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Applied Mathematics and Computation 131 (2002) 397–414 www.elsevier.com/locate/amc

A ratio-dependent predator–prey model with disease in the prey

Yanni Xiao *, Lansun Chen

Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, P.R. China

Abstract In this paper, a ratio-dependent predator–prey system with disease in the prey is formulated and analyzed. Mathematical analyses of the model equations with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. Specially, we shall show that ratio-dependent predator– prey models are rich in boundary dynamics, and most importantly, we shall show that a periodic solution can occur whether the system is permanent or not, that is, there are solutions which tend to disease-free equilibrium while bifurcating periodic solution exists. ? 2002 Elsevier Science Inc. All rights reserved.

Keywords: Predator–prey model; Global stability; Permanence; Hopf bifurcation

1. Introduction Since the pioneering work of Kermack–Mckendrick (1927) on SIRS (susceptible–infective–removal–susceptible), epidemiological models have received much attention from scientists. Relevant references in this context are also vast and we shall mention here a survey paper [1] and some books (see [2–4], to mention a few). In standard epidemiological models, only single-species models are considered, and some threshold results are obtained. However, the actual situation is not always like this. In the natural world, species does not exist alone while species spreads the disease, it also competes with the other species for space or food, or is predated by other species. Therefore, it is of more biological signi?cance to consider the e?ect of

*

Corresponding author. E-mail addresses: xyn@math03.math.ac.cn (Y. Xiao), lschen@math08.math.ac.cn (L. Chen).

0096-3003/02/$ - see front matter ? 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 1 5 6 - 4

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interacting species when we study the dynamical behaviors of epidemiological models. But little attention has been paid so far to merge these two areas of research [5–7]. In this paper we introduce the predator based on the basic epidemiological model, namely the SI, in order to investigate how the predation process in?uences the epidemics. For this purpose, we consider the simple case when the predator mainly eats the infected prey. This is in accordance with the fact that the infected individuals are less active and can be caught more easily, or the behavior of the prey individuals is modi?ed such that they live in parts of the habitat which are accessible to the predator (?sh and aquatic snails staying close to water surface, snails staying on the top of the vegetation rather than under the plant cover) [8]. Peterson and Page [9] have indicated that wolf attacks on moose are more often successful if the moose is heavily infected by ‘Echinococcus granulosus’. Recently, there is a growing explicit biological and physiological evidences [10–13] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator–prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and so should be the so-called predator functional responses. This is strongly supported by numerous ?eld and laboratory experiments and observations [12,13]. We have two populations: 1. The prey, whose total population density is denoted by N. 2. The predator, whose population density is denoted by Y. We make the following assumptions: ?H1 ?: In the absence of disease the prey population density grows according to a logistic curve with carrying capacity K ?K > 0?, with an intrinsic birth rate constant r ?r > 0?: dN N ? rN 1 ? : ?1:1? dt K ?H2 ?: In the presence of disease we assume that the total prey population N is composed of two population classes: one is the class of susceptible prey, denoted by S, and the other is the class of the infected prey, denoted by I. Therefore, at any time t, the total density of prey population is N ?t? ? S?t? ? I?t?: ?1:2?

(Note. In the following we are always referring to population densities, we may omit the word ‘density’ for the sake of simplicity.)

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399

?H3 ?: We assume that only susceptible prey S is capable of reproducing with logistic law (Eq. (1.1)) for simplicity (see [7,14]), i.e., the infected prey I is removed by death (say, its death rate is positive constant c) or by predation before having the possibility of reproducing. However, the infective population I still contributes with S to population growth toward the carrying capacity. ?H4 ?: We assume that the disease is spread among the prey population only and the disease is not genetically inherited. The infected populations do not recover or become immune. The incidence is assumed to be the simple mass action incidence bSI, where b > 0 is called the transmission coe?cient. Then the SI epidemic model yields: dS S?I ? rS 1 ? ? bSI; dt K ?1:3? dI ? bSI ? cI: dt ?H5 ?: We assume that the predator–prey eats only the infected prey with ratio-dependent Michaelis–Menten functional response function IY ?m > 0?: g?I; Y ? ? mY ? I It is assumed that the predator has a death rate constant d ?d > 0?, and predation coe?cients b ?b > 0?. The coe?cient in conversing prey into predator is k ?0 < k 6 1?. From the above assumptions, the model equations are: dS S?I ? rS 1 ? ? bSI; dt K dI bIY ? bSI ? cI ? ; dt mY ? I dY kbIY ? ?dY ? : dt mY ? I

?1:4?

Chattopadhyay and Arino [6] have considered a predator–prey model with disease in prey. They assumed that the sound prey population grows according to a logistic law involving the whole prey population, i.e., dS S?I ? r?S ? I? 1 ? ? bSI ? c?S?Y ; dt K where c?S? is the predator response function. They further assumed that the intrinsic growth rate r is large enough such that S ? I approaches the environment carrying capacity K, and using S ? I ? K, they reduced the threedimensional system to a two-dimensional system. Hence they studied the local stability, extinction and Hopf bifurcation in a two-dimensional system. By

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applying a Poincar map they observed the connection between the reduced e and the original system. System (1.4) is also similar to the one studied by Venturino [7]. However, there are two di?erences: (i) Venturino does not allow for density dependence in the host regulation except for the disease. (ii) In our model, the so-called ratio-dependent Michaelis–Menten predation process is considered. So we believe that this is the ?rst time that eco-epidemiology model has been formulated and analyzed. In analogy to the threshold result of epidemic theory (see for example [15]), we obtain the threshold parameter which governs the development of the disease. The objective of this paper is to perform a global qualitative study on system (1.4). Specially, we shall show that eco-epidemiological model with ratio-dependent Michaelis–Menten type functional response is rich in boundary dynamics. Even when there is no positive steady state, both infected prey and predator can become extinct (that is, some solutions tend to disease-free equilibrium). Such an extinction occurs in two cases. More importantly, the bifurcating solution exists whether the system is permanent or not. In the absence of predators it is well known [2] that for system (1.3) either endemic equilibrium or disease-free equilibrium is globally asymptotically stable. This means that the system has no nontrivial positive periodic solution. In this paper, by studying the in?uence of the predation process on epidemics, we show that periodic solution can occur. This suggests that predation process with ratio-dependent Michaelis–Menten functional response can make the behavior of epidemic models more complex. This paper is organized as follows. In Section 2, boundedness of solutions, ‘nonpersistent’ and permanence of solutions are presented. In Section 3, we obtain the existence and local stability of boundary and positive equilibria, and determine conditions for which the system enters a Hopf type bifurcation. The global stability of boundary equilibria of system is presented. In Section 4, we try to interpret our mathematical results in terms of their ecological implication and formulate our conclusion. We also point out some future research directions.

2. Preliminaries For the sake of simplicity, we put in dimensionless form the model equations (1.4) by rescaling the variables on the carrying capacity value K, i.e., s? S ; K i? I ; K y? Y ; K

and then using as dimensionless time, x ? bKt. This leads to the dimensionless equations:

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401

ds ? as?1 ? ?s ? i?? ? si; dx di liy ? si ? b2 i ? ; dx my ? i dy kliy ? ?b1 y ? ; dx my ? i where a? r ; bK b1 ? d ; bK b2 ? c ; bK l? b bK

?2:1?

?2:2?

are the dimensionless parameters. The initial conditions for (2.1) are R3 ? f?s; i; y? 2 R3 : s P 0; i P 0; y P 0g: ? ?2:3?

For convenience, in the following we replace x by t for the dimensionless time. Standard and simple arguments show that solutions of system (2.1) always exist and stay positive. 2.1. Boundedness of solutions Lemma 2.1. Assume that initial condition of Eq. (2.1) satisfies s0 ? i0 P 1. Then either (i) s?t? ? i?t? P 1 for all t P 0 and, therefore, as t ! ?1, ?s?t?; i?t?; y?t?? ! E10 ? ?1; 0; 0?, or (ii) there exists a t1 > 0 such that s?t? ? i?t? < 1 for all t > t1 . Finally (iii) if s0 ? i0 < 1, then s?t? ? i?t? < 1 for all t P 0. Proof. We consider ?rst s?t? ? i?t? P 1 for all t P 0. From the ?rst two equations of (2.1) we get d liy ?s?t? ? i?t?? ? as?1 ? ?s ? i?? ? b2 i ? : dt my ? i Hence, for all t P 0, we have that s0 ?t? ? i0 ?t? 6 0. Let

t!1

?2:4?

lim ?s?t? ? i?t?? ? n:

?2:5?

If n > 1, then by the Barblat Lemma, we have a d liy ?s?t? ? i?t?? ? lim as?t??1 ? s?t? ? i?t?? ? b2 i?t? ? 0 ? lim t!?1 dt t!?1 my ? i liy ? lim as?t??1 ? n? ? b2 i?t? ? t!?1 my ? i 6 ? minfa?n ? 1?; b2 g lim ?s?t? ? i?t??

t!?1

? ?n minfb2 ; a?n ? 1?g < 0:

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This contradiction shows that n ? 1, i.e.,

t!?1

lim ?s?t? ? i?t?? ? 1:

?2:6?

Let us denote by g?t? ? s?t? ? i?t? for t 2 ?0; ?1?. Of course, g?t? is di?erentiable and g0 ?t? uniformly continuous for t 2 ?0; ?1?. Thus, with Eq. (2.6) all the assumptions of Barblat Lemma hold true and, therefore, a d ?s?t? ? i?t?? ? 0: dt Since from the ?rst two equations of (2.1)

t!?1

lim

?2:7?

d li?t?y?t? ?s?t? ? i?t?? ? as?t??1 ? ?s?t? ? i?t??? ? b2 i?t? ? ; dt my?t? ? i?t? then Eq. (2.6) implies that d li?t?y?t? ?s?t? ? i?t?? ? ? lim b2 i?t? ? lim : t!?1 dt t!?1 my?t? ? i?t?

?2:8?

?2:9?

Hence, Eqs. (2.7) and (2.8) are in agreement if and only if limt!?1 i?t? ? 0, which jointly with Eq. (2.6) implies limt!?1 s?t? ? 1. From the third equation of (2.1), we have y?t? tends to zero as t ! ?1. This completes the case (i). Assume that assumption (i) is violated. Then there exists t0 > 0 at which for the ?rst time s?t0 ? ? i?t0 ? ? 1. According to Eq. (2.8) we have d li?t0 ?y?t0 ? ?s?t? ? i?t??jt?t0 ? ?b2 i?t0 ? ? < 0: dt my?t0 ? ? i?t0 ? ?2:10?

This implies that once a solution with s ? i has entered into the interval ?0; 1? then it remains bounded there for all t > t0 , i.e., s?t? ? i?t? < 1 for all t > t0 : ?2:11? Finally, if s0 ? i0 < 1, applying the previous argument it follows that s?t? ? i?t? < 1 for all t > 0, i.e., (iii) holds true. This completes the proof. ? Lemma 2.1 implies that for any ?s0 ; i0 ; y0 ? such that s0 ? i0 P 1, then either a time t0 > 0 exists for which s?t? ? i?t? < 1 for all t > t0 , or limt!?1 s?t? ? 1, limt!?1 i?t? ? 0. Furthermore, if s0 ? i0 < 1, then s?t? ? i?t? < 1 for all t > 0. Hence, in any case a nonnegative time, say t? , exists such that i?t? < 1, s?t? < 1 ? for all t > t? , where > 0 is su?ciently small. From the third equation of Eq. (2.1) _ y ?t? 6 ? b1 y?t? ? kli?t? kl 6 ? b1 y?t? ? : m m ?2:12?

There exists a positive constant M ? kl=mb1 such that y?t? < ?kl=mb1 ? ? for all large t. This shows that system (2.1) is dissipative. Let X be the following subset of R3 : ?

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X ? f?s; i; y? 2 R3 : s ? i 6 1; y 6 Mg: ?

?2:13?

Theorem 2.1. The set X is a global attractor in R3 and, of course, it is positively ? invariant. Proof. Due to Lemma 2.1 for all initial conditions in R3 such that ?s0 ; i0 ; y0 ? 62 X, ? either there exists a positive time, say T , T ? maxft1 ; t? g, such that the corresponding solution ?s?t?; i?t?; y?t?? 2 int X for all t > T , or the corresponding solution is such that ?s?t?; i?t?; y?t?? ! E10 ? ?1; 0; 0? as t ! ?1. But E10 2 oX. Hence the global attractivity of X in R3 has been proved. ? Assume now that ?s0 ; i0 ; y0 ? 2 int X. Lemma 2.1 implies that s?t? ? i?t? < 1 for all t > 0. Then we know that y?t? 6 M for all large t. Let us remark that if ?s0 ; i0 ; y0 ? 2 oX, because s0 ? i0 ? 1 or y0 ? M or both, then still the corresponding solution ?s?t?; i?t?; y?t?? must immediately enter int X or coincide with E10 . This completes the proof. ? We shall point out here that although ?0; 0; 0? and ?1; 0; 0? are de?ned for system (2.1), they cannot be linearized at. So, local stability of ?0; 0; 0? and ?1; 0; 0? cannot be studied. Indeed, these singularities at the origin or diseasefree equilibrium, while causing much di?culty in our analysis of the system, contribute signi?cantly to the richness of dynamics of the model (see also [16]). Except for equilibria E0 ? ?0; 0; 0?, E10 ? ?1; 0; 0?, system (2.1) has also equilibria a?1 ? b2 ? M ; 0 ??; ; ? and E? ? ?s? ; i? ; y ? ?; E20 ? b2 ; s i y 1?a where a?1 ? i; a kl ? b1 ? y? ? i: mb1 s? ? 1 ? i? ? a ?mk?1 ? b2 ? ? ?kl ? b1 ??; mk?a ? 1? ?2:14?

The boundary equilibrium E20 exists if b2 < 1 and the existence condition for the positive equilibrium E? is mk?1 ? b2 ? > kl ? b1 > 0: ?2:15?

It is clear that condition (2.15) is necessary for the existence of components of i? , y ? of the positive equilibrium. It is also to be noted that this condition implies the existence of E20 . Hence, we conclude that the existence of E? implies the existence of E20 , but the reverse is not true. In the following, we shall show that system (2.1) is not persistent and then there exist solutions which tend to ?1; 0; 0? even when the positive equilibrium exists.

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De?nition 2.1. System (2.1) is said to be not persistent if min lim inf s?t?; lim inf i?t?; lim inf y?t? < 0

t!?1 t!?1 t!?1

for some of its positive solutions. Theorem 2.2. If 1 ? b2 ? ?l=m? ? b1 < 0, then system (2.1) is not persistent. Proof. Since 1 ? b2 ? ?l=m? ? b1 < 0, there is a n such that l=?m ? n? ? 1 ? b2 ? b1 . Let d ? i?0?=y?0? < n and s?0? < 1. Then s?t? < 1 for t P 0 by Lemma 2.1. We claim that for all t > 0; i?t?=y?t? < n and limt!?1 i?t? ? 0. Otherwise, there is a ?rst time t1 , i?t1 ?=y?t1 ? ? n and for t 2 ?0; t1 ?, i?t?=y?t? < n. Then for t 2 ?0; t1 ?, we have di?t? l 6 i?t? 1 ? b2 ? dt m ? i=y l 6 i?t? 1 ? b2 ? ? ?b1 i?t?; m?n which implies that i?t? 6 x?0?e?b1 t . However, for all t P 0, _ y ?t? P ? b1 y?t?; which implies that y?t? P y?0?e?b1 t . This shows that for t 2 ?0; t1 ?, i?t? i?0? 6 ? d < n; y?t? y?0? a contradiction to the existence of t1 , proving the claim. This in turn implies that i?t? 6 i?0?e?b1 t for all t P 0. That is, limt!?1 i?t? ? 0. ? Note that under the assumption 1 ? b2 ? ?l=m? ? b1 < 1, system (2.1) may have positive steady state. This shows that system (2.1) can have both positive steady state and positive solutions that tend to the disease-free equilibrium. In fact, we also have: Theorem 2.3. If 1 ? b2 ? ?l=m? ? b1 < 0, then there exist positive solutions ?s?t?; i?t?; y?t?? of system (2.1) such that limt!?1 ?s?t?; i?t?; y?t?? ? ?1; 0; 0?. Proof. If kl 6 b1 , then this is obvious from the third equation of Eq. (2.1) that leads to Theorem 2.2. Assume that kl > b1 . Again, the argument leading to Theorem 2.2 shows that limt!?1 i?t? ? 0, and for t P 0, i?t?=y?t? 6 d, provided that d ? i?0?=y?0? < n, s?0? < 1, where n ? ?1=?1 ? b2 ? b1 ?? ? m. Let ?s?t?; i?t?; y?t?? be the solution of Eq. (2.1), with i?0?=y?0? < n and s?0? < 1. Since solution ?s?t?; i?t?; y?t?? of Eq. (2.1) is bounded, we have

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405

0 6 l1 ? lim sup y?t? < ?1;

t!?1

M

0 6 l2 ? lim inf y?t? < ?1:

t!?1

M

_ If l2 > 0, then we see that for large t, y ?t? < ?1b1 y?t?, which leads to 2 limt!?1 y?t? ? 0. This is a contradiction. So we must have l2 ? 0. Assume below that l1 > 0. Since limt!?1 i?t? ? 0, then there is a t1 such that for t > t1 ; i?t? 6 mb1 l1 =?2?kl ? b1 ??. From the assumption l1 > 0, we see that there is a t2 > t1 _ _ such that y?t2 ? > 1l1 and y ?t2 ? > 0. However, we note that y ?t2 ? > 0 implies 2 i?t2 ? > mb1 y?t2 ? mb1 l1 : > kl ? b1 2?kl ? b1 ?

This is a contradiction to the fact that i?t? 6 mb1 l1 =?2?kl ? b1 ?? for t > t1 . Hence we must have l1 ? 0. Hence we must have l1 ? 0, that is, limt!?1 y?t? ? 0. Since limt!?1 i?t? ? 0, there is a t3 > t2 such that for t > t3 ; i?t? < , > 0 su?ciently small. From the ?rst equation of Eq. (2.1), we know a?1 _ ? s < s < as?1 ? s?: as 1 ? a This clearly shows that limt!?1 s?t? ? 1. This completes the proof. ?

Note that under assumption 1 ? b2 ? ?l=m? ? b1 < 0, Eq. (2.1) may have positive steady state and positive solutions that tend to the E10 ? ?1; 0; 0?. Theorem 2.3 implies that if 1 ? b2 ? ?l=m? ? b1 < 0, the boundary equilibrium E20 ? ?; ; 0? of Eq. (2.1) is not globally asymptotically stable. Note that under s i the condition 1 ? b2 ? ?l=m? ? b1 < 0, system (2.1) can have no positive steady state and at the same time, ?; ; 0? is locally stable (just add the assumption s i kl < b1 ). In such a case, some solutions tend to ?1; 0; 0? and some tend to ?; ; 0?. Hence the above theorem shows that system (2.1) can have exhibited s i behavior similar to bistability. Theorem 2.4. System (2.1) is permanent provided kl > b1 , 1 ? b2 ? ?l=m? > 0. Proof. From the ?rst two equations of Eq. (2.1), we have: ds?t? ? as?t??1 ? s?t? ? i?t?? ? s?t?i?t?; dt di?t? l P s?t?i?t? ? b2 i?t? ? i?t?: dt m Consider the comparison equations: _ u1 ?t? ? au1 ?t??1 ? ?u1 ?t? ? u2 ?t??? ? u1 ?t?u2 ?t?; l _ u2 ?t? ? u1 ?t?u2 ?t? ? b2 u2 ?t? ? u2 ?t?: m ?2:17?

?2:16?

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It is easy to know that for 1 ? b2 ? ?l=m? > 0, l a l 1 ? b2 ? b2 ? ; m 1?a m is a unique positive equilibrium of system (2.17) which is globally asymptotically stable. Let u1 ?0? 6 s?0?, u2 ?0? 6 i?0?. If ?u1 ?t?; u2 ?t?? is a solution of (2.17) with initial value ?u1 ?0?; u2 ?0??, then by comparison theorem we have s?t? P u1 ?t?, i?t? P u2 ?t? for t > 0, and hence l M a l M 1 ? b2 ? ?i lim inf s?t? P b2 ? ?s; lim inf i?t? P t!?1 t!?1 m a?1 m ?2:18? provided 1 ? b2 ? ?l=m? > 0. Then, there is a T such that for any given > 0 su?ciently small i?t? P i ? ; t > T:

From the third equation of Eq. (2.1), we have _ y ?t? P ? b1 y?t? ? ? kly?t??i ? ? my?t? ? i ?

y?t? ??mb1 y?t? ? ?kl ? b1 ??i ? ??; my?t? ? i ?

which implies that

t!?1

lim inf y?t? P

?kl ? b1 ??i ? ? : mb1

Incorporating into Lemma 2.1, we know that system (2.1) is permanent. ?

3. Hopf bifurcation, global stability result ^ Let E ? ?^; ^; y ? be any arbitrary equilibrium. Then the characteristic equas i ^ ^ tion about E is given by a ? 2a^ ? ?a ? 1?^ ? k s i ??a ? 1?^ s 0 2 2 ml^ y l^ i ^ ^ ? b2 ? ?k ? i s det ?m^ ? ^?2 y i ?m^ ? ^?2 y i kml^2 y kl^2 i 0 ?b1 ? ? k ^?2 ^?2 ?m^ ? i y ?m^ ? i y

? 0: ?3:1?

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407

For equilibrium E20 , (3.1) reduces to ?k ? ?kl ? b1 ???k2 ? ka ? ?a ? 1?? ? 0: s si ?3:2?

Obviously, whenever the positive steady state E? exists, E20 ? ?; ; 0? is uns i stable. In the following, we always assume that E? exists. Before considering the local stability analysis of the positive equilibrium E? , we recall that the stability properties of E? depend on the susceptible dimensionless concentration s? , which we shall rename as n ? s? ; n 2 ?b2 ; 1?: ?3:3?

The characteristic equation (3.1) about E? gives k3 ? Q1 ?n?k2 ? Q2 ?n?k ? Q3 ?n? ? 0; where the coe?cients Qi ?n?; i ? 1; 2; 3 are: b1 1 Q1 ?n? ? ?kl ? b1 ? 1 ? ? an; mk kl b1 1 an ? an?1 ? n?; Q2 ?n? ? ?kl ? b1 ? 1 ? mk kl b1 Q3 ?n? ? ?kl ? b1 ?an?1 ? n?; n 2 ?b2 ; 1?: kl Obviously, Q3 ?n? > 0 for all n 2 ?b2 ; 1?. Q1 can be rewritten as b1 ?mk ? 1? ? a ?n ? b2 ? ? ab2 : Q1 ?n? ? kl By applying the result [17] we obtain the following theorem: Theorem 3.1. Assume ??b1 =kl?mk < a ? ?b1 =kl? < 0 and 2mk 6 b2 < 1. Then a simple Hopf bifurcation occurs at the unique value n0 2 ?b2 ; 1? for increasing n, i.e., the positive equilibrium E? is locally asymptotically stable in ?b2 ; n0 ? and unstable in ?n0 ; 1?. Proof. From the conditions of theorem, we know that Q1 ?n? > 0 for all n 2 ?b1 ; 1?. Now let us look at ?3:4?

?3:5?

?3:6?

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M?2? ?n? ? Q1 ?n?Q2 ?n? ? Q3 ?n? b1 1 b1 1 ? ?kl ? b1 ? 1 ? ?kl ? b1 ? 1 ? an ? an mk mk kl kl b1 ?kl ? b1 ? ? an?1 ? n? an ? k 2 lm 2 b1 b1 ?mk ? 1??n ? b2 ? an ? ?mk ? 1??n ? b2 ??an?2 ? kl kl ab1 2 n?1 ? n??n ? b2 ?: ? ?an? ?1 ? n? ? kl Obviously, M?2? ?b2 ? ? ?an? ?1 ? n? ? ?ab2 ? ?1 ? b2 ? > 0; ab1 b1 ?2? ?mk ? 1??1 ? b2 ? ?mk ? 1??1 ? b2 ? ? a < 0: M ?1? ? kl kl Clearly, a ? ?b1 =kl??mk ? 1? > 0 and mk < 1 imply that a ? ?b1 =kl??mk ? 1? ? ?1 ? b2 ? > 0. Since M?2? ?n? is continuous on ?b2 ; 1?, a value n0 2 ?b2 ; 1? must exist at which M?2? ?n0 ? ? 0. Now, we check the sign of d2 M?2? ?n?=dn2 for n 2 ?b2 ; 1?. 2 d2 M?2? ?n? b1 b1 ?mk ? 1? ?2n ? 4?n ? b2 ?? ? ?mk ? 1? ?a 2 kl kl dn ? ?4a2 n ? 2a2 ?n ? b2 ?? ? 2a2 ?1 ? n? ? 4a2 n ab1 ?2?1 ? n? ? 2?n ? b2 ? ? 2n? ? kl 2ab1 b1 ?mk ? 1??n ? b2 ? ? 4an ?mk ? 1? ? kl kl 2 b1 b1 ? ?mk ? 1? ? a ? 2ab2 ?mk ? 1? kl kl b1 b1 2ab1 b2 ? 2a?1 ? n? a ? ? 4an ?a ? kl kl kl 2ab1 4anb1 ?mk ? 1??n ? b2 ? ? ?mk ? 1? 6 kl kl 2 b1 b1 ? ?mk ? 1? ? a ? 2ab2 ?mk ? 1? kl kl b1 4anmkb1 2ab1 b2 ? ? 2a?1 ? n? ?a ? kl kl kl 4ab1 mk 2ab1 b2 ? < < 0; kl kl which implies that the value n0 is unique.

2 2

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409

Furthermore,

2 dM?2? ?n? b1 b1 jn?n0 ? 2an0 ?mk ? 1? ?n0 ? b2 ? ? ?mk ? 1??2a2 n2 ? a2 b2 n0 ? 0 dn kl kl ab1 ?n0 ?n0 ? b2 ? ? n0 ?1 ? n0 ?? ? a2 n0 ?1 ? n0 ? ? ?an0 ?2 ? kl 2ab1 n0 b1 ?mk ? 1??n0 ? b2 ? a ? ?mk ? 1? ? kl kl b1 b1 ? a2 b2 n0 ?mk ? 1? ? an0 ?1 ? n0 ? a ? kl kl b1 b1 b2 ?a ? ? an0 n0 kl kl 2ab1 n0 b1 ?mk ? 1??n0 ? b2 ? a ? ?mk ? 1? 6 kl kl 2 a b1 b2 n0 ?mk ? 1? ? kl b1 mkb1 b1 b2 ? an0 ? ? an0 ?1 ? n0 ? a ? kl kl kl ab1 n0 6 ?mk ? b2 ? < 0: kl

Hence M?2? ?n? > 0 in ?b2 ; n0 ? and, according to the Routh–Hurwitz criterion, E? is locally asymptotically stable in ?b2 ; n0 ?. Furthermore, according to the result [17] we have a simple Hopf bifurcation toward periodic solutions for increasing n, being M?2? ?n? < 0 in ?n0 ; 1?. Of course, E? is unstable in ?n0 ; 1?. This completes the proof. ? In Section 2, we have shown that whenever the parameter mk?1 ? b2 ? > kl ? b1 > 0, then the positive equilibrium E? is feasible and the boundary one E20 ? ?; ; ? is unstable. However, when kl ? b1 the positive equilibrium E? s i y collapses into E20 , whereas for kl < b1 the positive equilibrium is not feasible and boundary one E20 becomes locally asymptotically stable. The above results hold when b2 < 1. As b2 increases to 1 and when b2 ? 1 the boundary equilibrium E20 collapses into E10 ? ?1; 0; 0?, whereas for b2 > 1 the equilibrium E20 is not feasible. Although E1 cannot be linearized at, then local stability of E10 cannot be studied, we can prove its global stability by constructing Lyapunov function if b2 P 1. Now, we are going to establish the global stability of E10 and E20 . Theorem 3.2. If b2 P 1, then the disease-free equilibrium E10 ? ?1; 0; 0? is globally asymptotically stable.

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Y. Xiao, L. Chen / Appl. Math. Comput. 131 (2002) 397–414

Proof. Let X be set (2.13) of R3 . We proved that any solution of Eq. (2.1) ? starting outside X (in R3 ) either enters into X at some ?nite time, say t0 > 0, ? and then it remains in its interior X for all t > t0 or tends to the boundary equilibrium E10 2 oX. It is therefore su?cient to prove that E10 is asymptotically stable with respect to int X to prove the global asymptotic stability in R3 . ? Let R3 ? f?s; i; y? 2 R3 ; s > 0; i P 0; y P 0g and consider scalar function ?s ? V : R3 ! R ?s V ?t? ? s ? 1 ? ln s ? i: From system (2.1) we get _ V ?t? ? ?a?1 ? s??1 ? s ? i? ? ?1 ? b2 ?i ? 6 ? a?1 ? s??1 ? s ? i? ? ?1 ? b2 ?i: Clearly, the ?rst term on the right of the above formula is always negative in _ int X. If b2 > 1, then V ?t? is negative de?ned in int X and vanishes if and only if ?s; i? ? ?1; 0?. If b2 ? 1, then we have _ V ?t? 6 ? a?1 ? s??1 ? s ? i? 6 0 in X. However, in this case, we have the largest positively invariant subset of _ the set where V ?t? ? 0 is ?s; i? ? ?1; 0?. Hence, for all solutions of system (2.1) starting in int X, we know that limt!?1 s?t? ? 1, limt!?1 i?t? ? 0. Then for any given > 0 su?ciently small, there is a T such that i?t? < ; Hence, we have _ y ?t? 6 ? b1 y?t? ? kl kl i?t? < ?b1 y?t? ? ; m m t > T: liy my ? i ?3:7?

which implies that limt!?1 sup y?t? 6 kl=mb1 . Because of arbitrary smallness of and positivity of solutions, we have limt!?1 y?t? ? 0. This completes the proof. ? Theorem 3.3. If kl < b1 and 1 ? b2 ? ?l=m? > 0, then all solutions of system (2.1) starting in X approach the boundary equilibrium E20 ? ?; ; ? as t ! ?1. s i y Proof. From the proof of Theorem 2.4, we obtain a l M 1 ? b2 ? lim inf i?t? P ?i t!?1 1?a m

?3:8?

provided 1 ? b2 ? ?l=m? > 0. Obviously, limt!?1 sup y?t? 6 0 if kl < b1 . Incorporating into the positivity of solutions, we obtain that limt!?1 y?t? ? 0.

Y. Xiao, L. Chen / Appl. Math. Comput. 131 (2002) 397–414

411

Since 1 > b2 , choose > 0 su?ciently small such that 1 ? b2 ? l > 0: i? ?3:9?

From (3.8) we know that there is a T1 such that for t > T1 , i?t? > i ? . There is T2 > T1 such that for t > T2 , y?t? < . Then the second equation of Eq. (2.1) gives di?t? li?t?y?t? l P s?t?i?t? ? b2 i?t? ? P i?t? s?t? ? b2 ? : ?3:10? dt i? i? Consider the comparison equations ds1 ?t? ? as1 ?t??1 ? s1 ?t? ? i1 ?t?? ? s1 ?t?i1 ?t?; dt di1 ?t? l ? i1 ?t? s1 ?t? ? b2 ? ; dt i? and ds2 ?t? ? as2 ?t??1 ? s2 ?t? ? i2 ?t?? ? s2 ?t?i2 ?t?; dt di2 ?t? ? i2 ?t??s2 ?t? ? b2 ?: dt

?3:11?

?3:12?

Let s1 ?0? 6 s?0?, i1 ?0? 6 i?0?. If ?s1 ?t?; i1 ?t?? is a solution of Eq. (3.11) with initial value ?s1 ?0?; i1 ?0??, then by comparison theorem we have s1 ?t? 6 s?t?, i1 ?t? 6 i?t? for all t P 0. It is easy to know that if (3.9) holds true, l a l M 1 ? b2 ? ; ?s1? ; i1? ?? b2 ? i? a?1 i? is a unique positive equilibrium of Eq. (3.11) which is globally asymptotically stable. By comparison theorem, we have

t!?1

lim inf s?t? P lim inf s1 ?t? ? s1? ;

t!?1

t!?1

lim inf i?t? P lim inf i1 ?t? ? i1? :

t!?1

Similarly

t!?1

lim sup s?t? 6 lim sup s2 ?t? ? s2? ;

t!?1

t!?1

lim sup i?t? 6 lim sup i2 ?t? ? i2? ;

t!?1

where ?s2? ; i2? ? ? b2 ; a ?1 ? b2 ? a?1

is a unique positive equilibrium of Eq. (3.12).

412

Y. Xiao, L. Chen / Appl. Math. Comput. 131 (2002) 397–414

Because of arbitrary of , we obtain that

t!?1

lim s?t? ? b2 ? ; s

t!?1

lim i?t? ? ?

a ?1 ? b1 ? ? : i a?1

This completes the proof.

4. Discussion It is interesting to compare the three species eco-epidemiological model with ratio-dependent Michaelis–Menten type analyzed here, with the analogous di?erential equation model analyzed in [6], which has a functional response function c?S?. The existence and local stability of equilibria are discussed. They also show that Hopf bifurcation can occur when choosing c?S? ? S=?m ? S?. In [7], which belonged to the Lotka–Volterra type predator–prey model in the absence of disease, existence and local stability of equilibria are obtained, and no periodic solution exists. In this paper, we have shown that for eco-epidemic model with ratio-dependent Michaclis–Menten type functional response, these are rich in boundary dynamics, and most importantly, we show that even when there is no positive steady state, both infected prey and predator can become extinct. Such an extinction occurs in two cases. In one case, both infected prey and predator species become extinct regardless of the initial data. In the other case, both of them will die out only if initial prey/predator ratio is too low. Also we show that whether the system is permanent or not, periodic solutions bifurcating from positive equilibrium can occur provided conditions of Theorem 3.1 are satis?ed. That is, when system (2.1) is permanent, and hence the positive equilibrium E? exists, we obtained that periodic solution can occur (see Fig. 1); On the other hand, when the system (2.1) is not permanent, we can also choose appropriate parameters to satisfy the conditions of Theorem 3.1, then the

Fig. 1. a ? 0:7, b ? 1=7, b2 ? 0:8, m ? 0:5, k ? 0:65, l ? 0:1.

Y. Xiao, L. Chen / Appl. Math. Comput. 131 (2002) 397–414

413

Fig. 2. a ? 0:8, b ? 1=7, b2 ? 0:9064, m ? 1:5, k ? 0:2, l ? 0:8.

periodic solution can occur (see Fig. 2). In this case, we verify that condition of Theorem 2.3 holds true, then at this time there are some solutions which tend to disease-free equilibrium E10 ? ?1; 0; 0?. The threshold parameter r is found in this paper which controls the development of the disease. The parameter r governs whether or not the disease persists. In the absence of the disease, the prey population approaches the carrying capacity K. The threshold parameter 1 bK r? ? b2 c is the average number of adequate contacts, when the prey population is K, of an infective during the mean infectious period 1=c. In a totally susceptible population of K, the parameter r is the average number of new infections (secondary cases) produced per infective, so it is often called the basic reproduction number. We know that if r 6 1 (i.e., b2 P 1), the disease dies out, and of course the predator population goes to extinction, that is to say, the diseasefree equilibrium E10 is globally asymptotically stable; whereas r > 1 (i.e., b2 < 1) the infection does not tend to zero. There are still many interesting and challenging mathematical questions which need to be studied for system (2.1). For example, we cannot analyze system (2.1) in its all parameter regions, there is room for improvement. However, signi?cant improvements appear to be di?cult. Also, we are unable to show the local stability of origin and E10 , or to show that E? is globally asymptotically stable when E? is locally asymptotically stable. We conjure that the origin is always unstable. We can also consider the infective rate b?S=N ?I. We leave this for future work.

References

[1] H.W. Hethcote, A thousand and one epidemic models, in: S.A. Levin (Ed.), Frontiers in Mathematical Biology, Lecture Notes in Biomathematics, vol. 100, Springer, Berlin, 1994, p. 504.

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Y. Xiao, L. Chen / Appl. Math. Comput. 131 (2002) 397–414

[2] R.M. Anderso, R.M. May, Infectious Disease of Humans Dynamics and Control, Oxford University Press, Oxford, 1991. [3] N.J.T. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Gri?n, London, 1975. [4] O. Diekmann, J.A.P. Hecsterbeck, J.A.J. Metz, The legacy of Kermack and Mckendrick, in: D. Mollision (Ed.), Epidemic Models, Their Structure and Relation to Data, Cambridge University Press, Cambridge, 1994. [5] K.P. Hadeler, H.I. Freedman, Predator–prey population with parasitic infection, J. Math. Biol. 27 (1989) 609–631. [6] J. Chattopadhyay, O. Arino, A predator–prey model with disease in the prey, Nonlinear Anal. 36 (1999) 749–766. [7] E. Venturino, The in?uence of disease on Lotka–Volterra systems, Rocky Mountain J. Math. 24 (1994) 389–402. [8] J.C. Holmes, W.M. Bethel, Modi?cation of intermediate host behavior by parasites, in: E.V. Canning, C.A. Wright (Eds.), Behavioral Aspects of Parasite Transmission, Suppl. I to Zool. f.Linnean Soc. 51 (1972) 123–149. [9] R.O. Peterson, R.E. Page, Wolf density as a predictor of predation rate, Swedish Wildlife Research Suppl. 1 (1987) 771–773. [10] R. Arditi, H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology 73 (1992) 1544–1551. [11] R. Arditi, L.R. Gimzburg, H.R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent models, Amer. Nat. 138 (1991) 1287–1296. [12] R. Arditi, N. Perrin, H. Saiah, Functional response and heterogeneities: an experiment test with cladocerans, OIKOS 60 (1991) 69–75. [13] I. Hanski, The functional response of predator: worries bout scale, TREE 6 (1991) 141–142. [14] Y. Xiao, L. Chen, Modeling and analysis of a predator–prey model with disease in the prey, Math. Biosci. 171 (2001) 59–82. [15] R.M. Anderson, R.M. May, The population dynamics of microparasites and their invertebrates hosts, Proc. Roy. Soc. London B 291 (1981) 451–463. [16] Y. Kuang, E. Beretta, Global qualitative analysis of a ratio dependent predator prey system, J. Math. Biol. 36 (1998) 389–406. [17] W.M. Liu, Criterion of Hopf bifurcation without using eigenvalues, J. Math. Anal. Appl. 182 (1994) 250–255.

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