmω + Kμβ + α m dm>0 S = ωαβ K μ + mmω + Kμβ + α m dm β β ω d+αβ μω + β mμω + α m μω I = Y = β βk μ m μω β β ω d+αβ μω + β mμω + α m μω μββ K ωm mαmμω + βmω + Kμβ + α m dm. The linearized system of system.3ates I Y is du t = μu dt t A 0 u t τ +du t B 0 u 3 t τ du t = A dt 0 u t τ ωu t+b 0 u 3 t τ = C 0 u t τ +D 0 u 3 t τ mu 3 t du 3 t dt. A 0 = βy +αy B 0 = βs β C +αy 0 = +α I m Y and D 0 = β I +α I. The characteristic equation of system.is λ 3 + A λ + A λ + A 3 + B λ + B λ + B 3 e λτ + C λ + C λ + C 3 e λτ +D λ + D e λτ +τ = 0. A = μ + ω + m A = μω + μm + mω A 3 = μmω B = A 0 B = A 0 ω + m d B 3 = A 0 mω d C = D 0 C = D 0 μ ω C 3 = D 0 μω D = A 0 D 0 B 0 C 0 D = A 0 D 0 d B 0 C 0 μ A 0 D 0 ω. Next we consider the following four cases.

4 Li et al. Advances in Difference Equations 08 08:59 Page 4 of Case : τ =0τ =0. The characteristic equation.becomes λ 3 +A + B + C λ +A + B + C + D λ +A 3 + B 3 + C 3 + D =0..3 Let H A + B + C >0 A 3 + B 3 + C 3 + D >0 A + B + C A + B + C + D A 3 + B 3 + C 3 + D >0. According to the Routh Hurwitz criteria if conditions H andh hold then all the roots of.3 must have negative real parts. We have the following results. Theorem. Assume that H and H hold. If τ = τ =0then the positive equilibrium ES I Y of system.3 is locally asymptotically stable. Case : τ >0τ =0. The characteristic equation.reduces to λ 3 +A +C λ +A +C λ+a 3 +C 3 + [ B λ +B +D λ+b 3 +D ] e λτ = 0..4 Let λ = iω ω > 0 be the root of equation.4. Then we have: B + D ω cos ω τ +[B ω B 3 + D ] sin ω τ = ω 3 A + C ω B + D ω sin ω τ [B ω B 3 + D ] cos ω τ =A + C ω A 3 + C 3..5 It follows that ω 6 + E ω 4 + E ω + E 3 = 0.6 E =A + C A + C B E =A + C A + C A 3 + C 3 +B B 3 + D B + D E 3 =A 3 + C 3 B 3 + D. Let r = ω.thenequation.6becomes r 3 + E r + E r + E 3 = 0..7 Denote h r =r 3 + E r + E r + E 3..8 Thus dh r dr =3r +E r + E.

5 Li et al. Advances in Difference Equations 08 08:59 Page 5 of If E 3 =A 3 + C 3 B 3 + D <0thenh 0 < 0 and lim r + h r =+.Equation.7 has at least one positive root. If E 3 =A 3 + C 3 B 3 + D 0and = E 3E 0 then equation.7 hasno positive root for r [0 +. If E 3 =A 3 + C 3 B 3 + D 0and = E 3E > 0 then the equation 3r +E r + E =0 has two real roots r = E + and r 3 = E.Becauseh 3 r = >0and h r = <0equation.7 has at least one positive root if and only if r = E + >0andh 3 r 0 r and r are the local minimum and maximum of h r respectively. Without loss of generality we assume that.7 has three positive roots defined by r r andr 3 respectively.then.6 has three positive roots ω = r k =3.From.5weget [B + D B A + C ]ω 4 cos ω τ = [B ω B 3 + D ] +B + D ω A 3 + C 3 B 3 + D [B ω B 3 + D ] +B + D ω + [A + C B 3 + D +B A 3 + C 3 A + C B + D ]ω [B ω B 3 + D ] +B + D ω and τ j = { [B + D B A + C ]ω 4 arccos ω [B ω B 3 + D ] +B + D ω A 3 + C 3 B 3 + D [B ω B 3 + D ] +B + D ω + [A + C B 3 + D +B A 3 + C 3 A + C B + D ]ω [B ω B 3 + D ] +B + D ω } +jπ k =3j =0... Denote τ 0 = τ 0 0 = min k {3} { τ 0} ω0 = ω 0. Next we verify the transversality condition. Let λτ =α τ +iω τ be the root of equation.4nearτ = τ j satisfying j j α τ =0 ω τ = ω. Substituting λτ into.4 and taking the derivative with respect to τ wehave [ ] dλ = [3λ +A + C λ +A + C ]e λτ dτ λ[b λ +B + D λ +B 3 + D ]

6 Li et al. Advances in Difference Equations 08 08:59 Page 6 of By.9wehave [ Redλτ + B λ +B + D λ[b λ +B + D λ +B 3 + D ] τ λ..9 ] dτ τ =τ j [ [3λ +A + C λ +A + C ]e λτ ] = Re λ[b λ +B + D λ +B 3 + D ] [ ] B λ +B + D + Re λ[b λ +B + D λ +B 3 + D ] τ =τ j τ =τ j = { [ 3ω k +A + C ] [[ ω B ω B 3 + D ] sin ω τ j +B + D ω cos ω τ j ] A + C ω [[ B ω B 3 + D ] cos ω τ j B + D ω +B ω [ B3 + D ω B ω 3 B + D ω sin ω τ j ] ]} = { 3ω 6 + [ A + C A + C B ] ω 4 + [ A + C A + C A 3 + C 3 +B B 3 + D B + D ] ω } = { r 3r +E 3 r + E 3 } = r h r =B + D ω 4 +[B 3 + D ω B ω 3 ] >0.Noticethat >0r >0 {[ ] Redλτ sign dτ τ =τ j } = sign {[ Redλτ dτ ] τ =τ j j dre λτ and thus dτ has the same sign as h r. To investigate the root distribution of the transcendental equation.4 we introduce the result of Ruan and Wei [6]. Lemma. Consider the exponential polynomial P λ e λτ...e λτ m = λ n + p 0 λn + + p 0 n λ + p0 n + [ p λn + + p n λ + ] p n e λτ + + [ p m λ n + + p m n λ + ] pm n e λτ m τ i 0 i =...m and p i j i = 0...m; j =...n are constants. As τ τ...τ m vary the sum of the order of the zeros of Pλ e λτ...e λτ m on the open right half-plane can change only if a zero appears on or crosses the imaginary axis. According to this analysis we have the following results. }

7 Li et al. Advances in Difference Equations 08 08:59 Page 7 of Theorem. For τ >0and τ =0suppose that H and H hold. Then: i If E 3 0 and = E 3E 0 then all roots of equation.4 have negative real parts for all τ 0 and the positive equilibrium ES I Y is locally asymptotically stable for all τ 0. ii If either E 3 <0or E 3 0 = E 3E >0 r >0 and h r 0 then h r has at least one positive root and all roots of equation.4 have negative real parts for τ [0 τ 0 and the positive equilibrium ES I Y is locally asymptotically stable for all τ [0 τ 0. iii If ii holds and h r 0 then system.3 undergoes Hopf bifurcations at the positive equilibrium ES I Y for τ = τ j k =3;j =0... When τ =0andτ > 0 the stability of the equilibrium ES I Y and the existence of Hopf bifurcation can be obtained based on a similar discussion which we omit in this paper. Case 3: τ = τ = τ >0. When τ = τ = τ > 0 the characteristic equation.becomes λ 3 + A 3 λ + A 3 λ + A 33 + B 3 λ + B 3 λ + B 33 e λτ +C 3 λ + C 3 e λτ = 0.0 A 3 = A A 3 = A A 33 = A 3 B 3 = B + C B 3 = B + C B 33 = B 3 + C 3 C 3 = D C 3 = D. Both sides of equation.0 are multiple e λτ and we obviously get λ 3 + A 3 λ + A 3 λ + A 33 e λτ + B 3 λ + B 3 λ + B 33 +C3 λ + C 3 e λτ = 0.. Let λ = iω 3 ω 3 > 0 be the root of equation.. Separating the real and imaginary parts we obtain: ω3 3 C 3 + A 3 ω 3 cos ω 3 τ +A 3 ω3 A 33 + C 3 sin ω 3 τ = B 3 ω 3 ω3 3 +C 3 A 3 ω 3 sin ω 3 τ + A 3 ω3 + A 33 + C 3 cos ω 3 τ = B 3 ω3 B 33 from which it follows that cos ω 3 τ = l 34ω3 4+l 3ω3 +l 30 ω 6 3 +k 34ω 4 3 +k 3ω 3 +k 30 sin ω 3 τ = l 35ω3 5+l 33ω3 3+l 3ω 3 ω 6 3 +k 34ω 4 3 +k 3ω 3 +k l 30 = A 33 B 33 + C 3 B 33 l 3 = A 3 B 33 + B 33 C 3 A 33 B 3 A 3 B 3

8 Li et al. Advances in Difference Equations 08 08:59 Page 8 of l 3 = A 3 B 3 + A 3 B 33 + A 33 B 3 B 3 C 3 + B 3 C 3 l 33 = A 3 B 3 B 3 C 3 A 3 B 3 B 33 l 34 = B 3 A 3 B 3 l 35 = B 3 k 30 = A 33 C 3 k 3 = A 3 A 33 + A 3 C 3 k 34 = A 3 A 3. Since sin ω 3 τ + cos ω 3 τ =wehave ω 3 + E 3ω E 3ω E 33ω E 34ω E 35ω 3 + E 36 = 0.4 E 3 =k 34 l35 E 3 =k 3 + k34 l 33l 35 l34 E 33 =k 30 +k 3 k 34 l 3 l 35 l33 l 3l 34 E 34 =k 30 k 34 + k3 l 3l 33 l 30 l 34 l3 E 35 =k 30 k 3 l3 l 30l 3 E 36 = k30 l 30. Letting r 3 = ω 3 equation.4istransformedinto r E 3r E 3r E 33r E 34r 3 + E 35r 3 + E 36 = 0..5 If all the parameters of system.3 are given then it is easy to get the roots of equation.5 by using the Matlab software package. To give the main results in this paper we make the following assumption. H 3 equation.5 has at least one positive real root. Suppose that condition H 3 holds. Without loss of generality we assume that.4has six positive real roots say r 3 r 3...r 36.Then.3 has six positive real roots ω 3 = r 3 ω 3 = r... ω 3i = r 3i i =...6. Thus if we denote τ j 3k = { arccos ω 3k l 34 ω 4 3k + l 3ω 3k + l 30 ω 6 3k + k 34ω 4 3k + k 3ω 3k + k 30 } +jπ.6 for k =...6j =0...then±iω 3k is a pair of purely imaginary roots of.corresponding to τ j 3k.Define τ 30 = τ 3k0 = min k {...6} { τ 0} ω30 = ω 3k0. 3k Let λτ=α 3 τ+iω 3 τ be the root of equation.nearτ = τ j 3k satisfying j j α τ 3k =0 ω3 τ 3k = ω3k. Substituting λτ into. and taking the derivative with respect to τ we have [ ] dλ = 3λ +A 3 λ + A 3 e λτ +B 3 λ + B 3 + C 3 e λτ dτ λ[λ 3 + A 3 λ + A 3 λ + A 33 e λτ C 3 λ + C 3 e λτ ] τ λ..7

9 Li et al. Advances in Difference Equations 08 08:59 Page 9 of Letting λ = ±iω 3k at the roots of equation.atτ = τ j j d Reλτ 3k 3k weshouldcompute. dτ By calculation we get [ dλ Re dτ ] = P 33 + ip 34 τ τ=τ j P 3 + ip 3 λ 3k P 3 = ω3k 4 + A 3ω3k C 3ω3k cos ω3k τ j 3k + A 3 ω3k 3 + A 33ω 3k + C 3 ω 3k sin ω3k τ j 3k P 3 = A 3 ω3k 3 A 33ω 3k + C 3 ω 3k cos ω3k τ j 3k + ω3k 4 + A 3ω3k + C 3ω3k sin ω3k τ j 3k P 33 = 3ω 3k + A 3 + C 3 cos ω3k τ j 3k P 34 =A 3 ω 3k cos ω 3k τ j 3k So we have [ dλ Re dτ ] τ=τ j 3k = P 33P 3 + P 34 P 3 P3 +. P 3 A3 ω 3k sin ω 3k τ j 3k + B3 + 3ω 3k + A 3 C 3 sin ω3k τ j 3k +B3 ω 3k. Obviously if condition H 4 P 3 P 33 + P 3 P 34 0 d Re λτ holds then dτ λ=iω3k = Re[ dλτ ] dτ λ=iω 3k 0. Thus we have the following results. Theorem.3 For system.3 with τ = τ = τ >0let H H 4 hold. The equilibrium point ES I Y is asymptotically stable for τ [0 τ j j 3k and unstable for τ > τ 3k ; Hopf bifurcation occurs when τ = τ j 3k. Case 4: τ 0 τ 0 τ >0τ τ. We consider.withτ in its stable interval [0 τ 0 andτ considered as a parameter. Let λ = iω ω > 0 be a root of equation.. Separating real and imaginary parts leads to C ω + D ω cosω τ D sinω τ cosω τ +C ω C 3 D ω sinω τ D cos ω τ sinω τ = ω 3 A ω B ω cosω τ B ω sinω τ +B 3 sinω τ C ω + D ω cosω τ D sinω τ sinω τ.8 C ω C 3 D ω sinω τ D cos ω τ cosω τ = A ω A 3 B ω sinω τ +B ω cosω τ B 3 cosω τ. From equation.8 we can obtain cos ω τ h 4 ω = 4 + h 4 ω 3 + h 43 ω + h 44 ω + h 45 f 4 ω 4 + f 4 ω 3 + f 43 ω + f 44 ω + f.9 45

10 Li et al. Advances in Difference Equations 08 08:59 Page 0 of h 4 = C C A B C cos ω τ + D cos ω τ h 4 = B C sin ω τ D sin ω τ + B C sin ω τ + A D sin ω τ h 43 = A C B C cos ω τ A D cos ω τ B D + B D + A 3 C + B 3 C cos ω τ + A C 3 + B C 3 cos ω τ + A D cos ω τ h 44 = B 3 C sin ω τ + A D sin ω τ B C 3 sin ω τ A3 D sin ω τ h 45 = B 3 D A 3 C 3 B 3 C 3 cos ω τ A3 D cos ω τ f 4 = C f 4 = C D sin ω τ f 43 = C +C D cos ω τ + D C C 3 C C 3 cos ω τ f 44 =C 3 D sin ω τ C D sin ω τ f 45 = C3 + D +C 3D cos ω τ. From equation.8weobtain: ω 6 + E 4 ω 4 + E 4 ω + E 43 + E 44 sin ω τ + E45 cos ω τ = 0.0 E 4 = B A C + A E 4 = A +C C 3 A A 3 + B B B C D E 43 = B 3 D C 3 + A 3 E 44 = B ω 5 +B 3 A B + A B + C D ω 3 +C D + A 3 B C 3 D A B 3 ω E 45 =A B B ω 4 +A B A 3 B C D + C D A B 3 ω +A 3 B 3 C 3 D. Denote F ω =ω 6 + E 4 ω 4 + E 4 ω + E 43 + E 44 sinω τ +E 45 cosω τ. If E 43 = B 3 D C 3 + A 3 <0then F 0 < 0 lim F ω ω =+. + We can see that.0 has at most six positive roots ω ω...ω 6.Foreveryfixedω k k =...6for.9 the critical value τk j = { h4 ω 4 k + h 4 ω 3 k + h 43 ωk ωk arccos f 4 ωk 4 + f 4 ωk 3 + f 43 ωk + f 44 ωk + f 45 + h 44 ω k + h 45 } +jπ k =...6;j =0.... There exists a sequence {τ kj j =0...} such that.8holds.

11 Li et al. Advances in Difference Equations 08 08:59 Page of Let τ 0 = τ k 0 0 = min k {...6} { τ k 0 } ω 0 = ω k 0.. Substituting τ into. and taking the derivative with respect to τ wehave [ dλ dτ ] = λ Q + Q e λτ + Q 3 e λτ + Q 4 e λτ +τ.3 Q 5 e λτ +τ + Q 6 e λτ Q =3λ +A λ + A Q = B τ λ +B τ B λ + B τ B 3 Q 3 =C λ + C Q 4 = τ D λ +D τ D Q 5 = D λ + D Q 6 = C λ + C λ + C 3. By.3wehave [ dλτ j Re k dτ ] λ=iω k = Q 3Q 33 + Q 3 Q 34 Q 3 + Q 3 Q 3 = D ωk sin ωk τ + τkj D ω k cos ωk τ + τk j C ω k3 sin ωk τ k + C3 ωk sin ωk τ kj C ω k cos ωk τ kj Q 3 = D ωk cos ωk τ + τkj + D ω k sin ωk τ + τk j C ω k3 cos ωk τ kj + C 3 ωk cos ωk τ kj + C ω k sin ωk τ kj Q 33 = 3ω k + A +B ωk sin ωk τ τ B ωk sin ωk τ + B cos ωk τ + τ B ω k cos ωk τ τ B 3 cos ωk τ +C ωk sin ωk τ kj + C cos ωk τ kj τ D ωk sin ωk τ + τk j + D cos ωk τ + τkj τ D cos ωk τ + τkj Q 34 =A ωk +B ωk cos ωk τ τ B ωk cos ωk τ B sin ωk τ τ B ω k sin ωk τ + τ B 3 sin ωk τ +C ωk cos ωk τ k C sin ωk τ kj τ D ωk cos ωk τ + τk j D sin ωk τ + τkj + τ D sin ωk τ + τkj. We suppose that H 5 Q 3 Q 33 + Q 3 Q Then Re dλ dτ λ=iω k 0 and we have the following result on the stability and Hopf bifurcation in system.3. Theorem.4 For system.3 with τ [0 τ 0 suppose that H H and H 5 hold. If E 43 = B 3 D C 3 + A 3 <0then the positive equilibrium point ES I Y is locally

12 Li et al. Advances in Difference Equations 08 08:59 Page of asymptotically stable for τ [0 τ0 and unstable for τ > τ0. Hopf bifurcation occurs when τ = τ kj k =...6;j =0... When τ >0τ 0 τ 0 τ τ the stability of the equilibrium ES I Y andtheexistence of Hopf bifurcation can be obtained based on a similar discussion which we omit in this paper. 3 Direction and stability of the Hopf bifurcation In this section we employ the normal form method and center manifold theorem [7 9] to determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system.3withrespecttoτ for τ 0 τ 0. Without loss of generality we denote any of the critical values τ = τ kj k =...6;j =0...byτ 0. Let u = x S u = y I u 3 = z Y t = t/τ andτ = τ 0 + μ μ R3.Thenμ =0isthe Hopf bifurcation value of system.3 which may be written as a functional differential equation in C = C[ 0] R 3 ut=l μ u t +f μ u t 3. ut=xt yt zt T R 3 andl μ φ:c R 3 and f μ u t aregivenby L μ φ= φ 0 τ0 + μ A φ 0 + φ τ τ τ0 + μ B 0 φ τ τ0 + φ τ0 + μ C φ 3. φ 3 0 φ 3 τ τ0 φ 3 φ =φ φ φ 3 T C[ 0] R 3 and μ d 0 A 0 0 B A = 0 ω 0 B = A 0 0 B 0 C = m C 0 D 0 f μ φ= f τ0 + μ 3.3 f f 3 f = k φ τ + k 4 φ 3 3 τ 0 τ f = k φ τ + k 4 φ 3 3 τ 0 φ 3 τ τ 0 τ τ 0 τ 0 + k 5 φ τ φ 3 τ τ 0 + k φ3 τ τ 0 φ k φ 3 + k 5 φ τ τ 0 τ 0 τ τ φ 3 3 τ 0 + k 3 φ τ τ 0 τ τ 0 + τ 0 + k 3 φ τ + f 3 = k 3 φ φ 3 + k 3 φ + k 33φ φ 3 + k 34 φ 3 + k 35 φ 3 φ 3 τ 0 φ3 τ τ0 φ3 τ τ0

13 Li et al. Advances in Difference Equations 08 08:59 Page 3 of β k = + αy k αβs = + αy k αβ 3 3 = + αy 3 k 4 = α βs + αy k α β 4 5 = + αy 4 β k = + αy k αβs = + αy k αβ 3 3 = + αy 3 k 4 = α βs + αy k α β 4 5 = + αy 4 β k 3 = + α I k α β 3 = + α I 3 α k 34 = β + α I 4 m Y α k 35 = β + α I. 4 m Y k 33 = α β + α I 3 Obviously L μ φ is a continuous linear mapping from C[ 0] R 3 intor 3.BytheRiesz representation theorem there exists a 3 3matrixfunctionηθ μ θ 0 whose elements are of bounded variation such that 0 L μ φ= dηθ μφθ φ C [ 0] R In fact we can choose ηθ μ=τ0 + μ[a δθ+c δθ ++B δθ + τ τ0 ] δ is the Dirac delta function. For φ C [ 0] R 3 we define dφθ θ [ 0 dθ Aμφ = dηs μφs θ =0 0 and 0 θ [ 0 Rμφ = f μ φ θ =0. Then for θ =0system3.isequivalentto u t = Aμu t + Rμu t 3.5 u t = ut + θ=u t + θ u t + θ u 3 t + θ for θ [ 0]. The adjoint operator A of A is defined by A dψs s 0 ] ds ψs= 0 dηt t0ψ t s =0 associated with the bilinear form ψs φθ = ψ0φ0 0 θ ξ=0 ψξ θ dηθφξ dξ 3.6 ηθ =ηθ0 Denote A = A0. Then A and A are adjoint operators. From our discussion we see that ±iω 0 τ 0 are eigenvalues of A and they also are eigenvalues of A.

14 Li et al. Advances in Difference Equations 08 08:59 Page 4 of Suppose that qθ =q q 3 T e iω 0 τ 0 θ is the eigenvector of A corresponding to iω0 τ 0 and q s=d q q 3 eiω 0 τ 0 s is the eigenvector of A corresponding to iω0 τ 0.Thenby a simple computation we obtain q = iω 0 + μ d iω 0 + ω q 3 = iω 0 + μiω 0 ω A 0e iω 0 τ d iω 0 + w B 0 e iω 0 τ d iω 0 + w. q = iω 0 + μ + A 0e iω 0 τ q A 0 e iω 0 τ 3 = da 0e iω 0 τ iω0 + ω iω 0 + μ + A 0e iω 0 τ A 0 C 0 e iω 0 τ +τ0. From 3.6wehave q s qθ = D q q3 0 θ ξ=0 0 θ ξ=0 q q 3 T D q q 3 e iξ θω 0 τ dηθ q q 3 T e iξω 0 τ dξ D q q 3 e iξ θω 0 τ 0 dηθ q q 3 T e iξω 0 τ 0 dξ = D { +q q + q 3q 3 + τ e iω 0 τ q q 3 B q q 3 T + τ0 eiω 0 τ 0 q q 3 C q q 3 T} = D { +q q + q 3q 3 + τ e iω 0 τ [ A 0 q A 0 + q 3 B0 q B ] 0 + τ0 e iω 0 τ 0[ q 3 C 0q + D 0 q 3 ]}. Thus we can choose D = { +q q + q 3q 3 + τ e iω 0 τ [ A 0 q A 0 + q 3 B0 q B ] 0 + τ 0 e iω 0 τ 0[ q 3 C 0q + D 0 q 3 ]} 3.7 such that q s qθ =and q s qθ =0. NextweusethesamenotationsasthoseinHassard[9] and firstly compute the coordinates to describe the center manifold C 0 at μ =0.Letu t be the solution of equation 3. with μ =0.Define zt= q u t Wt θ=ut θ Re { ztqθ }. 3.8 On the center manifold C 0 wehave Wt θ=w zt zt θ = W 0 0 z + W θz z + W 0 θ z + W 30θ z3 6 + z and z are the local coordinates for center manifold C 0 in the directions of q and q.notethatw is real if u t is real. For the solution u t C 0 of 3. since μ =0wehave ż = iω 0 τ 0 z + q θ f 0 W zt zt θ +Re { ztqθ } = iω 0 τ 0 z + q 0f 0 W zt zt 0 +Re { ztq0 }

15 Li et al. Advances in Difference Equations 08 08:59 Page 5 of = iω0 τ 0 z + q 0f 0 z z iω0 τ 0 z + gz z 3.9 Then z gz z = q 0f 0 z z=g 0 + g z z z + g 0 + g zz gz z = q 0f 0 z z = Dτ 0 q q 3 = Dτ 0 {[k u t τ + k 3 u t τ + k 4 u 3 3t + q τ 0 τ τ 0 f 0 f 0 f 0 T 3 τ 0 u 3t [k u t τ + k 3 u t τ + k 4 u 3 3t τ 0 τ τ 0 u 3t τ τ τ 0 τ 0 + k 5 u t τ τ 0 u 3t u 3t τ τ τ 0 τ 0 τ 0 + k 5 u t τ τ 0 + k u 3t τ τ0 u 3 3t τ + k u 3t τ 0 ] + τ τ 0 u 3 3t τ ] τ0 + + q 3[ k3 u t u 3t + k 3 u t + k 33u t u 3t + k 34 u 3 t + k 35 u 3 t u 3t ]}. Since u t θ =u t θ u t θ u 3t θ T = Wt θ +zqθ + z qθ and qθ =q q 3 T e iω 0 τ 0 θ wehave u t τ τ0 = ze iω 0 τ + ze iω 0 τ + W 0 τ τ0 + W 0 τ z τ0 z + W + O z z 3 τ z τ0 z u t τ τ0 = q ze iω 0 τ + q ze iω 0 τ + W 0 + W τ τ0 z z + W 0 τ τ0 u 3t τ τ0 = q 3 ze iω 0 τ + q 3 ze iω 0 τ + W W 3 0 τ z τ0 + O z z 3 τ τ 0 τ τ0 z z + O z z 3 z + W 3 u t = ze iω 0 τ 0 + ze iω 0 τ 0 + W 0 z + W z z τ τ 0 z z

16 Li et al. Advances in Difference Equations 08 08:59 Page 6 of + W z 0 + O z z 3 u t = q ze iω 0 τ 0 + q ze iω 0 τ 0 + W 0 z + W z z + W z 0 + O z z 3 u 3t = q 3 ze iω 0 τ 0 + q 3 ze iω 0 τ 0 + W 3 0 z + W 3 z z + W 3 z 0 + O z z 3. Comparing the coefficients with 3.0 we have g 0 = Dτ 0{[ k q 3 e iω 0 τ + k q3 e iω 0 τ ] [ + q k q 3 e iω 0 τ + k q3 e iω 0 τ ] + q 3[ k3 q q 3 e iω 0 τ 0 + k 3 q e iω 0 τ 0]} g = Dτ 0{[ ] ] k q 3 + q 3 +k q 3 q 3 + q [ k q 3 + q 3 +k q 3 q 3 ]} + q 3[ k3 q q 3 + q q 3 +k 3 q q g 0 = Dτ 0{[ k q 3 e iω 0 τ + k q 3 e iω 0 τ ] + q [ k q 3 e iω 0 τ + k q 3 e iω 0 τ ] + q 3[ k3 q q 3 e iω 0 τ 0 + k 3 q e iω 0 τ 0]} g = Dτ 0 {[k e iω 0 τ W 3 +q 3 e iω 0 τ W τ τ0 + k q 3 e iω 0 τ W 3 0 τ τ τ 0 τ 0 + e iω 0 τ W 3 0 τ τ 0 +4q 3 e iω 0 τ W 3 τ + k 3 4q3 q 3 e iω 0 τ +q3 e iω 0 τ +6k 4 q3 q 3 e iω 0 τ + q [k e iω 0 τ W 3 τ + q 3 e iω 0 τ W 0 τ τ 0 τ 0 + e iω 0 τ W 3 0 τ +q 3 e iω 0 τ W τ 0 τ τ 0 τ 0 ] + q 3 e iω 0 τ W 0 τ τ0 + k q 3 e iω 0 τ W 3 0 τ τ0 +4q 3 e iω 0 τ W 3 τ τ0 + k 3 4q3 q 3 e iω 0 τ +q3 e iω 0 τ ] +6k 4 q3 q 3 e iω 0 τ [ + q 3 k3 q e iω 0 τ 0W 3 + q e iω 0 τ 0W q 3 e iω 0 τ 0W 0 + q 3e iω 0 τ 0W + k 3 4q e iω 0 τ 0W + q e iω 0 τ 0W 0 + k 33 4q q q 3 e iω 0 τ 0 +q q 3e iω 0 τ 0 +6k 34 q q e iω 0 τ 0] }

17 Li et al. Advances in Difference Equations 08 08:59 Page 7 of W 0 θ= ig 0 ω0 τ 0 q0e iω 0 τ 0 θ + iḡ 0 3ω0 τ 0 q0e iθω 0 τ 0 + E e iθω 0 τ 0 W θ= ig ω0 τ 0 q0e iθω 0 τ 0 + iḡ ω0 τ 0 q0e iθω 0 τ 0 + E and iω0 + μ + A 0e iω 0 τ d B 0 e iω 0 τ E = A 0 e iω 0 τ iω0 ω B 0e iω 0 τ 0 C 0 e iω 0 τ 0 iω0 D 0e iω 0 τ 0 + m μ A 0 d B 0 N E = A 0 ω B 0 N 0 C 0 D 0 m N 3 M M M 3 M = k q 3 e iω 0 τ + k q 3 e iω 0 τ M = k q 3 e iω 0 τ + k q 3 e iω 0 τ M 3 = k 3 q q 3 e iω 0 τ 0 + k 3 q e iω 0 τ 0 N = k q 3 + q 3 +k q 3 q 3 N = k q 3 + q 3 +k q 3 q 3 N 3 = k 3 q q 3 + q q 3 +k 3 q q. Thus we can calculate the following values: i C 0 = g ω0 τ 0 g 0 g g 0 + g 3 μ = ReC 0 Reλ τ0 T = Im C 0 + μ Im λ τ0 ω0 τ 0 β =Re C 0. Based on this discussion we obtain the following results. Theorem 3. i μ determines the direction of the Hopf bifurcation. If μ >0μ <0 then the Hopf bifurcation is supercritical subcritical. ii β determines the stability of the bifurcating periodic solutions. If β <0β >0 then the bifurcating periodic solutions are stable unstable. iii T determines the period of the bifurcating periodic solutions. If T >0T <0 then the period of the bifurcating periodic solutions increases decreases. 4 Numerical simulation In this section we give some numerical simulations supporting our theoretical predictions. The selection of parameter values refers to [4] and references therein. The same parameters as in [4] are adopted: μ =0.k = 63 β = 0.0 β = 0.0 γ = 0.0. According to [4] and references therein the other parameters α α d m are appropriately chosen

18 Li et al. Advances in Difference Equations 08 08:59 Page 8 of to system.3. As an example we consider the following system: ds dt = S 0.0Y t τ +0.0Y t τ St τ + 0.0I di dt = 0.0Y t τ +0.0Y t τ St τ 0.04I dy = 0.0It τ dt +0.0It τ 0 0. Y t τ 0.Y. 4. Obviously hypotheses H andh hold: H mω + Kμβ + α m dm = αβ K μ + mmω + Kμβ + α m dm = 0.00 > 0 β βk μ m μω =6.6e 04 αmμω + βmω + Kμβ + α m dm= 9.0e 05 > 0; H A + B + C = > 0 A 3 + B 3 + C 3 + D = 0.00 > 0 A + B + C A + B + C + D A 3 + B 3 + C 3 + D = > 0. Then E = is a unique positive equilibrium of system 4.. When τ = τ =0 the positive equilibrium E = of system 4. is locally asymptotically stable. When τ =0and τ 0 the characteristic equation is λ λ λ+.305e λ λ e λτ =0. By a simple computation we can easily get E 3 = e 06 < 0 ω 0 = 0.86 τ 0 = ByTheorem.ifτ =3<τ 0 = or τ =5.79<τ 0 = then the positive equilibrium E is asymptotically stable. If τ =5.88>τ 0 = then the positive equilibrium E is unstable and system 4. undergoesa Hopf bifurcation at E and a family of periodic solutions bifurcate from the positive equilibrium E. This property can be illustrated by in Figs. 3.Furtherwecan compute the values c 0 = i μ = 0.34 T =.368 β = Since μ >0and β <0 the bifurcating periodic solution from E is supercritical and asymptotically stable at τ = τ. 3 When τ >0and τ >0letτ = and chose τ as a parameter. Then the characteristic equation is λ λ λ +4.0e λ λ e 04 e 5.79λ = λ λ e 05 e λτ λ e λ5.79+τ We obtain E 43 =.3868e 06 < 0 ω0 = τ 0 = and condition H 5 Q 3 Q 33 + Q 3 Q 34 = is satisfied. From Theorem.4 we know that the positive equilibrium E is asymptotically stable for τ [0 τ0. As τ continues to increase the positive equilibrium E will lose stability and a Hopf bifurcation occurs once τ > τ0. The corresponding numerical simulation results are shown Figs. 4 5.

19 Li et al. Advances in Difference Equations 08 08:59 Page 9 of Figure The trajectories of system 4.withτ =3<τ 0 = τ = 0. The positive equilibrium E is asymptotically stable. The initial value is Figure The trajectories of system 4.withτ =5.79<τ 0 = τ = 0. The positive equilibrium E is asymptotically stable. The initial value is Further we get c 0 = i μ = 0.47 T = β = 0.0. Therefore from Theorem 3. we know that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable. 5 Conclusion In this paper we study the dynamics of plant virus propagation model with two delays. First we obtain sufficient conditions for the stability of positive equilibrium E and the ex-

20 Li et al. Advances in Difference Equations 08 08:59 Page 0 of Figure 3 The trajectories of system 4.withτ =5.88>τ 0 = τ = 0. A Hopf bifurcation occurs from the positive equilibrium E. The initial value is Figure 4 The trajectories of system 4.withτ = 5.79 τ =<τ0 = The positive equilibrium E is asymptotically stable. The initial value is istence of Hopf bifurcation when τ >0τ =0andτ 0 τ 0 τ >0τ τ respectively. Next when τ τ by using the center manifold and normal form theory regarding τ as a parameter we investigate the direction and stability of the Hopf bifurcation. We derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally a numerical example supporting our theoretical predictions is given.

21 Li et al. Advances in Difference Equations 08 08:59 Page of Figure 5 The trajectories of system 4.withτ = 5.79 τ =5.>τ0 = A Hopf bifurcation occurs from the positive equilibrium E. The initial value is Acknowledgements The authors are grateful to the editor and anonymous reviewers for their valuable comments and suggestions. Funding Qinglian Li is supported by the National Natural Science Foundation of China No ; Yunxian Dai is supported by the National Natural Science Foundation of China No and No Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details Department of Applied Mathematics Kunming University of Science and Technology Kunming People s Republic of China. School of Mathematics and Statistics Central South University Changsha People s Republic of China. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: April 08 Accepted: 7 July 08 References. Jeger M.J. van den Bosch F. Madden L.V. Holt J.: A model for analysing plant-virus transmission characteristics and epidemic development. Math. Med. Biol Jeger M.J. Madden L.V. van den Bosch F.: The effect of transmission route on plant virus epidemic development and disease control. J. Theor. Biol Varney E.: Plant diseases: epidemics and control. Amer. J. Potato Res Jackson M. Chen-Charpentier B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. Math Liu J. Sun L.: Dynamical analysis of a food chain system with two delays. Qual. Theory Dyn. Syst Shi R. Yu J.: Hopf bifurcation analysis of two zooplankton phytoplankton model with two delays. Chaos Solitons Fractals Deng L. Wang X. Peng M.: Hopf bifurcation analysis for a ratio-dependent predator prey system with two delays and stage structure for the predator. Appl. Math. Comput Wang J. Wang Y.: Study on the stability and entropy complexity of an energy-saving and emission-reduction model with two delays. Entropy Ma J. Si F.: Complex dynamics of a continuous Bertrand duopoly game model with two-stage delay. Entropy

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