Applied Mathematics and Computation 224 (2013) 372–386

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Modelling and analysis of a delayed predator–prey model withdisease in the predator

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.08.067

⇑ Corresponding author.E-mail address: rxu88@163.com (R. Xu).

Rui Xu ⇑, Shihua ZhangInstitute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, Hebei, PR China

a r t i c l e i n f o

Keywords:Eco-epidemiological modelDelayHopf bifurcationLaSalle’s invariance principleGlobal stability

a b s t r a c t

In this paper, we study a predator–prey model with a transmissible disease spreading inthe predator population and a time delay representing the gestation period of the predator.By analyzing corresponding characteristic equations, the local stability of each of feasibleequilibria and the existence of Hopf bifurcations at the disease-free equilibrium and thecoexistence equilibrium are established, respectively. By means of Lyapunov functionalsand LaSalle’s invariance principle, sufficient conditions are derived for the global stabilityof the predator-extinction equilibrium and the disease-free equilibrium and the globalattractiveness of the coexistence equilibrium of the system, respectively. Numerical simu-lations are carried out to support the theoretical analysis.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Since the pioneering work of Anderson and May [1], great attention has been paid to the modelling and analysis of eco-epidemiological systems recently (see, for example, [1,2,4,9–14,18,19,21,24,25,27–30,32–35]). An increasing number ofworks are devoted to the study of the relationships between demographic processes among different populations and dis-eases. Most of these works dealt with predator–prey models with disease in the prey (see, for example, [4,10,14,21,27–30,34]). In [4], Chattopadhyay and Arino considered a predator–prey model with disease in the prey. They assumed thatthe sound prey population grows according to a logistic law involving the whole prey population. In [27], Xiao and Chen for-mulated and analyzed a three species (namely, sound prey (susceptible), infected prey (infective), and predator) eco-epide-miological system. It was assumed that the disease spreads among the prey population only and the disease is notgenetically inherited. The infected populations do not recover or become immune. They considered the case where the pred-ator mainly eats only the infected prey. This is in accordance with the fact that the infected individuals are less active and canbe caught more easily, or the behavior of the prey is modified such that they live in parts of the habitat which are accessibleto the predator.

Recently, attention has been paid to the modelling and analysis of eco-epidemiological predator–prey system by assum-ing that the predator population suffer a transmissible disease (see, for example, [6,9,12,22,23,25,30–32]). In [25], afterreviewing the classical Lotka–Volterra type predator–prey model and SIS epidemic model, Venturino formulated two eco-epidemiological models with disease in the predators and mass action and standard incidence rates, respectively. In thetwo models, it was assumed that the disease spreads among predators only and that the infected individuals do not repro-duce, only sound ones do. Analysis of the long-term behavior of solutions of the two models was carried out to show thatwhether and how the presence of the disease in the predator species affects the behavior of the ecological system, but alsowhether the introduction of a sound prey can affect the dynamics of the disease in the predator population. Following the

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 373

work of Venturino [25], in [31], Zhang and Sun considered a predator–prey model with disease in the predator and generalfunctional response. Sufficient conditions were derived for the permanence of the eco-epidemiological system. In [6], Guoet al. considered the following eco-epidemiological model:

_xðtÞ ¼ xðtÞðr � a11xðtÞÞ � a12xðtÞSðtÞ1þmxðtÞ ;

_SðtÞ ¼ a21xðtÞSðtÞ1þmxðtÞ � r1SðtÞ � bSðtÞIðtÞ;

_IðtÞ ¼ bSðtÞIðtÞ � r2IðtÞ;

ð1:1Þ

where xðtÞ; SðtÞ and IðtÞ represent the densities of the prey, susceptible (sound) predator and the infected predator populationat time t, respectively. The parameters a11; a12; a21;m; r; r1; r2 and b are positive constants. In system (1.1), the followingassumptions were made:

(A1) In the absence of predation, the prey population xðtÞ grows logistically with the intrinsic growth rate r > 0 and car-rying capacity r=a11.

(A2) The total predator population N is composed of two population classes: one is the class of susceptible (sound) pred-ator, denoted by S, and the other is the class of infected predator, denoted by I.

(A3) The disease spreads among the predator species only by contact and the disease can not be transmitted vertically.The infected predator population do not recover or become immune. The disease incidence is assumed to be the sim-ple mass action incidence bSI, where b > 0 is called the disease transmission coefficient.

(A4) Only the susceptible predators have ability to capture prey with Holling type-II response function x=ð1þmxÞ;m > 0is the half saturation rate of the predator and the infected predator are unable to catch prey due to a high infection.The parameter a12 is the capturing rate of the sound predator, a21=a12 is the conversion rate of nutrients into thereproduction of the predator by consuming prey, r1 is the natural death rate of the sound predator, r2 is the naturaland disease-related mortality rate of the infected predator. Here, r1 6 r2.

It is well known that time delays of one type or another have been incorporated into mathematical models of populationdynamics by many researchers. We refer to the monographs of Gopalsamy [5], Kuang [15] and MacDonald [16] for generaldelayed biological systems and to Beretta and Kuang [3] and Wangersky and Cunningham [26] and references cited thereinfor studies on delayed predator–prey systems. In general, delay differential equations exhibit much more complicateddynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstableand cause the population to fluctuate. Time delay due to gestation is a common example, because generally the consumptionof prey by predator throughout its past history governs the present birth rate of the predator. In [26], Wangersky and Cunn-ingham proposed and studied the following non-Kolmogorov-type predator–prey model:

_xðtÞ ¼ xðtÞðr1 � axðtÞ � a1yðtÞÞ;_yðtÞ ¼ a2xðt � sÞyðt � sÞ � r2yðtÞ:

ð1:2Þ

This model assumes that a duration of s time units elapses when an individual prey is killed and the moment when the cor-responding addition is made to the predator population.

In this paper, motivated by the works of Guo et al. [6], Venturino [25] and Wangersky and Cunningham [26], we are con-cerned with the combined effects of a transmissible disease spreading in predator population by contact and a time delaydue to the gestation of the predator on the global dynamics of a predator–prey system with Holling type-II functional re-sponse. To this end, we consider the following delay differential equations:

_xðtÞ ¼ xðtÞðr � a11xðtÞÞ � a12xðtÞSðtÞ1þmxðtÞ ;

_SðtÞ ¼ a21xðt � sÞSðt � sÞ1þmxðt � sÞ � r1SðtÞ � bSðtÞIðtÞ;

_IðtÞ ¼ bSðtÞIðtÞ � r2IðtÞ;

ð1:3Þ

where the parameters a11; a12; a21;m; r; r1; r2 and b are the same as that defined in system (1.1), s P 0 represents the timedelay due to the gestation of the sound predator.

The initial conditions for system (1.3) take the form

xðhÞ ¼ /1ðhÞ; SðhÞ ¼ /2ðhÞ; IðhÞ ¼ /3ðhÞ;/1ð0Þ > 0; /2ð0Þ > 0; /3ð0Þ > 0;

ð1:4Þ

where ð/1ðhÞ;/2ðhÞ;/3ðhÞÞ 2 Cð½�s;0�;R3þ0Þ, the space of continuous functions mapping the interval ½�s;0� into R3

þ0, hereR3þ0 ¼ fðx1; x2; x3Þ : xi P 0; i ¼ 1;2;3g.

374 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

It is well known by the fundamental theory of functional differential equations [8], system (1.3) has a unique solutionðxðtÞ; SðtÞ; IðtÞÞ satisfying the initial conditions (1.4). It is easy to show that all solutions of system (1.3) with initial conditions(1.4) are defined on ½0;þ1Þ and remain positive for all t P 0.

To the best of our knowledge, there have been few works in the literature studying the global stability of the coexistenceequilibrium of an eco-epidemiological model. In the present paper, our primary goal is to carry out a complete mathematicalanalysis of system (1.3) and establish its global dynamics. The strategy of proofs utilizes global Lyapunov functionals andLaSalle’s invariance principle that are motivated by the work in McCluskey [17].

The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations,we study the local stability of each of feasible equilibria of system (1.3) and the existence of Hopf bifurcations of system (1.3)at the disease-free equilibrium and the coexistence equilibrium, respectively. In Section 3, by means of Lyapunov functionalsand LaSalle’s invariance principle, we establish sufficient conditions for the global stability of the predator-extinction equi-librium and the disease-free equilibrium and the global attractiveness of the coexistence equilibrium of system (1.3), respec-tively. Numerical simulations are carried out in Section 4 to support the main theoretical results. A brief discussion is givenin Section 5 to conclude this work.

2. Local stability and Hopf bifurcations

In this section, we study the local stability of each of feasible equilibria of system (1.3) by analyzing the correspondingcharacteristic equations, respectively.

System (1.3) always has a trivial equilibrium E0ð0;0;0Þ and a predator-extinction equilibrium E1ðr=a11;0;0Þ. If the follow-ing holds:

(H1) a21r > r1ða11 þmrÞ,

then system (1.3) has a disease-free equilibrium E2ðx1; S1;0Þ, where

x1 ¼r1

a21 �mr1; S1 ¼

a21½a21r � r1ða11 þmrÞ�a12ða21 �mr1Þ2

: ð2:1Þ

Further, it is easy to show that if bS1 > r2, system (1.3) has a coexistence equilibrium E�ðx�; S�; I�Þ, where

x� ¼mr � a11 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðmr � a11Þ2 þ 4ma11ðr � a12S�Þ

q2ma11

;

S� ¼ r2

b; I� ¼ 1

ba21x�

1þmx�� r1

� �:

ð2:2Þ

It is easy to prove that the equilibrium E0ð0;0;0Þ is always unstable.The characteristic equation of system (1.3) at the equilibrium E1ðr=a11;0;0Þ is of the form

ðkþ rÞðkþ r2Þ kþ r1 �a21r

a11 þmre�ks

� �¼ 0: ð2:3Þ

Eq. (2.3) always has two negative real roots: k1 ¼ �r; k2 ¼ �r2. All other roots of Eq. (2.3) are determined by the followingequation

kþ r1 �a21r

a11 þmre�ks ¼ 0: ð2:4Þ

Denote

f ðkÞ ¼ kþ r1 �a21r

a11 þmre�ks:

If (H1) holds, it is easy to show that, for k real,

f ð0Þ ¼ r1 �a21r

a11 þmr< 0; lim

k!þ1f ðkÞ ¼ þ1:

Hence, f ðkÞ ¼ 0 has a positive real root. Therefore, if (H1) holds, the equilibrium E1ðr=a11;0;0Þ is unstable.If a21r < r1ða11 þmrÞ, we claim that E1ðr=a11;0;0Þ is locally asymptotically stable. Otherwise, there is a root k satisfying

Rek P 0. It follows from (2.4) that

Rek ¼ a21ra11 þmr

e�sRek cosðsImkÞ � r1 6a21r

a11 þmr� r1 < 0;

which is a contradiction. Hence, if a21r < r1ða11 þmrÞ, the equilibrium E1ðr=a11;0;0Þ is locally asymptotically stable.The characteristic equation of system (1.3) at the equilibrium E2ðx1; S1;0Þ takes the form

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 375

ðkþ r2 � bS1Þ½k2 þ p1kþ p0 þ ðq1kþ q0Þe�ks� ¼ 0; ð2:5Þ

where

p0 ¼ r1 �r þ 2a11x1 þa12S1

ð1þmx1Þ2

!;

p1 ¼ �r þ 2a11x1 þa12S1

ð1þmx1Þ2þ r1;

q0 ¼ �r1ð�r þ 2a11x1Þ;q1 ¼ �r1:

Clearly, Eq. (2.5) always has a root k1 ¼ bS1 � r2. All other roots of Eq. (2.5) are determined by the following equation

k2 þ p1kþ p0 þ ðq1kþ q0Þe�ks ¼ 0: ð2:6Þ

When s ¼ 0, Eq. (2.6) reduces to

k2 þ ðp1 þ q1Þkþ p0 þ q0 ¼ 0: ð2:7Þ

It is easy to show that

p0 þ q0 ¼a12r1S1

ð1þmx1Þ2;

p1 þ q1 ¼ �r þ 2a11x1 þa12S1

ð1þmx1Þ2:

Hence, if bS1 < r2 and the following hold:

(H2) �r þ 2a11x1 þ a12S1=ð1þmx1Þ2 > 0,

the equilibrium E2 is locally asymptotically stable when s ¼ 0.

If ixðx > 0) is a solution of (2.6), separating real and imaginary parts, we have

x2 � p0 ¼ q0 cos xsþ q1x sinxs;p1x ¼ q0 sin xs� q1x cos xs:

ð2:8Þ

Squaring and adding the two equations of (2.8), it follows that

x4 þ p21 � 2p0 � q2

1

� �x2 þ p2

0 � q20 ¼ 0: ð2:9Þ

By calculation, we derive that

p21 � 2p0 � q2

1 ¼ �r þ 2a11x1 þa12S1

ð1þmx1Þ2

!2

;

p0 � q0 ¼ 2r1 �r þ 2a11x1 þa12S1

2ð1þmx1Þ2

" #:

ð2:10Þ

Note that if p0 > q0, then (H2) holds and p21 � 2p0 � q2

1 > 0. Hence, if p0 > q0, Eq. (2.9) has no positive real roots. Accord-ingly, by Theorem 3.4.1 in Kuang [15] we see that if p0 > q0 and bS1 < r2, then E2 is locally asymptotically stable for all s P 0.If p0 < q0, then Eq. (2.9) has a unique positive root x0. That is, Eq. (2.6) has a pair of purely imaginary roots of the form �ix0.Denote

sn ¼1x0

arccosq0 x2

0 � p0

� �� p1q1x2

0

q20 þ q2

1x20

þ 2npx0

; n ¼ 0;1;2; . . .

By Theorem 3.4.1 in Kuang [15] we see that if bS1 < r2, p0 < q0 and (H2) hold, then E2 remains stable for s < s0.We now claim that

dðRekÞds

js¼s0> 0:

This will show that there exists at least one eigenvalue with positive real part for s > s0. Moreover, the conditions for theexistence of a Hopf bifurcation [8] are then satisfied yielding a periodic solution. To this end, differentiating Eq. (2.6) withrespect s, it follows that

376 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

dkds

� ��1

¼ 2kþ p1

�kðk2 þ p1kþ p0Þþ q1

kðq1kþ q0Þ� s

k:

Hence, a direct calculation shows that

signdðRekÞ

ds

� �k¼ix0

¼ sign Redkds

� ��1( )

k¼ix0

¼ sign2x2

0 þ p21 � 2p0

ðp0 �x20Þ

2 þ p21x2

0

� q21

q20 þ q2

1x20

( ):

We derive from (2.8) that

p0 �x20

� �2 þ p21x

20 ¼ q2

0 þ q21x

20:

Hence, it follows that

signdðRekÞ

ds

� �k¼ix0

¼ sign2x2

0 þ p21 � 2p0 � q2

1

q20 þ q2

1x20

� �> 0:

Therefore, the transversal condition holds and a Hopf bifurcation occurs at x ¼ x0; s ¼ s0.In conclusion, we have the following results.

Theorem 1. For system (1.3), the following results hold true:

(i) If a21r < r1ða11 þmrÞ, then the equilibrium E1ðr=a11;0;0Þ is locally asymptotically stable; if a21r > r1ða11 þmrÞ, then E1 isunstable.(ii) Assume that bS1 < r2. If p0 > q0, then the equilibrium E2ðx1; S1;0Þ is locally asymptotically stable for all s P 0; if p0 < q0

and (H2) hold, then there exists a positive constant s0 such that E2 is locally asymptotically stable if 0 < s < s0 and is unstableif s > s0. Further, system (1.3) undergoes a Hopf bifurcation at E2 when s ¼ s0.

The characteristic equation of system (1.3) at the coexistence equilibrium E� is of the form

k3 þ P2k2 þ P1kþ P0 þ ðQ 2k

2 þ Q 1kÞe�ks ¼ 0; ð2:11Þ

where

P0 ¼ r2bI� �r þ 2a11x� þ a12S�

ð1þmx�Þ2

!;

P1 ¼ r2bI� þ a21x�

1þmx��r þ 2a11x� þ a12S�

ð1þmx�Þ2

!;

P2 ¼ �r þ 2a11x� þ a12S�

ð1þmx�Þ2þ a21x�

1þmx�;

Q 1 ¼ �a21x�

1þmx�ð�r þ 2a11x�Þ;

Q 2 ¼ �a21x�

1þmx�:

ð2:12Þ

When s ¼ 0, Eq. (2.11) reduces to

k3 þ ðP2 þ Q 2Þk2 þ ðP1 þ Q 1Þkþ P0 ¼ 0:

It is easy to show that if

(H3) �r þ 2a11x� þ a12S�=ð1þmx�Þ2 > 0,

then we have that

P1 þ Q 1 ¼ r2bI� þ a12a21x�S�

ð1þmx�Þ3> 0;

ðP1 þ Q 1ÞðP2 þ Q 2Þ � P0 ¼a12a21x�S�

ð1þmx�Þ3�r þ 2a11x� þ a12S�

ð1þmx�Þ2

!> 0:

Hence, by Routh–Hurwitz criterion we know that if (H3) holds, the equilibrium E� is locally asymptotically stable whens ¼ 0.

Substituting k ¼ ixðx > 0) into (2.11) and separating the real and imaginary parts, one obtains that

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 377

� P2x2 þ P0 ¼ Q 2x2 cos xs� Q 1x sinxs;x3 � P1x ¼ Q2x2 sinxsþ Q1x cos xs:

ð2:13Þ

Squaring and adding the two equations of (2.13), it follows that

x6 þ px4 þ qx2 þ r ¼ 0; ð2:14Þ

where

p ¼ P22 � 2P1 � Q 2

2; q ¼ P21 � 2P0P2 � Q 2

1; r ¼ P20: ð2:15Þ

Letting z ¼ x2, Eq. (2.14) can be written as

hðzÞ :¼ z3 þ pz2 þ qzþ r ¼ 0: ð2:16Þ

Denote D ¼ p2 � 3q. It is easy to show that if D 6 0, the function hðzÞ is strictly monotonically increasing. If D > 0 and

z� ¼ffiffiffiffiDp� p

=3 < 0 or D > 0; z� ¼

ffiffiffiffiDp� p

=3 > 0 but hðz�Þ > 0, then hðzÞ has always no positive roots. Hence, under these

conditions, if (H3) holds, Eq. (2.11) has no purely imaginary roots for any s > 0 and accordingly, the equilibrium E� is locallyasymptotically stable for all s P 0.

In what follows, we assume that

(H4) D > 0; z� ¼ffiffiffiffiDp� p

=3 > 0;hðz�Þ 6 0.

In this case, by Lemma 2.2 in [20], we see that Eq. (2.16) has at least one positive root. Without loss of generality, weassume that (2.16) has three positive roots, namely, z1; z2 and z3, respectively. Accordingly, Eq. (2.14) has three positive rootsxk ¼

ffiffiffiffiffizkp ðk ¼ 1;2;3Þ.

For k ¼ 1;2;3, from (2.13) one can get the corresponding sjk > 0 such that (2.11) has a pair of purely imaginary roots �ixk

given by

sjk ¼

1xk

arccos �ðP2Q 2 � Q 1Þx2k þ P1Q 1 � P0Q 2

Q22x2

k þ Q 21

" #þ 2pj

xk; j ¼ 0;1; . . . : ð2:17Þ

Let kðsÞ ¼ vðsÞ þ ixðsÞ be a root of Eq. (2.11) satisfying vðsjkÞ ¼ 0;xðsj

kÞ ¼ xk.Differentiating the two sides of (2.11) with respect to s, it follows that

dkds

� ��1

¼ 3k2 þ 2P2kþ P1

�kðk3 þ P2k2 þ P1kþ P0Þ

þ 2Q 2kþ Q 1

kðQ2k2 þ Q1kÞ

� sk: ð2:18Þ

After some algebra, one obtains that

signdRek

ds

� �s¼sj

k

¼ sign Redkds

� ��1( )

s¼sjk

¼ sign �P1 � 3x2

k

� �x2

k � P1� �

þ 2P2 P0 � P2x2k

� �x3

k � P1xk� �2 þ P0 � P2x2

k

� �2 � Q21 þ 2Q 2

2x2k

ðQ2x2kÞ

2 þ Q 21x2

k

( ): ð2:19Þ

We derive from (2.13) that

x3k � P1xk

� �2 þ P0 � P2x2k

� �2 ¼ Q 2x2k

� �2 þ Q 21x

2k :

Hence, it follows that

signdðRekÞ

ds

� �s¼sj

k

¼ sign3x4

k þ 2 P22 � 2P1 � Q 2

2

x2

k þ P21 � 2P0P2 � Q2

1

Q22x4

k þ Q 21x2

k

24

35 ¼ sign

h0ðzkÞQ 2

2x4k þ Q 2

1x2k

" #:

From what has been discussed above, we have the following results.

Theorem 2. Let p; q and r be defined in (2.15). Assume that bS1 > r2 and ðH3Þ hold. Then the following results hold true:

(i) If D 6 0 or D > 0 and z� ¼ffiffiffiffiDp� p

=3 < 0 or D > 0; z� ¼

ffiffiffiffiDp� p

=3 > 0 and hðz�Þ > 0, then the equilibrium E� of system

(1.3) is locally asymptotically stable for all s P 0.(ii) If (H4) holds, then hðzÞ has at least one positive root zk, and all roots of (2.11) have negative real parts for s 2 ½0; s0

kÞ, and theequilibrium E� of system (1.3) is locally asymptotically stable for s 2 ½0; s0

kÞ.(iii) If all conditions as stated in (ii) hold true and h0ðzkÞ – 0, then system (1.3) undergoes a Hopf bifurcation at E� whens ¼ sj

kðj ¼ 0;1; . . .Þ.

378 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

3. Global stability

In this section, we are concerned with the global stability of the coexistence equilibrium E�ðx�; S�; I�Þ, the disease-freeequilibrium E2ðx1; S1;0Þ and the predator-extinction equilibrium E1ðr=a11;0;0Þ of system (1.3), respectively. The strategyof proofs is to use Lyapunov functional and LaSalle’s invariance principle.

We first give a result on the upper bound of positive solutions of system (1.3) with initial conditions (1.4).

Lemma 3.1. There are positive constants M1 and M2 such that for any positive solution ðxðtÞ; SðtÞ; IðtÞÞ of system (1.3) with initialconditions (1.4),

lim supt!þ1

xðtÞ < M1; lim supt!þ1

SðtÞ < M2; lim supt!þ1

IðtÞ < M2: ð3:1Þ

Proof. Let ðxðtÞ; SðtÞ; IðtÞÞ be any positive solution of system (1.3) with initial conditions (1.4). Define

VðtÞ ¼ xðt � sÞ þ a12

a21ðSðtÞ þ IðtÞÞ:

Calculating the derivative of VðtÞ along positive solutions of system (1.3), it follows that

ddt

VðtÞ ¼ xðt � sÞðr � a11xðt � sÞÞ � a12

a21ðr1SðtÞ þ r2IðtÞÞ ¼ �r1VðtÞ þ xðt � sÞðr þ r1 � a11xðt � sÞÞ þ a12

a21ðr1 � r2ÞIðtÞ

6 �r1VðtÞ þ ðr þ r1Þ2

4a11;

which yields lim supt!þ1VðtÞ 6 ðr þ r1Þ2=ð4a11r1Þ. If we choose

M1 ¼ðr þ r1Þ2

4a11r1; M2 ¼

a21ðr þ r1Þ2

4a11a12r1; ð3:2Þ

then (3.1) follows. This completes the proof. h

Lemma 3.2. For any positive solution ðxðtÞ; SðtÞ; IðtÞÞ of system (1.3) with initial conditions (1.4), we have that

lim inft!þ1

xðtÞ > x :¼ r � a12M2

a11; ð3:3Þ

where M2 is defined in (3.2).

Proof. Let ðxðtÞ; SðtÞ; IðtÞÞ be any positive solution of system (1.3) with initial conditions (1.4). By Lemma 3.1 it follows thatlim supt!þ1SðtÞ 6 M2. Hence, for e > 0 being sufficiently small, there is a T0 > 0 such that if t > T0; SðtÞ < M2 þ e. Accordingly,for e > 0 being sufficiently small, we derive from the first equation of system (1.3) that, for t > T0,

_xðtÞP xðtÞðr � a11xðtÞ � a12ðM2 þ eÞÞ;

which yields

lim inft!þ1

xðtÞP x :¼ r � a12M2

a11:

The proof is complete. h

We are now in a position to state and prove our result on the global stability of the coexistence equilibrium E�ðx�; S�; I�Þ ofsystem (1.3).

Theorem 3. Assume that bS1 > r2. Then the coexistence equilibrium E�ðx�; S�; I�Þ of system (1.3) is globally attractive providedthat

(H5) x > r=ð2a11Þ.

Here, x > 0 is defined in (3.3).

Proof. Let ðxðtÞ; SðtÞ; IðtÞÞ be any positive solution of system (1.3) with initial conditions (1.4). Define

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 379

V11ðtÞ ¼ k1 x� x� � x� lnxx�

þ S� S� � S� ln

SS�þ I � I� � I� ln

II�; ð3:4Þ

where k1 is a positive constant to be determined later.Calculating the derivative of V11ðtÞ along positive solutions of system (1.3), we derive that

ddt

V11ðtÞ ¼ k1 1� x�

x

� �xðtÞðr � a11xðtÞÞ � a12xðtÞSðtÞ

1þmxðtÞ

� �þ 1� S�

S

� �a21xðt � sÞSðt � sÞ

1þmxðt � sÞ � r1SðtÞ � bSðtÞIðtÞ� �

þ 1� I�

I

� �½bSðtÞIðtÞ � r2IðtÞ�: ð3:5Þ

On substituting x�ðr � a11x�Þ ¼ a12x�S�=ð1þmx�Þ into (3.5), it follows that

ddt

V11ðtÞ ¼ k1 1� x�

x

� �xðtÞðr � a11xðtÞÞ � x�ðr � a11x�Þ þ a12x�S�

1þmx�

� �� k1 1� x�

x

� �a12xðtÞSðtÞ1þmxðtÞ

þ a21xðt � sÞSðt � sÞ1þmxðt � sÞ � r1SðtÞ � a21S�xðt � sÞSðt � sÞ

SðtÞð1þmxðt � sÞÞ þ r1S� � bI�SðtÞ þ r2I�: ð3:6Þ

We rewrite (3.6) as follows

ddt

V11ðtÞ ¼ k1 1� x�

x

� �xðtÞðr � a11xðtÞÞ � x�ðr � a11x�Þ þ a12x�S�

1þmx�

� �� k1ð1þmx�Þ a12xðtÞSðtÞ

1þmxðtÞ þ k1a12x�SðtÞ

þ a21xðt � sÞSðt � sÞ1þmxðt � sÞ � r1SðtÞ � a21S�xðt � sÞSðt � sÞ

SðtÞð1þmxðt � sÞÞ þ r1S� � bI�SðtÞ þ r2I�: ð3:7Þ

Define

V1ðtÞ ¼ V11ðtÞ þ V12ðtÞ; ð3:8Þ

where

V12ðtÞ ¼ a21

Z t

t�s

xðuÞSðuÞ1þmxðuÞ �

x�S�

1þmx�� x�S�

1þmx�lnð1þmx�ÞxðuÞSðuÞx�S�ð1þmxðuÞÞ

� �du: ð3:9Þ

A direct calculation shows that

ddt

V12ðtÞ ¼ a21xðtÞSðtÞ

1þmxðtÞ �xðt � sÞSðt � sÞ1þmxðt � sÞ þ

x�S�

1þmx�ln

xðt � sÞSðt � sÞð1þmxðtÞÞxðtÞSðtÞð1þmxðt � sÞÞ

� �: ð3:10Þ

Letting k1a12ð1þmx�Þ ¼ a21, we derive from (3.7)–(3.10) that

ddt

V1ðtÞ ¼ k1 1� x�

x

� �xðtÞðr � a11xðtÞÞ � x�ðr � a11x�Þ þ a12x�S�

1þmx�

� �� a21S�xðt � sÞSðt � sÞ

SðtÞð1þmxðt � sÞÞ þ r1S� þ r2I�

þ a21x�S�

1þmx�ln

xðt � sÞSðt � sÞð1þmxðtÞÞxðtÞSðtÞð1þmxðt � sÞÞ

¼ k1 1� x�

x

� �½xðtÞðr � a11xðtÞÞ � x�ðr � a11x�Þ� þ a21x�S�

1þmx�1� x�ð1þmxðtÞÞ

xðtÞð1þmx�Þ

� �

� a21x�S�

1þmx�ð1þmx�Þxðt � sÞSðt � sÞ

x�SðtÞð1þmxðt � sÞÞ þa21x�S�

1þmx�þ a21x�S�

1þmx�lnð1þmx�Þxðt � sÞSðt � sÞ

x�SðtÞð1þmxðt � sÞÞ þ lnx�ð1þmxðtÞÞxðtÞð1þmx�Þ

� �

¼ k1ðx� x�Þ2

x½r � a11ðxðtÞ þ x�Þ� � a21x�S�

1þmx�x�ð1þmxðtÞÞxðtÞð1þmx�Þ � 1� ln

x�ð1þmxðtÞÞxðtÞð1þmx�Þ

� �

� a21x�S�

1þmx�ð1þmx�Þxðt � sÞSðt � sÞ

x�SðtÞð1þmxðt � sÞÞ � 1� lnð1þmx�Þxðt � sÞSðt � sÞ

x�SðtÞð1þmxðt � sÞÞ

� �: ð3:11Þ

Note that the function gðxÞ ¼ x� 1� ln x is always non-negative for any x > 0, and gðxÞ ¼ 0 if and only if x ¼ 1. Hence, ifxðtÞ > r=ð2a11Þ for t P T, we have

ðx� x�Þ2

x½r � a11ðxþ x�Þ� 6 0;

with equality if and only if x ¼ x�. This, together with (3.11), implies that if xðtÞ > r=ð2a11Þ for t P T , V 01ðtÞ 6 0, with equalityif and only if x ¼ x�; ð1þmx�Þxðt�sÞSðt�sÞ

x�SðtÞð1þmxðt�sÞÞ ¼ 1. We now look for the invariant subset M within the set

M ¼ ðx; S; IÞ : x ¼ x�;ð1þmx�Þxðt � sÞSðt � sÞ

x�SðtÞð1þmxðt � sÞÞ ¼ 1� �

:

380 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

Since x ¼ x� on M and consequently, 0 ¼ _xðtÞ ¼ x�ðr � a11x� � a12SðtÞ=ð1þmx�ÞÞ, which yields SðtÞ ¼ S�. It follows from thesecond equation of system (1.3) that 0 ¼ _SðtÞ ¼ a21x�S�=ð1þmx�Þ � r1S� � bS�IðtÞ, which leads to I ¼ I�. Hence, the onlyinvariant set in M is M¼ fðx�; S�; I�Þg. Therefore, the global attractiveness of E� follows from LaSalle’s invariance principlefor delay differential systems (see, for example, Haddock and Terjéki [7]). This completes the proof. h

Theorem 4. Assume that bS1 < r2. If (H1) and (H5) hold, then the disease-free equilibrium E2ðx1; S1;0Þ of system (1.3) is globallyasymptotically stable.

Proof. It is easy to see that if (H5) holds, then x1 > r=ð2a11Þ. It follows from (2.10) that p0 > q0 holds. By Theorem 1, wesee that if bS1 < r2 and (H5) hold, then the equilibrium E2ðx1; S1;0Þ is locally asymptotically stable. Hence, it suffices toshow that all positive solutions of system (1.3) with initial conditions (1.4) converge to E2. We achieve this by con-structing a global Lyapunov functional. Let ðxðtÞ; SðtÞ; IðtÞÞ be any positive solution of system (1.3) with initial conditions(1.4).

Define

V21ðtÞ ¼ k2 x� x1 � x1 lnxx1

� �þ S� S1 � S1 ln

SS1þ I; ð3:12Þ

where k2 > 0 is a constant to be determined later.Calculating the derivative of V21ðtÞ along positive solutions of system (1.3), it follows that

ddt

V21ðtÞ ¼ k2 1� x1

x

xðtÞ r � a11xðtÞ � a12SðtÞ

1þmxðtÞ

� �� �þ 1� S1

S

� �a21xðt � sÞSðt � sÞ

1þmxðt � sÞ � r1SðtÞ � bSðtÞIðtÞ� �

þ bSðtÞIðtÞ � r2IðtÞ: ð3:13Þ

On substituting x1ðr � a11x1Þ ¼ a12x1S1=ð1þmx1Þ into (3.5), we derive that

ddt

V21ðtÞ ¼ k2 1� x1

x

xðtÞðr � a11xðtÞÞ � x1ðr � a11x1Þ þ

a12x1S1

1þmx1

� �� k2 1� x1

x

a12xðtÞSðtÞ1þmxðtÞ

þ a21xðt � sÞSðt � sÞ1þmxðt � sÞ � r1SðtÞ � a21S1xðt � sÞSðt � sÞ

SðtÞð1þmxðt � sÞÞ þ r1S1 þ ðbS1 � r2ÞIðtÞ: ð3:14Þ

Eq. (3.14) can be rewritten as

ddt

V21ðtÞ ¼ k2 1� x1

x

xðtÞðr � a11xðtÞÞ � x1ðr � a11x1Þ þ

a12x1S1

1þmx1

� �� k2ð1þmx1Þ

a12xðtÞSðtÞ1þmxðtÞ þ k2a12x1SðtÞ

þ a21xðt � sÞSðt � sÞ1þmxðt � sÞ � r1SðtÞ � a21S1xðt � sÞSðt � sÞ

SðtÞð1þmxðt � sÞÞ þ r1S1 þ ðbS1 � r2ÞIðtÞ: ð3:15Þ

Letting k2ð1þmx1Þa12 ¼ a21, it follows from (3.15) that

ddt

V21ðtÞ ¼ k2 1� x1

x

½xðtÞðr � a11xðtÞÞ � x1ðr � a11x1Þ� þ k2 1þmx1 �

x1ð1þmxðtÞÞx

� �a12x1S1

1þmx1

� a21x1S1

1þmx1

ð1þmx1Þxðt � sÞSðt � sÞx1SðtÞð1þmxðt � sÞÞ þ

a21x1S1

1þmx1þ ðbS1 � r2ÞIðtÞ: ð3:16Þ

Define

V2ðtÞ ¼ V21ðtÞ þ V22ðtÞ; ð3:17Þ

where

V22ðtÞ ¼ a21

Z t

t�s

xðuÞSðuÞ1þmxðuÞ �

x1S1

1þmx1� x1S1

1þmx1lnð1þmx1ÞxðuÞSðuÞx1S1ð1þmxðuÞÞ

� �du: ð3:18Þ

By calculation we have that

ddt

V22ðtÞ ¼ a21xðtÞSðtÞ

1þmxðtÞ �xðt � sÞSðt � sÞ1þmxðt � sÞ þ

x1S1

1þmx1ln

xðt � sÞSðt � sÞð1þmxðtÞÞxðtÞSðtÞð1þmxðt � sÞÞ

� �: ð3:19Þ

It therefore follows from (3.16)–(3.19) that

Fig. 1.m ¼ 0:2

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 381

ddt

V2ðtÞ ¼ k2 1� x1

x

½xðtÞðr � a11xðtÞÞ � x1ðr � a11x1Þ� þ

a21x1S1

1þmx11� x1ð1þmxðtÞÞð1þmx1ÞxðtÞ

� �

� a21x1S1

1þmx1

ð1þmx1Þxðt � sÞSðt � sÞx1SðtÞð1þmxðt � sÞÞ þ

a21x1S1

1þmx1þ ðbS1 � r2ÞIðtÞ

þ a21x1S1

1þmx1lnð1þmx1Þxðt � sÞSðt � sÞ

x1SðtÞð1þmxðt � sÞÞ þ lnx1ð1þmxðtÞÞxðtÞð1þmx1Þ

� �

¼ k2ðxðtÞ � x1Þ2

xðtÞ ½r � a11ðxðtÞ þ x1Þ� �a21x1S1

1þmx1

x1ð1þmxðtÞÞð1þmx1Þx

� 1� lnx1ð1þmxðtÞÞð1þmx1Þx

� �

� a21x1S1

1þmx1

ð1þmx1Þxðt � sÞSðt � sÞx1SðtÞð1þmxðt � sÞÞ � 1� ln

ð1þmx1Þxðt � sÞSðt � sÞx1SðtÞð1þmxðt � sÞÞ

� �þ ðbS1 � r2ÞIðtÞ: ð3:20Þ

It follows from (3.20) that if bS1 � r2 < 0, (H1) and (H5) hold true, then V 02ðtÞ 6 0, with equality if and only ifx ¼ x1; I ¼ 0; ð1þmx1Þxðt�sÞSðt�sÞ

x1SðtÞð1þmxðt�sÞÞ ¼ 1. We now look for the invariant subset M within the set

M ¼ ðS; IÞ : x ¼ x1; I ¼ 0;ð1þmx1Þxðt � sÞSðt � sÞ

x1SðtÞð1þmxðt � sÞÞ ¼ 1� �

:

Since x ¼ x1 onM and consequently, 0 ¼ _xðtÞ ¼ x1ðr � a11x1 � a12SðtÞ=ð1þmx1ÞÞ, which yields S ¼ S1. Hence, the only invari-ant set in M isM¼ fðx1; S1; 0Þg. Using LaSalle’s invariance principle for delay differential systems, the global asymptotic sta-bility of the equilibrium E2 of system (1.3) follows. h

Theorem 5. If a21r 6 r1ða11 þmrÞ, the predator-extinction equilibrium E1ðr=a11; 0;0Þ of system (1.3) is globally asymptoticallystable.

0 50 100 150 200 250 300 350 400

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

t−x plane

x(t)

0 50 100 150 200 250 300 350 400

0.4

0.5

0.6

0.7

0.8

0.9

1

t−S plane

S(t)

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t−I plane

I(t)

0.20.4

0.60.8

11.2

0.20.4

0.60.8

10

0.1

0.2

0.3

0.4

0.5

x(t)S(t)

I(t)

The temporal solution and phase portrait found by numerical integration of system (1.3) with r ¼ 1:5; a11 ¼ 0:8; a12 ¼ 1:5; a21 ¼ 1;; r1 ¼ 0:5; r2 ¼ 0:5; b ¼ 0:5; s ¼ 1:6; ð/1;/2;/3Þ � ð0:5;0:5;0:5Þ.

0 100 200 300 400 500

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

t−x plane

x(t)

0 100 200 300 400 500

0.4

0.5

0.6

0.7

0.8

0.9

1

t−S plane

S(t)

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t−I plane

I(t)

00.5

11.5

0.20.4

0.60.8

10

0.1

0.2

0.3

0.4

0.5

x(t)S(t)

I(t)

Fig. 2. The temporal solution and phase portrait found by numerical integration of system (1.3) with r ¼ 1:5; a11 ¼ 0:8; a12 ¼ 1:5; a21 ¼ 1;m ¼ 0:2;r1 ¼ 0:5; r2 ¼ 0:5;b ¼ 0:5; s ¼ 2:3; ð/1;/2;/3Þ � ð0:5;0:5;0:5Þ.

382 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

Proof. Since the local stability of E1 is established in Theorem 1, it suffices to show that all positive solutions of system (1.3)with initial conditions (1.4) converge to E1. Let ðxðtÞ; SðtÞ; IðtÞÞ be any positive solution of system (1.3) with initial conditions(1.4).

Denote x0 ¼ r=a11. Define

V3ðtÞ ¼ k3 x� x0 � x0 lnxx0

� �þ Sþ I þ a21

Z t

t�s

xðuÞSðuÞ1þmxðuÞdu; ð3:21Þ

where k3 > 0 is a constant to be determined later.Calculating the derivative of V3ðtÞ along positive solutions of system (1.3), we derive that

ddt

V3ðtÞ ¼ k3 1� x0

xðtÞ

� �xðtÞðr � a11xðtÞÞ � a12xðtÞSðtÞ

1þmxðtÞ

� �þ a21xðtÞSðtÞ

1þmxðtÞ � r1SðtÞ � r2IðtÞ: ð3:22Þ

On substituting r ¼ a11x0 into (3.22), one obtains that

ddt

V3ðtÞ ¼ k3 1� x0

xðtÞ

� ��a11xðtÞðxðtÞ � x0Þ �

a12xðtÞSðtÞ1þmxðtÞ

� �þ a21xðtÞSðtÞ

1þmxðtÞ � r1SðtÞ � r2IðtÞ

¼ �k3a11ðx� x0Þ2 � k3ð1þmx0Þa12xðtÞSðtÞ1þmxðtÞ þ k3a12x0SðtÞ þ a21xðtÞSðtÞ

1þmxðtÞ � r1SðtÞ � r2IðtÞ: ð3:23Þ

Letting k3ð1þmx0Þa12 ¼ a21, we derive from (3.23) that

ddt

V3ðtÞ ¼ �k3a11ðxðtÞ � x0Þ2 þa21r � r1ða11 þmrÞ

a11 þmrSðtÞ � r2IðtÞ: ð3:24Þ

Let M be the largest invariant subset of fV 03ðtÞ ¼ 0g. Clearly, if a21r < r1ða11 þmrÞ, it follows from (3.24) that V 03ðtÞ 6 0,with equality if and only if x ¼ x0; S ¼ 0; I ¼ 0. If a21r ¼ r1ða11 þmrÞ, we obtain from (3.24) that V 03ðtÞ ¼ 0 if and only if

0 100 200 300 400 500 600 700 8000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t−x plane

x(t)

0 100 200 300 400 500 600 700 8000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

t−S(t)

S(t)

0 100 200 300 400 500 600 700 8000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

t−I plane

I(t)

0.20.4

0.60.8

1

00.2

0.40.6

0.80.4

0.5

0.6

0.7

0.8

x(t)S(t)

I(t)

Fig. 3. The temporal solution and phase portrait found by numerical integration of system (1.3) with r ¼ 0:55; a11 ¼ 0:125; a12 ¼ 1:8; a21 ¼ 1:35;m ¼ 0:01; r1 ¼ 0:17; r2 ¼ 0:25;b ¼ 0:95; s ¼ 0:1, ð/1;/2;/3Þ � ð0:5;0:5;0:5Þ.

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 383

x ¼ x0; I ¼ 0. Noting that M is invariant, for each element in M, we have xðtÞ ¼ x0. It therefore follows from the firstequation of system (1.3) that 0 ¼ _xðtÞ ¼ �a12x0SðtÞ=ð1þmx0Þ, which yields SðtÞ ¼ 0. Hence, V 03ðtÞ ¼ 0 if and only ifðxðtÞ; SðtÞ; IðtÞÞ ¼ ðx0;0;0Þ. Therefore, the global asymptotic stability of the equilibrium E1ðr=a11;0;0Þ follows. This com-pletes the proof. h

4. Numerical simulations

In this section, we give some examples to illustrate the main results in Sections 2 and 3.

Example 1. In system (1.3), let r ¼ 1:5; a11 ¼ 0:8; a12 ¼ 1:5; a21 ¼ 1;m ¼ 0:2; r1 ¼ 0:5; r2 ¼ 0:5; b ¼ 0:5. It is easy to show that(H1)-(H2) hold true and p0 � q0 � �0:1361 < 0. Hence, system (1.3) has a disease-free equilibrium E2ð5=9;190=243;0Þ. ByTheorem 1 we see that there exists a positive constant s0 ¼ 1:8766 such that E2 is locally asymptotically stable if 0 < s < s0

and is unstable if s > s0. Further, system (1.3) undergoes a Hopf bifurcation at E2 when s ¼ s0. An investigation of system(1.3) with the coefficients above can be conducted via a numerical integration using the standard MATLAB algorithm (see,Figs. 1 and 2).

Example 2. In system (1.3), let r ¼ 0:55; a11 ¼ 0:125; a12 ¼ 1:8; a21 ¼ 1:35;m ¼ 0:01; r1 ¼ 0:17; r2 ¼ 0:25; b ¼ 0:95. By calcu-lation we have S1 � 0:2972 and bS1 � r2 � 0:0323. In this case, system (1.3) has a unique coexistence equilibriumE�ð0:6344;0:2632;0:7169Þ. By Theorem 2 we see that there exists a positive constant s0 ¼ 0:1688 such that E� is locallyasymptotically stable if 0 < s < s0 and is unstable if s > s0. Further, system (1.3) undergoes a Hopf bifurcation at E� whens ¼ s0. Numerical simulation illustrates the result above (see, Figs. 3 and 4).

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

t−x plane

x(t)

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

t−S plane

S(t)

0 100 200 300 400 500 6000

0.5

1

1.5

t−I plane

I(t)

01

23

0

1

2

30

0.5

1

1.5

x(t)S(t)

I(t)

Fig. 4. The temporal solution and phase portrait found by numerical integration of system (1.3) with r ¼ 0:55; a11 ¼ 0:125; a12 ¼ 1:8; a21 ¼ 1:35;m ¼ 0:01; r1 ¼ 0:17; r2 ¼ 0:25;b ¼ 0:95; s ¼ 0:3, ð/1;/2;/3Þ � ð0:5;0:5;0:5Þ.

384 R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386

Example 3. In system (1.3), let r ¼ 3; a11 ¼ 5; a12 ¼ 0:3; a21 ¼ 0:3;m ¼ 1; r1 ¼ 0:1; r2 ¼ 0:15; b ¼ 0:2. It is easy to show thatsystem (1.3) always has a trivial equilibrium E0ð0;0;0Þ, a predator-extinction equilibrium E1ð0:6;0;0Þ and a disease-freeequilibrium E2ð0:5;2:5;0Þ. Clearly, bS1 � r2 ¼ 0:35. Hence, system (1.3) has a unique coexistence equilibriumE�ð0:5714;0:75;0:0454Þ. The equilibria E0; E1 and E2 are always unstable. A direct calculation shows that Eq. (2.14) has threeroots: �7:8679 and 0:0010� 0:0009i. Therefore, the equilibrium E� is locally asymptotically stable for all s P 0. It is easy toshow that x ¼ 0:3117 > 0:3 ¼ r=ð2a11Þ. By Theorem 3, we see that the equilibrium E� is globally stable. Numerical simulationillustrates the result above (see, Fig. 5).

5. Discussion

In this paper, we have investigated the global dynamics of a predator–prey model with a disease that can be transmittedby contact spreading among the predator population and a time delay representing the gestation period of the predator. Byanalyzing the corresponding characteristic equations, the local stability of each of feasible equilibria has been established,respectively. It has been shown that, under some conditions, the time delay due to the gestation of the predator may desta-bilize both the disease-free equilibrium and the coexistence equilibrium of the eco-epidemiological system and cause thepopulation to fluctuate. By comparison argument, a priori lower bound of the density of the prey population was derived.By means of Lyapunov functional and LaSalle’s invariance principle, sufficient conditions were obtained for the globalasymptotic stability of the coexistence equilibrium, the disease-free equilibrium and the predator-extinction equilibriumof system (1.3), respectively. By Theorem 3 we see that if the prey population is always abundant enough and the diseasetransmission coefficient b is sufficiently large, the coexistence equilibrium is a global attractor of the system (1.3). In thiscase, the disease spreading in the predator becomes endemic and the prey, sound predator and the infected predator coexist.By Theorem 4 we see that if the disease transmission coefficient b is sufficiently small and the prey population is alwaysabundant enough, the disease among the predator population dies out and in this case, the prey and the sound predator

0 2000 4000 6000 8000 100000

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S(t)

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Fig. 5. The temporal solution and phase portrait found by numerical integration of system (1.3) with r ¼ 3; a11 ¼ 5; a12 ¼ 0:3; a21 ¼ 0:3;m ¼ 1;r1 ¼ 0:1; r2 ¼ 0:15; b ¼ 0:2; s ¼ 2; ð/1;/2;/3Þ � ð0:01;0;01;0;01Þ.

R. Xu, S. Zhang / Applied Mathematics and Computation 224 (2013) 372–386 385

coexist. We rewrite a21r 6 r1ða11 þmrÞ as a21 6 r1ðmþ a11=rÞ. By Theorem 5 wee see that if the carrying capacity of the preyand the conversion rate of the predator are sufficiently small, and the death rate of the sound predator and the half satura-tion rate of the predator are sufficiently large, the prey population persists and the predator population goes to extinction.

There have been a few works on the effects of transmissible disease spreading among predator population and time delaydue to gestation on the dynamics of predator–prey systems (see, for example, [30,32]). In these works, by regarding the delayas the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at thecoexistence equilibrium, the local asymptotic stability of the coexistence equilibrium and the existence of Hopf bifurcationwere investigated. Little attention has been paid to the global stability of the coexistence equilibrium. To the best of ourknowledge, mathematically, Theorem 3 in the present work is the first result on the global stability of a unique coexistenceequilibrium for eco-epidemiological predator–prey models with disease in the predator and time delay due to gestation ofthe predator. The proof relies on the construction of a global Lyapunov functional. Establishing global stability is crucial forour study, since local stability cannot rule out the possibility of periodic solutions far away from equilibria. Mathematically,the Lyapunov functionals in this paper would be successfully applied in the other biological dynamic models with discretedelays and even with distributed delays.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11071254), the Scientific ResearchFoundation for the Returned Overseas Chinese Scholars, State Education Ministry and the Natural Science Foundation ofYoung Scientist of Hebei Province (No. A2013506012).

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