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Annali di Matematica DOI 10.1007/s10231-012-0264-z

Bifurcation analysis of an autonomous epidemic predator–prey model with delay

Changjin Xu · Maoxin Liao

Received: 22 September 2011 / Accepted: 16 February 2012 ? Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2012

Abstract In this paper, a class of an autonomous epidemic predator–prey model with delay is considered. Its linear stability and Hopf bifurcation are investigated. Applying the normal form theory and center manifold theory, the explicit formulas for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included. Keywords Predator–prey model · Time delay · Stability · Hopf bifurcation · Periodic solution Mathematics Subject Classi?cation (2000) 34K20 · 34C25

1 Introduction After the seminal work of Volterra and Lotka in the mid-1920s, the dynamics properties (including stable, unstable, persistent, and oscillatory behavior) of the predator–prey models that have significant biological background have been one of the most active areas of research

This work is supported by National Natural Science Foundation of China (No. 10771215 and No. 10771094), the Doctoral Foundation of Guizhou College of Finance and Economics (2010), the Science and technology Program of Hunan Province (No. 2010FJ6021) and the soft Science and Technology Program of Guizhou Province (No. 2011LKC2030). C. Xu (B ) Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, People’s Republic of China e-mail: xcj403@126.com M. Liao School of Mathematics and Physics, Nanhua University, Hengyang 421001, People’s Republic of China

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and have attracted great attention of many researchers. Many excellent and interesting results have been obtained [4–6,10,12–14,18]. In 2009, Tian et al. [17] investigated the periodic and almost periodic solution of the following non-autonomous epidemic predator–prey system with time delay: ? ˙ (t ) = X (t )[r1 (t ) ? a (t ) X (t ) ? b1 (t ) S (t ) ? b2 (t ) I (t )], ?X ˙ S (t ) = c(t ) X (t ? τ ) S (t ? τ ) + S (t )[?r2 (t ) ? d1 (t )( S (t ) + I (t )) ? e(t ) I (t )], ?˙ I (t ) = I (t )[e(t ) S (t ) ? d2 (t )( S (t ) + I (t ))],

(1.1)

where X (t ) denotes the density of the prey, S (t ) and I (t ) denote the density of the susceptible predator and the infected predator, respectively; ri (t )(i = 1, 2) denotes the intrinsic rate of natural increase, and the minus before r2 (t ) means that the susceptible predator is dependent on the prey, that is, if there is no prey, then the predator will be extinct. a (t ) means coef?cient of the density dependence, di (t )(i = 1, 2) means the competitive coef?cient between the predator, bi (t )(i = 1, 2) means the preying capacity for the susceptible and the infected predator, c(t ) means the relative preying capacity of the susceptible predator, e(t ) means the touching rate between the susceptible predator and the infected predator, while τ > 0 is the time required for the gestation of the susceptible predator. a (t ), bi (t ), c(t ), di (t ), e(t ), ri (t )(i = 1, 2) are continuous and strictly positive functions. It is well known that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation, is one of the most important theme on the population dynamics. To obtain a deep and clear understanding of dynamics of predator–prey system with time delay, In the present paper, we go no to investigate the model (1.1) under the following assumptions: the coef?cients are independent of the time of t , that is, ri (t ) = ri , bi (t ) = bi , di (t ) = di (i = 1, 2), a (t ) = a , c(t ) = c, e(t ) = e, and ri , bi , di , a , c, e(i = 1, 2) are all positive constants. Further, considering the biological meaning of model (1.1), we think that the coef?cient e(t ) in the second equation and the coef?cient e(t ) in the third equation of model (1.1) shall be different. So we denote the coef?cient e(t ) in the second equation and the coef?cient e(t ) in the third equation of model (1.1) by e1 and e2 , respectively. Then, system (1.1) becomes the following autonomous predator–prey system: ? ˙ (t ) = X (t )[r1 ? a X (t ) ? b1 S (t ) ? b2 I (t )], ?X ˙ S (t ) = cX (t ? τ ) S (t ? τ ) + S (t )[?r2 ? d1 ( S (t ) + I (t )) ? e1 I (t )], ?˙ I (t ) = I (t )[e2 S (t ) ? d2 ( S (t ) + I (t ))],

(1.2)

The more detail biological meaning of the coef?cients of the system (1.2) is same as that in [17]. In this paper, we study the stability, the local Hopf bifurcation for system (1.2). We would like to mention that there are a lot of papers on the Hopf bifurcation of predator–prey models [1,3,9,16,19–24]. To the best of our knowledge, it is the ?rst time to deal with the research of Hopf bifurcation for model (1.2). We believe that our results obtained in this paper are a good complement to the earlier publications about model (1.1). The remainder of the paper is organized as follows. In Sect. 2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Sect. 3, the direction and stability of the local Hopf bifurcation are established. In Sect. 4, numerical simulations are carried out to support analytical ?ndings. Some main conclusions are drawn in Sect. 5.

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2 Stability of the positive equilibrium and local Hopf bifurcations In this section, we shall study the stability of the positive equilibrium and the existence of local Hopf bifurcations. It is easy to see that if the following condition: (H1) 0 < K < r1 , e1 > d1 , e2 > d2 holds, where K = r1 [d1 d2 + (d1 + e1 )(e1 ? d1 )] + r2 [b1 d2 + b2 (e2 ? d2 )] , a [d1 d2 + (d1 + e1 )(e1 ? d1 )] + c[b1 d2 + b2 (e2 ? d2 )]

then system (1.2) has an unique positive equilibrium E 0 ( X ? , S ? , I ? ), where X? = r1 [d1 d2 + (d1 + e1 )(e1 ? d1 )] + r2 [b1 d2 + b2 (e2 ? d2 )] , a [d1 d2 + (d1 + e1 )(e1 ? d1 )] + c[b1 d2 + b2 (e2 ? d2 )] ? d2 (r1 ? a X ) e2 ? d 2 ? , I? = S . S? = b1 d2 + b2 (e2 ? d2 ) d2

(2.1)

? (t ) = S (t ) ? S ? , I?(t ) = I (t ) ? I ? and still denote X ? (t ), S ? (t ), I?(t ) ? (t ) = X (t ) ? X ? , S Let X by X (t ), S (t ), I (t ), respectively, then (1.2) becomes ? ˙ (t ) = m 1 X (t ) + m 2 S (t ) + m 3 I (t ) ? a X 2 (t ) ? b1 X (t ) S (t ) ? b2 X (t ) I (t ), X ? ? ?˙ S (t ) = n 1 S (t ) + n 2 I (t ) + n 3 X (t ? τ ) + n 4 S (t ? τ ) (2.2) ? d1 S 2 (t ) ? (d1 + e1 ) S (t ) I (t ) + cX (t ? τ ) S (t ? τ ), ? ? ?˙ 2 I (t ) = l1 S (t ) + l2 I (t ) + (e2 ? d2 ) S (t ) I (t ) ? d2 I (t ), where m 1 = r ? 2a X ? ? b1 S ? ? b2 I ? , m 2 = ?b1 X ? , m 3 = ?b2 X ? , n 1 = ? d1 S ? + r2 + d2 ( S ? + I ? ) + e1 I ? , n 2 = ?(d1 + e1 ) S ? , n 3 = cS ? , n 4 = cX ? , l1 = e2 ? d2 , l2 = e2 S ? ? 2d2 I ? ? d2 S ? . The linearization of Eq. (2.2) at (0, 0, 0) is ? ˙ (t ) = m 1 X (t ) + m 2 S (t ) + m 3 I (t ), ?X ˙ (t ) = n 1 S (t ) + n 2 I (t ) + n 3 X (t ? τ ) + n 4 S (t ? τ ), S ?˙ I (t ) = l1 S (t ) + l2 I (t ) whose characteristic equation is λ3 + A1 λ2 + A2 λ + A3 ? B1 λ2 + B2 λ + B3 e?λτ = 0, where A1 = ?(m 1 + l2 + n 1 ), A2 = n 1 (m 1 + l2 ) + m 1 l2 ? l1 n 2 , A3 = m 1 n 2 l1 ? m 1 n 1 l2 , B1 = n 4 , B2 = m 2 n 3 ? n 4 (m 1 + l2 ), B3 = m 1 l2 n 4 + m 3 n 3 l1 ? m 2 n 3 l2 . (2.4)

(2.3)

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Denote P (λ) = λ3 + A1 λ2 + A2 λ + A3 , Q (λ) = ?( B1 λ2 + B2 λ + B3 ). Then, (2.4) takes the form P (λ) + Q (λ)e?λτ = 0. (2.5)

In order to investigate the distribution of roots of the transcendental equation (2.5), the following Lemma is useful. Lemma 2.1 [15] For the transcendental equation

(0) P λ, e?λτ1 , . . . , e?λτm = λn + p1 λn ?1 + · · · + pn ?1 λ + pn (1) e?λτ1 + · · · + p1 λn ?1 + · · · + pn ?1 λ + pn (m ) e?λτm = 0, + p1 λn ?1 + · · · + pn ?1 λ + pn (m ) (m ) (1) (1) (0) (0)

as (τ1 , τ2 , τ3 , . . . , τm ) vary, the sum of orders of the zeros of P (λ, e?λτ1 , . . . , e?λτm ) in the open right half complex plane can change, if and only if a zero appears on or crosses the imaginary axis. For τ = 0, (2.5) becomes λ3 + ( A1 ? B1 )λ2 + ( A2 ? B2 )λ + A3 ? B3 = 0. (2.6)

A set of necessary and suf?cient conditions for all roots of (2.6) to have a negative real part is given by the well-known Routh–Hurwitz criteria in the following form: (H2) ( A1 ? B1 )( A2 ? B2 ) ? ( A3 ? B3 ) > 0, A3 ? B3 > 0. Assume that i ω(ω > 0) is a root of (2.5). Following the line of Beretta and Kuang [2], ω must be the solution of the system of equations: ? ? sin ωτ = Im P (i ω) , Q (i ω) (2.7) ? cos ωτ = ? Re P (i ω) , Q (i ω) Namely, ω must be a positive root of the function F (ω) = | P (i ω)|2 ? Q (i ω)|2 . (2.8) We denote the root of F (ω) = 0 by ωk . Then, the characteristic roots λ = ±i ωk occur at the delay values τk

( j)

=

θk + 2 j π , ωk

j = 0 , 1, 2 , . . . ,

(2.9)

where θk ∈ [0, 2π) is the solution of ? ? sin θk = Im ? cos θk = De?ne τ0 = τk 0 =

(0) k ∈{1,2,3}

P (i ωk ) Q (i ωk ) , P (i ωk ) ? Re Q (i ωk )

.

(2.10)

min

τk

(0)

, ω0 = ωk 0 .

(2.11)

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In view of (2.8), we have

2 4 2 2 2 2 2 ω6 + A2 1 ? 2 A2 ? B1 ω + A2 ? 2 A1 A3 + 2 B1 B3 ? B2 ω + A3 ? B3 = 0. (2.12)

Let z = ω2 , then (2.13) become z 3 + r 1 z 2 + r 2 z + r 3 = 0, where

2 r1 = A2 1 ? 2 A2 ? B1 , 2 r2 = ? A2 2 ? 2 A1 A3 + 2 B1 B3 ? B2 , 2 r3 = A2 3 ? B3 .

(2.13)

Denote h ( z ) = z 3 + r 1 z 2 + r 2 z + r 3 = 0,

2 where r = r2 ? 1 3 r1 , q = 2 3 27 r1

(2.14)

?1 3 r1 r2 + r3 . Then, h (z ) = 3z 2 + 2r1 z + r2 . (2.15)

According to Beretta and Kuang [2], we derive sign

( j)

d Reλ dτ

( j) τ =τk

= sign

d F (ω) dω

ω=ωk

= sign[h (z k )],

(2.16) √ zk >

where τk are the delay values (2.9) at which the characteristic roots λ = ±i ωk ωk = 0, k = 1, 2, 3) occur. We assume that (H3) h (z k ) = 0.

Without loss of generality, we assume that (2.14) has three distinct positive roots, say z 1 , z 2 , z 3 such that √ √ √ ω1 = z 1 < ω2 = z 2 < ω 3 = z 3 . Since lim z →+∞ h (z ) = +∞, the only possibility for the signs of h (z ) at the roots z k is that h (z 1 ) > 0, h (z 2 ) < 0, h (z 3 ) > 0. In the following, we assume that the θk values are such that τ3

(0)

=

θ3 θ2 θ1 (0) (0) (0) (1) < τ2 = < τ1 = and τ1 < τ3 . ω3 ω2 ω1

Then under the assumption (H3), it is easy to know that all the characteristic roots λ of (2.5) (0) have Reλ < 0 in the delay interval [0, τ0 = τ3 ). Because of h (z 3 ) > 0, in the interval (0) (0) (τ0 = τ3 , τ2 ) there will be two characteristic roots with positive real parts that, thanks (0) to h (z 2 ) < 0, will cross the imaginary axis toward negative real parts at τ2 , whereas at (0) τ1 , because of h (z 1 ) > 0, other two characteristic roots cross the imaginary axis assuming ( j) ( j) ( j) positive real parts. Similar analysis to τ3 , τ2 , τ1 , we have the following Theorem 2.1 by the results of Kuang [11] and Hale [7].

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Theorem 2.1 If ( H 1) and ( H 2) hold, then the equilibrium E 0 ( X ? , S ? , I ? ) of system (1.2) is asymptotically stable for τ ∈ [0, τ0 ). In addition to the conditions ( H 1) and ( H 2), we further assume that ( H 3) holds, then the positive equilibrium E 0 ( X ? , S ? , I ? ) is asymptotically stable when 0, τ3 and unstable when τ3 , τ2

(0) (0) (0) (0)

∪ τ2 , τ1

(0)

(0)

(0)

∪ τ1 , +∞ .

(0)

The stability switches occur at τ3 , τ2 , τ1 and system (1.2) undergoes a Hopf bifurcation ( j) at the positive equilibrium E 0 ( X ? , S ? , I ? ) when τ = τk , k = 1, 2, 3; j = 0, 1, 2, ldots. 3 Direction and stability of the Hopf bifurcation In the previous section, we have obtained conditions for Hopf bifurcation to occur when ( j) τ = τk , k = 1, 2, 3; j = 0, 1, 2, . . . . In this section, we shall derive the explicit formulae for determining the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium E 0 ( X ? , S ? , I ? ) at these critical value of τ , by using techniques from normal form and center manifold theory [8]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcation at the positive equilibrium E 0 ( X ? , S ? , I ? ) for ( j) τ = τk , k = 1, 2, 3; j = 0, 1, 2, . . . ., and then ±i ω0 are corresponding purely imaginary roots of the characteristic equation at the positive equilibrium E 0 ( X ? , S ? , I ? ). ( j) For convenience, let x1 (t ) = X (τ t ), x2 (t ) = S (τ t ), x3 (t ) = I (τ t ), and τ = τk + μ, ( j) where τk is de?ned by (2.10) and μ ∈ R , then system (2.2) can be written as an FDE in C = C ([?1, 0]), R 3 ) as x ˙ (t ) = L μ xt + f (μ, xt ), (3.1) where x (t ) = (x1 (t ), x2 (t ), x3 (t ))T ∈ C , and xt (θ ) = x (t + θ ) = (x1 (t + θ ), x2 (t + θ ), x3 (t + θ ))T ∈ C , and L μ : C → R , f : R × C → R are given by ? ? ?? m2 m3 m1 φ1 (0) ( j) n1 n 2 ? ? φ2 (0) ? L μ φ = τk + μ ? 0 φ3 (0) 0 l1 l2 ?? ? ? φ1 (?1) 0 0 0 ( j) n4 0 ? ? φ2 (?1) ? (3.2) + τk + μ ? n 3 0 0 0 φ3 (?1) and f (μ, φ) = τk

( j)

(0)

?

? ? 2 ? +μ ? ? ?d1 φ2 (0) ? (d1 + e1 )φ2 (0)φ3 (0) + cφ1 (?1)φ2 (?1) ? , 2 (0) (e2 ? d2 )φ2 (0)φ3 (0) ? d2 φ3

2 (0) ? b φ (0)φ (0) ? b φ (0)φ (0) ?a φ1 1 1 2 2 1 3

? (3.3)

respectively, where φ(θ ) = (φ1 (θ ), φ2 (θ ), φ3 (θ ))T ∈ C . From the discussion in Sect. 2, we know that if μ = 0, then system (3.1) undergoes a Hopf bifurcation at the positive equilibrium E 0 ( X ? , S ? , I ? ) and the associated characteristic ( j) equation of system (3.1) has a pair of simple imaginary roots ±ω0 τk .

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Bifurcation analysis of epidemic predator–prey model

By the representation theorem, there is a matrix function with bounded variation components η(θ, μ), θ ∈ [?1, 0] such that

0

L μφ =

?1

dη(θ, μ)φ(θ ), for φ ∈ C .

(3.4)

In fact, we can choose m2 m1 ( j) n1 η(θ, μ) = τk + μ ? 0 0 l1 ? 0 0 ( j) n4 + τk + μ ? n 3 0 0 ? ? m3 n 2 ? δ(θ ) l2 ? 0 0 ? δ(θ + 1), 0

(3.5)

where δ is the Dirac delta function. For φ ∈ C ([?1, 0], R 3 ), de?ne ? d φ(θ ) ? ?1 ≤ θ < 0, ? dθ , 0 A(μ)φ = ? θ =0 ? dη(s , μ)φ(s ),

?1

(3.6)

and Rφ = 0, ?1 ≤ θ < 0, f (μ, φ), θ = 0. (3.7)

Then (3.1) is equivalent to the abstract differential equation x ˙t = A(μ)xt + R (μ)xt , where xt (θ ) = x (t + θ ), θ ∈ [?1, 0]. For ψ ∈ C ([0, 1], ( R 3 )? ), de?ne ? d ψ(s ) ? s ∈ (0, 1], ? ? ds , ? 0 A ψ(s ) = T ? ? dη (t , 0)ψ(?t ), s = 0.

?1

(3.8)

For φ ∈ C ([?1, 0], R 3 ) and ψ ∈ C ([0, 1], ( R 3 )? ), de?ne the bilinear form

0 θ

ψ, φ = ψ(0)φ(0) ?

?1 ξ =0

ψ T (ξ ? θ )dη(θ )φ(ξ )dξ,

where η(θ ) = η(θ, 0), the A = A(0) and A? are adjoint operators. By the discussions in the ( j) Sect. 2, we know that ±i ω0 τk are eigenvalues of A(0), and they are also eigenvalues of A? ( j) ( j) corresponding to i ω0 τk and ?i ω0 τk respectively. By direct computation, we can obtain q (θ ) = (1, α, β)T ei ω0 τk

( j)

θ

, q ? (s ) = M (1, α ? , β ? )ei ω0 τk

( j)

s

,

M=

1 , B

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where β(i ω0 ? l2 ) , l1 (i ω0 ? m 1 )l1 β= , m 2 (i ω0 ? l2 ) + m 3 l1 i ω0 + m 1 , α? = ( j) n 3 e?i ω0 τk α= β? = n 2 (i ω0 + m 1 ) ? m 3 n 3 e?i ω0 τk n 3 (i ω0 + l2 )e?i ω0 τk

( j)

( j) ( j)

,

( j)

? ? i ω0 τk ? ? + τ α( B = 1 + αα ? ? + ββ . k ? n 3 α + n 4 β )e

Furthermore, < q ? (s ), q (θ ) >= 1 and < q ? (s ), q ? (θ ) >= 0. Next, we use the same notations as those in Hassard et al. [8] and we ?rst compute the coordinates to describe the center manifold C0 at μ = 0. Let xt be the solution of Eq. (3.8) when μ = 0. De?ne z (t ) = q ? , xt , W (t , θ ) = xt (θ ) ? 2 Re{z (t )q (θ )}. on the center manifold C0 , and we have W (t , θ ) = W (z (t ), z ? (t ), θ ), where W (z (t ), z ? (t ), θ ) = W (z , z ? ) = W20 (θ ) where ? ? (1) ? ? (1) ? (1) W20 (θ ) W11 (θ ) W (θ ) ? ? (2) ? ? (2) ? ? 02 (2) W20 (θ ) = ? W20 (θ ) ? , W11 (θ ) = ? W11 (θ ) ? , W20 (θ ) = ? W02 (θ ) ? (3) (3) (3) W20 (θ ) W11 (θ ) W02 (θ ) ? ? ? . Noting and z and z ? are local coordinates for center manifold C0 in the direction of q ? and q that W is also real if xt is real, we consider only real solutions. For solutions xt ∈ C0 of (3.8), z ˙ (t ) = i ω0 τk z + q ? ? (θ ) f (0, W (z , z ? , θ )) + 2 Re{zq (θ )} = i ω0 τk z + q ? ? (θ ) f 0 .

def ( j) ( j)

(3.9)

(3.10)

z2 z ?2 ? + W02 (θ ) + · · · , + W11 (θ )z z 2 2

(3.11)

That is z ˙ (t ) = i ω0 τk z + g (z , z ? ), where g (z , z ? ) = g20 z2 z ?2 z2 z ? ? + g02 + g21 + g11 z z + ··· . 2 2 2

( j)

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Bifurcation analysis of epidemic predator–prey model

Hence, we have g (z , z ?) = q ? ? (0) f 0 (z , z ? ) = f (0, xt ) ? τ ( j ) ? (a + b1 α + b2 β) + α =M ? ? d1 + (d1 + e1 )αβ + cα e?2i ω0 τk k ? ? (e2 ? d2 )αβ ? d2 β 2 +β ?τ z2 + 2 M k

( j)

( j)

a + b1 Re{α } + b2 Re{β } zz ?

? ? (e2 ? d2 ) Re{αβ ? } + c Re{α }] + β ? } + d2 |β |2 +α ? ? [d2 + (d1 + e1 ) Re{αβ ?τ +M k

( j)

? + cα ?+α a + b1 α ? + b2 β ?β ? e?2i ω0 τk ? ? d1 + (d1 + e1 )α ?τ z ?2 + M k

( j)

( j)

? ? [(e2 ? d2 ) +β

?2 ? + d2 β ×α ?β

a W20 (0) + 2W11 (0) + b1

(1)

(1)

1 (1) α ? W20 (0) 2

1 1 1 (2) (3) (2) ? W (3) (0) + W (3) (0) β + W20 (0) + α W11 (0) + W11 (0) + b2 20 2 2 2 20 ? ? d1 W20 (0) + 2W11 (0) + (d1 + e1 ) +β W11 (0) + W11 (0) + α 1 1 (3) (2) (3) ? W (2) (0) + α β ? W20 (0) + β W11 (0) +α W11 (0) 20 2 2 ( j) ( j) 1 1 i ω0 τ ( j ) (1) (2) (1) α ? e k W20 (?1) + ei ω0 τk W20 (?1) + α ei ω0 τk W11 (?1) +c 2 2 ( j) 1 1 (2) (3) (2) ? W (2) (0) + α +e?i ω0 τk W11 (?1) + β?? (e2 ? d2 ) β ? W20 (0)β W11 (0) 20 2 2

(3) ? W (3) (0) + 2β W (3) (0) +α W11 (0) + d2 β 20 11 (1) (3) (1) (1)

? + h.o.t. z2 z

and we obtain ? τ ( j ) ? (a + b1 α + b2 β) + α g20 = 2 M ? ? d1 + (d1 + e1 )αβ + cα e?2i ω0 τk k ? ? (e2 ? d2 )αβ ? d2 β 2 , +β ?τ g11 = 2 M k

( j)

( j)

? ? [d2 + (d1 + e1 ) Re{αβ ? } + c Re{α }] a + b1 Re{α } + b2 Re{β } + α ,

( j)

? ? (e2 ? d2 ) Re{αβ ? } + d2 |β |2 +β ?τ g02 = 2 M k

( j)

?+α ? + cα ?? d1 + (d1 + e1 )α a + b1 α ? + b2 β ?β ? e?2i ω0 τk ,

(1) (1)

? ? [(e2 ? d2 ) +β

?2 ? + d2 β ×α ?β ?τ g21 = 2 M k

( j)

a W20 (0) + 2W11 (0) + b1

(2)

1 1 (2) (1) α ? W20 (0) + W20 (0) 2 2

+α W11 (0) + W11 (0) + b2

(3)

1 1 ? W (3) (0) + W (3) (0) + β W (1) (0) + W (3) (0) β 20 11 11 2 2 20

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?? d1 W (1) (0) + 2W (1) (0) + (d1 + e1 ) +α 20 11 +α W11 (0) + c

(3)

1 1 (3) (2) ? W (2) (0) + α β ? W20 (0) + β W11 (0) 20 2 2

( j) ( j) 1 i ω0 τ ( j ) (1) 1 (2) (1) α ? e k W20 (?1) + ei ω0 τk W20 (?1) + α ei ω0 τk W11 (?1) 2 2 ( j) 1 1 (2) (3) (2) ? W (2) (0) + α +e?i ω0 τk W11 (?1) + β?? (e2 ? d2 ) β ? W20 (0)β W11 (0) 20 2 2

(3) ? W (3) (0) + 2β W (3) (0) +α W11 (0) + d2 β 20 11 (i ) (i ) ( j) ( j)

.

For unknown W20 (0), W11 (0), W20 (?1), W11 (?1), (i = 1, 2, 3; j = 1, 2) in g21 , we still need to compute them. From (3.8) and (3.9), we have W = where z2 z ?2 ? + H02 (θ ) + · · · . + H11 (θ )z z 2 2 Comparing the coef?cients, we obtain H (z , z ? , θ ) = H20 (θ ) A ? 2i τk ω0 W20 = ? H20 (θ ), AW11 (θ ) = ? H11 (θ ). We know that for θ ∈ [?1, 0), ? (θ ) = ?g (z , z ? )q (θ ) ? g ? (z , z ? )q ? (θ ). (3.16) H (z , z ? , θ ) = ?q ? ? (0) f 0 q (θ ) ? q ? (0) f?0 q Comparing the coef?cients of (3.16) with (3.13) gives that H20 (θ ) = ?g20 q (θ ) ? g ? 02 q ? (θ ). H11 (θ ) = ?g11 q (θ ) ? g ? 11 q ? (θ ). From (3.14, 3.17) and the definition of A , we get ˙ 20 (θ ) = 2i ω0 τ ( j ) W20 (θ ) + g20 q (θ ) + g? ? (θ ). W 02 q k Noting that q (θ ) = q (0)e W20 (θ ) =

(1)

( j) i ω0 τk θ

?1 ≤ θ < 0, def AW ? 2 Re{q ? ? (0) f?q (θ )}, = AW + H (z , z ? , θ ), (3.12) AW ? 2 Re{q ? ? (0) f?q (θ )} + f , θ =0

(3.13)

( j)

(3.14) (3.15)

(3.17) (3.18)

(3.19)

, we have

( j)

ig20

( j) ω0 τk (2)

q (0)ei ω0 τk

(3) T

θ

+

ig ? 02

( j) 3ω0 τk

q ? (0)e?i ω0 τk

( j)

θ

+ E 1 e2i ω0 τk

( j)

θ

,

(3.20)

where E 1 = E 1 , E 1 , E 1 ∈ R 3 is a constant vector. Similarly, from (3.15, 3.18) and the definition of A, we have ˙ 11 (θ ) = g11 q (θ ) + g? ? (θ ), W 11 q ( j) ( j) ig11 ig ? 11 q (0)ei ω0 τk θ + q ? (0)e?i ω0 τk θ + E 2 . W11 (θ ) = ? ( j) ( j) ω0 τk ω0 τk where E 2 = E 2 , E 2 , E 2

(1) (2) (3) T

(3.21) (3.22)

∈ R 3 is a constant vector.

123

Bifurcation analysis of epidemic predator–prey model

In what follows, we shall seek appropriate E 1 , E 2 in (3.20, 3.22), respectively. It follows from the definition of A, (3.17) and (3.18) that

0

dη(θ )W20 (θ ) = 2i ω0 τk W20 (0) ? H20 (0)

?1

( j)

(3.23)

and

0

dη(θ )W11 (θ ) = ? H11 (0),

?1

(3.24)

where η(θ ) = η(0, θ ). From (3.12), we have H20 (0) = ?g20 q (0) ? g? ? (0) + 2τk ( H1 , H2 , H3 )T , 02 q H11 (0) = ?g11 q (0) ? g? ? (0) + 2τk ( P1 , P2 , P3 )T , 11 (0)q where H1 = ?(a + b1 α + b2 β), H2 = ? d1 + (d1 + e1 )αβ + cα e?2i ω0 τk H3 = (e2 ? d2 )αβ ? d2 β 2 , P1 = a + b1 Re{α } + b2 Re{β }, P2 = d1 + (d1 + e1 ) Re{αβ ? } + c Re{α }, P3 = (e2 ? d2 ) Re{αβ ? } + d 2 |β |2 . Noting that ? ?i ω0 τ ( j ) I k ? ??i ω0 τ ( j ) I k ?

?1

( j)

( j)

(3.25) (3.26)

( j)

,

0

? e

i ω0 τk θ

( j)

?

?1 0

dη(θ )? q (0) = 0, ?

e

( j) ?i ω0 τk θ

? (0) = 0 dη(θ )? q

and substituting (3.20) and (3.25) into (3.23), we have ? ? 0 ?2i ω0 τ ( j ) I ? k

?1

e2i ω0 τk

( j)

θ

( j) dη(θ )? E 1 = 2τk ( H1 , H2 , H3 )T .

That is ? ? 2i ω0 ? m 1 ?m 2 ?m 3 ? E 1 = 2( H1 , H2 , H3 )T . ? ?n e?2i ω0 τk( j ) 2i ω ? n ? n e?2i ω0 τk( j ) ?n 3 0 1 4 2 0 ?l1 2i ω0 ? l2 It follows that E1 =

(1) 11 1 (2) 12 1 (3) 13 1

,

E1 =

,

E1 =

,

(3.27)

123

C. Xu, M. Liao

where ?

1

11

12

13

?, = det ? ?n 3 e 2i ω0 ? n 1 ? n 4 e ?n 2 0 ?l1 2i ω0 ? l2 ? ? H1 ?m 2 ?m 3 ( j) ?, = 2 det ? H2 2i ω0 ? n 1 ? n 4 e?2i ω0 τk ?n 2 H3 ?l1 2i ω0 ? l2 ? ? 2i ω0 ? m 1 H1 ?m 3 ( j) ?, = 2 det ? ?n 3 e?2i ω0 τk H2 ?n 2 0 H3 2i ω0 ? l2 ? ? 2i ω0 ? m 1 ?m 2 H1 ( j) ( j) = 2 det ? ?n 3 e?2i ω0 τk 2i ω0 ? n 1 ? n 4 e?2i ω0 τk H2 ? . 0 ?l1 H3

2i ω0 ? m 1

( j) ?2i ω0 τk

?m 2

( j) ?2i ω0 τk

?m 3

?

Similarly, substituting (3.21) and (3.26) into (3.24), we have ? ?

?1 0

? dη(θ )? E 2 = 2τk ( P1 , P2 , P3 )T , .

( j)

That is ? m3 m1 m2 ? n 3 n 1 + n 4 n 2 ? E 2 = 2(? P1 , ? P2 , ? P3 )T . 0 l1 l2 ? It follows that E2 = where = ? m3 m1 m2 det ? n 3 n 1 + n 4 n 2 ? , 0 l1 l2 ? ? m3 ? P1 m 2 2 det ? ? P2 n 1 + n 4 n 2 ? , ? P3 l1 l2 ? ? m 1 ? P1 m 3 2 det ? n 3 ? P2 n 2 ? , 0 ? P3 l2 ? ? ? P1 m1 m2 2 det ? n 3 n 1 + n 4 ? P2 ? . 0 l1 ? P3 ?

(1) 21 2

,

E2 =

(2)

22 2

,

E2 =

(3)

23 2

,

(3.28)

2

21

=

22

=

23

=

123

Bifurcation analysis of epidemic predator–prey model

From (3.20,3.22,3.27) and (3.28), we can calculate g21 and derive the following values: c1 (0) = i

( j) 2ω0 τk

g20 g11 ? 2|g11 |2 ? ,

|g02 |2 3

+

g21 , 2

μ2 = ?

Re{c1 (0)} Re λ (τk )

( j)

β2 = 2 Re(c1 (0)), T2 = ? Im{c1 (0)} + μ2 Im λ (τk ) ω0 τk

( j) ( j)

.

These formulaes give a description of the Hopf bifurcation periodic solutions of (3.1) at ( j) τ = τk , (k = 1, 2, 3; j = 0, 2, 3, . . .) on the center manifold. From the discussion above, we have the following result: Theorem 3.1 The periodic solution is supercritical (subcritical) if μ2 > 0(μ2 < 0); the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if β2 < 0(β2 > 0); the periods of the bifurcating periodic solutions increase (decrease) if T2 > 0(T2 < 0). Remark 3.2 A τ T -periodic solution of (3.1) is a T -periodic solution of (2.2).

4 Numerical examples In this section, we present some numerical results of system (1.2) to verify the analytical predictions obtained in the previous section. From Sect. 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcation periodic solutions. Let us consider the following system: ? ˙ ? ? X (t ) = X (t )[2 ? 0.5 X (t ) ? 0.6 S (t ) ? 0.82 I (t )], ˙ (t ) = 2 X (t ? τ ) S (t ? τ ) + S (t )[?3 ? 0.4( S (t ) + I (t )) + 0.9 I (t )], (4.1) S ? ?˙ I (t ) = I (t )[4 S (t ) ? 2( S (t ) + I (t ))], which has a positive equilibrium E 0 ( X ? , S ? , I ? ) ≈ (1.4559, 0.8818, 0.8818) and satis?es the conditions indicated in Theorem 2.1. When τ = 0, the positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818) is asymptotically stable. Take j = 0, for example, by some complicated computation by means of Matlab 7.0, we get ω0 ≈ 0.8940, τ0 ≈ 0.2551, λ (τ0 ) ≈ 1.0038 ? 6.7531i . Thus, we can calculate the following values: c1 (0) ≈ ?2.1230 ? 6.1422i , μ2 ≈ 2.1150, β2 ≈ ?4.2460, T2 ≈ 89.5441. Furthermore, it follows that μ2 > 0 and β2 < 0. Thus, the positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818) is stable when τ < τ0 as is illustrated by the computer simulations (see Fig. 1). When τ passes through the critical value τ0 , the positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from the positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818).

123

C. Xu, M. Liao

3 2.5 2 2.5 2 1.5

X(t)

1.5 1 0.5 0 2 1.8 1.6 1.4

S(t)

1 0.5 0 0 20 40 60 80 100 120

20

40

60

80

100

120

t

t

2 1.5

I(t)

I(t)

1.2 1 0.8 0.6 0.4 0.2 0 20 40 60 80 100 120

1 0.5 0 3 2 1 1 1.5 2 2.5 3

S(t)

0 0.5

X(t)

t

Fig. 1 Behavior and phase portrait of system (4.1) with τ = 0.08 < τ0 ≈ 0.2551. The positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818) is asymptotically stable. The initial value is (0.5,0.5,0.5)

Since μ2 > 0 and β2 < 0, the direction of the Hopf bifurcation is τ > τ0 , and these bifurcating periodic solutions from E 0 ≈ (1.4559, 0.8818, 0.8818) at τ0 are stable, which are depicted in Fig. 2.

5 Conclusions In this paper, we have investigated local stability of the positive equilibrium E 0 ( X ? , S ? , I ? ) and local Hopf bifurcation in an autonomous epidemic predator–prey model with delay. We have showed that if the conditions (H1) and (H2) hold, the positive equilibrium E 0 ( X ? , S ? , I ? ) of system (1.2) is asymptotically stable for all τ ∈ [0, τ0 ). This means that the density of the prey, the density of the susceptible predator and the infected predator will tend to be stable, that is, the density of the prey, the density of the susceptible predator and the infected predator will tend to X ? , S ? , I ? , respectively, for all τ ∈ [0, τ0 ). Under the conditions (H1) and (H2), if the condition (H3) holds, as the delay τ increases, the positive equilibrium loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E 0 ( X ? , S ? , I ? ), that is, a family of periodic orbits bifurcates from the the positive equilibrium E 0 ( X ? , S ? , I ? ). This shows that the density of the prey, the density of the susceptible predator and the infected predator may keep in an oscillatory mode near the positive equilibrium E 0 ( X ? , S ? , I ? ). Applying the normal form theory and the center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating

123

Bifurcation analysis of epidemic predator–prey model

3 2.5 2 2 1.8 1.6 1.4 1.2

X(t)

S(t)

50 100 150 200 250

1 0.8 0.6 0.4 0.2 0 0

1.5 1 0.5 0 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 50 100 150 200 250

50

100

150

200

250

t

t

2 1.5

I(t)

I(t)

1 0.5 0 2 1.5 1 0.5 1 1.5 2 2.5

3

S(t)

0 0.5

X(t)

t

Fig. 2 Behavior and phase portrait of system (4.1) with τ = 0.3 > τ0 ≈ 0.2551. Hopf bifurcation occurs from the positive equilibrium E 0 ≈ (1.4559, 0.8818, 0.8818). The initial value is (0.5,0.5,0.5)

periodic orbits are discussed. A numerical example verifying our theoretical results is also included.

References

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C. Xu, M. Liao 11. Kuang, Y.: Delay Differential Equations with Applications in Populations Dynamics. Academic Press, INC, Massachusetts (1993) 12. Kuang, Y., Takeuchi, Y.: Predator-prey dynamics in models of prey dispersal in two-patch environments. Math. Biosci. 120(1), 77–98 (1994) 13. Li, B., Wang, M.X.: Stationary patterns of the stage-structured predator- prey model with diffusion and cross-diffusion. Math. Comput. Model. 54(5–6), 1380–1393 (2011) 14. Oeda, K.: Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone. J Differ. Equ. 250(10), 3988–4009 (2011) 15. Ruan, S.G., Wei, J.J.: On the zero of some transcendential functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discret. Impuls. Syst. Ser. A 10(2), 863–874 (2003) 16. Song, Y.L., Wei, J.J.: Local Hopf bifurcation and global existence of periodic solutions in a delayed predator-prey system. J. Math. Anal. Appl. 301(1), 1–21 (2005) 17. Tian, B.D., Qiu, Y.H., Chen, N.: Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay. Appl. Math. Comput. 215(2), 779–790 (2009) 18. Vayenas, D.V., Aggelis, G., Tsagou, V., Pavlou, S.: Dynamics of a two-prey-one-predator system with predator switching regulated by a catabolic repression control-like mode. Ecol. Model. 186(3), 345– 357 (2005) 19. Xu, C.J., Liao, M.X., He, X.F.: Stability and Hopf bifurcation analysis for a Lokta-Volterra predator-prey model with two delays. Int. J. Appl. Math. Comput. Sci. 21(1), 97–107 (2011) 20. Xu, C.J., Tang, X.H., Liao, M.X.: Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments. Appl. Math. Comput. 216(10), 2920–2936 (2010) 21. Xu, C.J., Tang, X.H., Liao, M.X., He, X.F.: Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays. Nonlinear Dyn. 66(1–2), 169–183 (2011) 22. Xu, R., Ma, Z.E.: Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure. Chaos Solitons Fract. 38(3), 669–684 (2008) 23. Yan, X.P., Zhang, C.H.: Hopf bifurcation in a delayed Lokta-Volterra predator-prey system. Nonlinear Anal. Real World Appl. 9(1), 114–127 (2008) 24. Zhou, X.Y., Shi, X.Y., Song, X.Y.: Analysis of non-autonomous predator-prey model with nonlinear diffusion and time delay. Appl. Math. Comput. 196(1), 129–136 (2008)

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