DYNAMICS OF A DELAY-DIFFUSION PREY-PREDATOR MODEL WITH DISEASE IN THE PREY

J. Appl. Math. & Computing Vol. 17(2005), No. 1-2, pp. 361-377 DYNAMICS OF A DELAY-DIFFUSION PREY-PREDATOR MODEL WITH DISEASE IN THE PREY B. MUHOPADHYAY AND R. BHATTACHARYYA Abstract. A mathematical model dealing with a prey-predator system with disease in the prey is considered. The functional response of the predator is governed by a Hoilling type-ii function. Mathematical analysis of the model regarding stability and persistence has been performed. The effect of delay and diffusion on the above system is studied. The role of diffusivity on stability and persistence criteria of the system has also been discussed. AMS Mathematics Subject Classification : Primary 92D30. eywords and phrases : Prey-predator system, persistence, impermanence, discrete time delay, diffusivity. 1. Introduction Mathematical modeling has been playing an important role in epidemiology, the study of spread and control of infectious diseases. Mathematical models contributes in two major aspects of epidemiology- (i) to build and test epidemiological theories and (ii) to plan, implement and evaluate detection, control and prevention programmes. Formally, epidemiological modeling refers to dynamic modeling, where, the population is divided into compartments based on their health status, namely, susceptible, infectious, recovered etc. The movement between compartments by becoming infected, progressing, recovering or migrating are specified by differential equations. The advantage of mathematical modeling of infectious diseases is the economy, clarity and precision of mathematical formulation. Moreover, there are many mathematical techniques available for determining the threshold, equilibrium, periodic solutions and stability behaviour of the system. Received October 8, 2003. Corresponding author. c 2005 orean Society for Computational & Applied Mathematics and orean SIGCAM. 361

362 Mukhopadhyay and Bhattacharyya The mathematical analysis can identify important combinations of parameters and essential aspects or variables in the model. In the natural world, species do not exist alone. Together with spreading the disease, they also compete with other species for food and space and is also predated by some other species, simultaneously. Thus while studying the dynamical behaviour of epidemiological models, it is of more biological significance to consider the effect of interaction between various species. The majority of models relating to the transmission of infectious diseases arise from the classical SIR model of ermack and Mcendrick [23]. Recently, epidemiological models have received much attention from scientists. The survey paper of Hethcote [18] is a vast reference of such models. Freedman [14] has studied a prey-predator system in which some members of the prey population and all predators are subject to infection by parasites and obtained conditions for persistence of all populations and global stability of the positive equilibrium. Holmes and Bethel [20] and Dobson [10] discussed situations where the behaviour of infected individuals of a prey population as a host is modified by the action of a parasite. Anderson and May [3] showed that invasion of a resident predator-prey or host-parasite system by a new strain of parasites could cause destabilization and exhibit limit cycles. Hadeler and Freedman [17] observed a similar phenomenon. Mukherjee [25] analysed a prey-predator model with parasite infection and obtained conditions for persistence and impermanence. Chattopadhyay et.al. [8] considered a prey-predator model with non-selective harvesting and infection in the prey population. Another important aspect which should be kept in mind while formulating an epidemiological model is the fact that the reproduction of predators after predation will not be instantaneous, but will be mediated by some discrete time lag required for gestation of predator. Das et.al. [9] considered an autotrophherbivore system with nutrient recycling where the effect of time delay is studied. Xiao and Chen [29] studied the dynamical behaviour in a prey-predator model with disease in the prey, taking into account, the discrete time lag due to gestation of the predator. Since in an epidemiological model, the population is assumed to be divided into compartments according to health status and there is a continuous movement of different individuals between the compartments, diffusion will also play an important role in such type of modeling. In the present paper, we consider a mathematical model consisting of susceptible and infected prey and the predators (namely an SI model). Our model is a modified form of the model considered by Xiao and Chen [29]. The predation functional response is taken as a Holling type-ii function. We have incorporated a discrete time delay due to gestation of predator into our model. The effect of diffusion due to the migration of the individuals between various compartments

Dynamics of a delay-diffusion prey-predator model 363 of the model system is also considered. The main objective of this paper is to discuss the equilibria and their stability. Persistence criteria of the system are derived. The effect of delay and diffusion on stability and persistence criteria of the system are also studied. 2. Description of the model In the present study, our model consists of two populations- (i) the prey, whose total density is denoted by N(t). (ii) the predator whose population density is denoted by Y (t). We make the following assumptions in our model system. (1) In the absence of disease, the prey population grows according to the logistic law with intrinsic growth rate r(r >0) and carrying capacity ( >0). (2) In the presence of disease, the prey population is divided into two classes, namely, the susceptible prey (S) and the infected prey (I), that is, at any time t, we have, N(t) =S(t)+I(t). (3) Only the susceptible prey is assumed to be capable of reproducing with logistic law. The infected prey is removed by death with positive death rate c or by predation, before having the possibility of reproducing. However, the infected population I also contributes with S to population growth toward the carrying capacity. (4) We also assume that the disease is spread among the prey population only and the disease is not genetically inherited. The infected population do not recover or become immune. (5) The predator has a death rate constant d(d >0) and predation coefficients p 1,p 2 (> 0). The coefficients of conversing prey into predator is q(0 < q 1). The functional response of the predator is assumed to be a Holling type-ii function, namely, f(s) = S m + S. We give a mathematical realization of all these assumptions into the following model. ( ds dt = rs 1 S + I ) SI p 1SY m + S, di dt = SI ci p 2IY, (1) dy YS = dy + qp 1 dt m + S + qp 2IY with the initial conditions S(0) = S 0 > 0; I(0) = I 0 > 0; Y (0) = Y 0 > 0; 0 <p 1,p 2 < 1.

364 Mukhopadhyay and Bhattacharyya We will study the stability of different equilibrium points and the criteria of persistence and impermanence. Biologically, persistence means the survival of all populations in future time. Mathematically, persistence means that the strictly positive solutions do not have omega limit points on the boundary of the nonnegative cone. For various definitions of persistence, see Gard [15] for a version of weak persistence, Freedman and Waltman [12, 13] for persistence and Hofbauer [19], Hutson and Vickers [21] for uniform persistence. An explicit condition for persistence in a three-trophic-level food chain where the trophic levels are assumed to be characterised by increasing and quite diversified time responses can be found in Muratori and Rinaldi [26]. A population x(t) is said to be uniformly persistent if there exists a δ>0, independent of x(0) > 0 such that lim inf t x(t) >δ. We say that a system persists uniformly whenever each component persists uniformly. 3. Boundedness, boundary equilibria and persistence To establish the biological validity of the model system, first we have to show that the solutions of system (1) are bounded. Theorem 1. All the solutions of the system (1) are bounded. Proof. Let W = S + I + Y. Then, [ ( dw = S r 1 S + I ) I p ] 1y + I[S c p 2 Y ] dt m + S ( +Y d + qp ) 1S m + S + qp 2I ( Sr 1 S ) ci dy [ ( = S r 1 S ) ] +1 S ci dy S(r +1) S ci dy ˆ(r +1) mw, where ˆ = max{s(0),}. Thus, dw dt + mw ˆ(r +1).

Dynamics of a delay-diffusion prey-predator model 365 Applying a theorem of differential inequalities [6], we get, and for t, Hence the theorem. 0 W ˆ(r +1) m + W (S 0,I 0,Y 0 ) e mt 0 W ˆ(r +1) m. Moreover, if (S(t),I(t),Y(t)) be any solution with the initial conditions S 0 > 0, I 0 > 0 and Y 0 > 0, then we can conclude from ( ds dt Sr 1 S ) and a standard comparison theorem that lim sup S(t) ˆ, t where ˆ = max {S(0),}. Hence the system (1) is dissipative. The equilibrium points of the system (1) are E 0 (0, 0, 0) [trivial equilibrium], E 1 (, 0, 0) [axial equilibrium], E 12 (S,I,0) and E 13 (Ŝ,0, Ŷ ) [boundary equilibria]. E 12 and E 13 are the boundary equilibria. E 12 represents the equilibrium for which the predator population will die out and E 13 represents the case for which the infected population will die out. The extinction of infected population corresponds to the end of the epidemic and consequently, the stability of E 13 plays an important role in controlling the epidemic. Now the community matrix of the system (1) will be given by V = r 1 S+I I p 1Y m+s rs + p1ys (m+s) S ( r 2 + ) p1s m+s I S c p 2 Y p 2 I qp 1Ym (m+s) qp 2 2 Y d + qp1s m+s + qp 2I So, the community matrix at E 0 is given by V (E 0 )= r 0 0 0 c 0 0 0 d The corresponding eigen values are r, c and d. Thus, two eigen values are negative and one is positive. So the trivial equilibrium point E 0 is unstable.

366 Mukhopadhyay and Bhattacharyya The boundary equilibrium points are E 12 and E 13. Now, E 12 (S,I,0), where S = c and I = r c. + r Therefore, E 12 exists if > c. (2) Also, E 13 (Ŝ,0, Ŷ ), where d + qp 1Ŝ m + Ŝ =0, ( rŝ 1 Ŝ ) p 1ŜŶ =0. (3) m + Ŝ Therefore, E 13 exists if d<qp 1. (4) The community matrix evaluated at E 1 is, V (E 1 )= r r p1 +m 0 c 0 0 0 d Since, V (E 1 ) has two negative and one positive root if E 12 exists, thus E 1 has stable and unstable manifolds. The community matrix about E 12 is given by V (E 12 )= rs ( r+ ) S p 1S m+s. I 0 p 2 I 0 0 d + qp1s m+s 2I The eigen values of V (E 12 ) are given by λ = d + q m + S + p 2I, λ ± = rs ± r 2 S 2 4IS(r + ). (5) 2 E 12 is locally asymptotically stable in S I plane as the sign of real parts of λ ± are always negative. E 12 is asymptotically stable or unstable in the Y-direction according as d> or <q m + S + p 2I..

Dynamics of a delay-diffusion prey-predator model 367 The community matrix about E 13 is given by { p1 Ŷ r V (E 13 )= (m+ŝ)2 }Ŝ Ŝ ( r + ) 0 Ŝ c p 2 Ŷ 0 qp1ŷ m qp 2 Ŷ 0 (m+ŝ)2 p1ŝ m+ŝ. The eigenvalues of V (E 13 ) are given by ν = Ŝ c p 2Ŷ [ (6) ν ± = 1 { p1 Ŷ Ŝ 2 (m + r } { p1 ± Ŝ Ŝ)2 2 Ŷ (m + r } ] 2 4qp2 1Ŷ Ŝm. Ŝ)2 (m + Ŝ)3 E 13 is asymptotically stable in the S Y plane if d2 +4dqp 1 d Ŝ>. (7) 2p 1 q E 13 is asymptotically stable or unstable in the direction of I according as, Ŝ c p 2Ŷ < or > 0, that is, Ŝ< or > p 2Ŷ + c. (8) Persistence of a system corresponds to the survival of all populations after a long time, no matter what the initial populations are. Theorem 2. Let (i) > c, { ( } (ii) d<min q m + S + p 2I ), qp 1, (iii) Ŝ>p 2Ŷ + c. If there exists no limit cycle in the S Y plane, then the system (1) is uniformly persistent. Proof. We have already shown that E 0 is a saddle point as > c and d<p 1 q. Also since, > c and d<p 1 q, the boundary equilibrium points E 12 and E 13 exists. Again, d<q m + S + p 2I

368 Mukhopadhyay and Bhattacharyya implies E 12 is a saddle point. Finally, the condition Ŝ> p 2Ŷ + c ensures that E 13 is also a saddle point. As there exists no limit cycle in the S Y plane and the system (1) is dissipative, by the main theorem of Bulter et. al. [7], the system is uniformly persistent. A community is impermanent if there is at least one semi-orbit which tends to the boundary. Now, we state a theorem on impermanence. Theorem 3. Let (i) > c, (ii) q m + S + p 2I <d<p 1 q, (iii) Ŝ<p 2Ŷ + c. Then the system (1) is impermanent. Proof. The condition d>q m + S + p 2I implies that Ẏ Y (E 12) < 0 that is the equilibrium point E 12 is saturated on the boundary. Consequently, there exists at least one orbit in the interior that converges to the boundary [19]. The condition Ŝ< p 2Ŷ + c implies, I I (E 13) < 0 that is, E 13 is a strictly saturated equilibrium point. Hence the theorem. The above analysis shows that the death rate of predator and the average number of adequate contacts, σ ( = ) c, play an important role in shaping the dynamics of the system (1). 4. Effects of delay on the model

Dynamics of a delay-diffusion prey-predator model 369 Let us now consider the system (1) with time delay due to gestation of the predator. The modified model will take the form, ( ds dt = rs 1 S + I ) SI p 1SY m + S, di dt = SI ci p 2IY, (9) dy Y (t τ)s(t τ) = dy + qp 1 + qp 2 I(t τ)y (t τ), dt m + S(t τ) where τ>0is the time required for the gestation of the predator. The initial conditions are, S(t) =S 0 (t) > 0, I(t) =I 0 (t) > 0, Y(t) =Y 0 (t) > 0, where S 0 (t), I 0 (t) and Y 0 (t) are given non-negative functions on τ <t<0. There exists four steady states on the boundary of R+, 3 namely, E 0 (0, 0, 0); E 1 (, 0, 0); E 12 (S,I,0); E 13 (Ŝ,0, Ŷ ). Obviously, the origin is unstable, and the existence of E 12 and E 13 imply the unstability of E 1. The linearised system about E 12 with the linearised variables s, i and y corresponding to S, I and Y is given by ds ( r ) dt = rs s S + i + p 1S m + S y, di dt = Is p 2Iy, (10) dy dt = dy + q m + S + p 2I y(t τ). The characteristic equation for the system (10) will be [ ( ][ λ + d q m + S + p 2I )e λτ λ 2 + rs ( r ) ] λ + SI + = 0 (11) Since all the roots of λ 2 + rs λ + SI ( r + ) =0 have negative real parts, E 12 is asymptotically stable in the S I plane. The root corresponding to the first factor of equation (11) will be positive if d<q m + S + p 2I

370 Mukhopadhyay and Bhattacharyya Consequently, E 12 is unstable in the direction of Y if d<q m + S + p 2I Let us now study the behaviour of the equilibrium point E 13. The corresponding linearised equations will be [ ds dt = rŝ + p 1Ŷ Ŝ ] ( r ) s (m + Ŝ Ŝ)2 + i p 1Ŝ m + Ŝ y, di dt =(Ŝ c p 2Ŷ )i, (12) dy qp1ŷm = dt (m + [s(t τ)] + qp 2Ŷ [i(t τ)] dy + qp 1Ŝ [y(t τ)]. Ŝ)2 m + Ŝ The characteristic equation corresponding to (12) will be ( r (Ŝ c p 2Ŷ [λ λ) 2 + λ Ŝ p 1Ŷ Ŝ (m + d + qp ) 1Ŝ Ŝ)2 m + Ŝ e λτ + qp2 1Ŷ Ŝm ] e λτ =0. (13) (m + Ŝ)3 If Ŝ>c + p 2Ŷ, then equation (13) will have one positive root. Let λ = µ + iω (µ 0), where λ is a root of ( r λ 2 + λ Ŝ p 1Ŷ Ŝ (m + d + qp ) 1Ŝ Ŝ)2 m + Ŝ e λτ + qp2 1Ŷ Ŝm (m + e λτ =0. (14) Ŝ)3 The imaginary part of the left hand side of (14) after substituting the value of λ will be ( r 2µ + Ŝ p 1Ŷ Ŝ (m + d + qp 1Ŝ Ŝ)2 m + Ŝ e µτ cos ωτ µqp 1Ŝ sin ωτ τe µτ m + Ŝ ωτ qp2 1Ŷ Ŝ (m + Ŝ)3 e µτ τ sin ωτ ωτ > r Ŝ p 1Ŷ Ŝ (m + d + qp 1Ŝ Ŝ)2 m + Ŝ (e µτ cos ωτ) [ µqp1 Ŝ m + Ŝ + qp2 1Ŷ Ŝ ] µτ sin ωτ e (m + Ŝ)3 ω ( r = p ) 1Ŷ Ŝ d (m + Ŝ)2 )

Dynamics of a delay-diffusion prey-predator model 371 for large τ. Therefore, for sufficiently large values of τ, E 13 will be stable in the S Y plane if [ r Ŝ p ] 1Ŷ >d (m + Ŝ)2 Theorem 4. Assume that the system (9) is dissipative. Further assume that { ( r d<min q m + S + p 2I,qp 1, Ŝ p )} 1Ŷ (m + Ŝ)2 and all other conditions of Theorem (2) hold. Then the delayed system (9) will be uniformly persistent. It is also interesting to note that if ( r d<ŝ p ) 1Ŷ (m + Ŝ)2 and Ŝ<c + p 2Ŷ then the equilibrium point E 13 will be asymptotically stable and, in this case, the infected population will die out after a long time., 5. Analysis of the delayed model with diffusion In a prey-predator system, diffusion plays an important role in shaping the dynamical behaviour of the system and in pattern formation [4, 5, 11, 22, 27, 28]. In this section, we will consider the effect of diffusion in the delayed SI model. The linearised model about E 12 with diffusion is ( rs + S ) i + s t = rs s p 1S m + S y + D 2 s S x 2, i t = Is +(S c)i p 2 i 2Iy + D I x 2, (15) y t = dy + q m + S + p 2 y 2I y(t τ)+d Y x 2, where D S, D I and D Y represent the diffusion coefficient of the susceptible prey, infected prey and predator respectively. We take the solutions of (15) in the form s = α 1 e λt cos px, i = α 2 e λt cos px, (16) y = α 3 e λt cos px,

372 Mukhopadhyay and Bhattacharyya where p is the wave number of perturbation. The characteristic equation of (15) will be [ ( ] λ + d + D Y p 2 q m + S + p 2I )e λτ [ ( + λ 2 + λ D I p 2 + D S p 2 + rs ) (17) +D S D I p 4 + rs D Ip 2 + IS( r + ) ] =0. Equation (17) has a positive eigen value if d + D Y p 2 <q m + S + p 2I. (18) All other roots of (17) have negative real parts as [ (D I + D S )p 2 + rs ] < 0 and consequently, E 12 is asymptotically stable in the S I plane. If (18) is satisfied, then E 12 is unstable in direction of Y and hence it will be a saddle point. Let us now consider the system about the equilibrium point E 13. The corresponding equations in this case will be [ s t = rŝ + p 1Ŷ Ŝ ] ( r ) s (m + Ŝ Ŝ)2 + i p 1Ŝ m + Ŝ y + D 2 s S x 2, i t =(Ŝ c p 2Ŷ )i + D 2 i I x 2, (19) y t = qp 1mŶ (m + s(t τ)+qp2ŷi(t τ) dy + qp 1Ŝ Ŝ)2 m + Ŝ y(t τ)+d 2 y Y x 2. The characteristic equation will be [ λ Ŝ + c + p 2 Ŷ + D I p 2][ λ 2 + λ{ r Ŝ } m + Ŝ e λτ (D S + D Y )p 2 + qp 1Ŝ { qp1 Ŝ m + Ŝ e λτ d D Y p 2 } { r Ŝ + qp2 1Ŷ Ŝm e λτ (m + Ŝ)3 p 1Ŷ Ŝ (m + Ŝ)2 d p 1Ŷ Ŝ } (m + + D Sp 2 ] Ŝ)2 =0. (20)

Dynamics of a delay-diffusion prey-predator model 373 E 13 will be stable or unstable in the direction of I according as Let λ = µ + iω be a root of [ { r λ 2 + λ Ŝ { r Ŝ + qp2 1Ŷ Ŝm e λτ (m + Ŝ)3 Ŝ< or > c + p 2Ŷ p 1Ŷ Ŝ (m + d + qp 1Ŝ Ŝ)2 + D I p2. (21) } m + Ŝ e λτ (D S + D Y )p 2 p 1Ŷ Ŝ }{ } (m + + D Sp 2 qp1 Ŝ Ŝ)2 m + Ŝ e λτ d D Y p 2 ] =0. (22) The imaginary part of (22) after substituting the value of λ will be 2µ + rŝ p 1ŜŶ (m + d + p 1qŜ Ŝ)2 m + Ŝ e µτ cos ωτ µ qp 1Ŝ sin ωτ e µτ m + Ŝ ω ( (D S + D Y )p 2 p1 Ŷ Ŝ (m + r ) Ŝ)2 Ŝ D Sp 2 qp1 Ŝ sin ωτ e µτ m + Ŝ ω qp2 1Ŷ Ŝm sin ωτ e µτ (m + Ŝ)3 ω > rŝ p 1ŜŶ (m + d (D Ŝ)2 S + D Y )p 2 + qp 1Ŝ m + Ŝ e µτ cos ωτ [ µqp1 Ŝ m + Ŝ + qp { 1Ŝ p1 Ŷ Ŝ m + Ŝ (m + r } Ŝ)2 Ŝ D Sp 2 = rŝ p 1ŜŶ (m + d (D Ŝ)2 S + D Y )p 2 + qp2 1 mŷ Ŝ (m + Ŝ)3 ] sin ωτ ω e ωτ for large τ. Ifd< rŝ p 1ŜŶ (m + (D S+D Y )p 2, then E 13 will be asymptotically Ŝ)2 stable is the S Y plane. Theorem 5. Assume that the system (15) is dissipative. Moreover if { ( } d<min q p1s + p m+s 2I D Y p 2 r, qp 1, Ŝ p1ŷ (D S + D Y )p 2 and Ŝ> c + p 2Ŷ + D I p2 (m+ŝ)2 )

374 Mukhopadhyay and Bhattacharyya and if all the other conditions of Theorem (2) holds, then the system (15) will be uniformly persistent. It will be of interest to note that if { r d<ŝ p } 1Ŷ (D S + D Y )p 2 (m + Ŝ)2 and Ŝ< 1 (c + p 2Ŷ + D Ip 2 ) the equilibrium point E 13 will be asymptotically stable. This analysis shows that stability of the system depends upon the diffusivities of different populations and the wave number of perturbation (p). If the diffusivity of the infected prey population (D I ) is high and that of the susceptible prey (D S ) and predator (D Y ) is low, then the equilibrium point E 13 will be asymptotically stable which biologically signifies that the infected population will die out after a long time. 6. Discussion Epidemiological models are now widely used as more epidemiologists realise the role that modeling can play in basic understanding and policy development of the subject. Even though vaccines are available for many infectious diseases, these diseases still cause suffering and mortality all over the world. The transmission mechanism from an infective to susceptible is understood for nearly all infectious diseases, and the spread of diseases through a chain of infections is known. However, the transmission interactions in a population are so complex that it is difficult to comprehend the large scale dynamics of the disease spread without the formal structure of a mathematical model. An epidemiological model uses a microscopic description( the role of an infectious individual) to predict the macroscopic behaviour of disease spread through a population. In the present paper, we have considered an eco-epidemiological model (namely an SI model). The predation functional response is assumed to be a Holling type-ii function. Since the predator feeding is not instantaneous, we have considered time-delay due to gestation. We have also incorporated the effect of diffusion due to migration of different individuals of the population. The system is analysed for its equilibria and their stability. Our analysis revealed that the death rate of predator (d) plays an important role in shaping the dynamics of the system. The main results of our investigation can be summarised as follows. 1. The analysis in section 3 revealed that the death rate of predators (d) and the parameter σ (σ = /c) determine whether the disease persists or not. σ is the average number of adequate contacts (when the prey population is ) of an

Dynamics of a delay-diffusion prey-predator model 375 infective during the mean infection period 1/c. The parameter σ is also called the basic reproductive (number) rate [1, 2, 24]. Our investigation also showed that the persistence of the infected population depends upon the equilibrium density of susceptible prey(ŝ). 2. In section 4, we have modified the original model (1) by considering the delay effect due to gestation. The analysis of the delayed model showed that the death rate of predator (d), the basic reproductive rate (σ) and the equilibrium density of susceptible prey are factors determining the dynamical behaviour of the system. The asymptotic stability of the disease-free equilibrium and endemic equilibrium also depend upon these parameters. 3. In section 5, we have studied the effect of diffusion of different populations on the delayed model system. The study of the resulting delay-diffusion model established that the diffusivity of the susceptible prey(d S ), the infected prey (D I ) and the wave number of perturbation (p), together with the parameters mentioned in section 4, govern the dynamical behaviour of the system. A high value of diffusivity of the infected prey population (D I ) and low values for the diffusivities of susceptible prey (D S ) and predator (D Y ) will be the guiding factors which ensure the stability of the disease-free equilibrium. Acknowledgement The authors are grateful to Prof. C. G. Chakrabarti, S. N. Bose Professor, Department of Applied Mathematics, University of Calcutta, for his continuous help and guidance throughout the preparation of the paper. References 1. R. M. Anderson and R. M. May, The population dynamics of macroparasites and their invertebrate hosts, Phil.Trans. Roy. Soc. London B291 (1981), 451-524. 2. R. M. Anderson and R. M. May, Directly transmitted infectious diseases : control by vaccination, Science, 215 (1982), 1053-1060. 3. R. M. Anderson and R. M. May, The invasion and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. Lond. B314 (1986), 533-570. 4. R. Bhattacharyya, M. Bandyopadhyay and S. Banerjee, Stability and bifurcation in a diffusive prey-predator system : non-linear bifurcation analysis, J. Appl. Math. & Computing 10 (2002), 17-26. 5. R. Bhattacharyya, B. Mukhopadhyay and M. Bandyopadhyay, Diffusion driven stability analysis of a prey-predator system with Holling type-iv functional response, System Analysis Modelling Simulation, 43(8) (2003), 1085-1093. 6. G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn. and Co., 1982.

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Dynamics of a delay-diffusion prey-predator model 377 Banibrata Mukhopadhyay is continuing his research work for the Ph.D. degree in the University of Calcutta. His research interests are Dynamical modeling involving diffusion and delay differential equations with application to complex biological systems. Department of Applied Mathematics, University of Calcutta, 92, A. P. C. Road, olkata - 700 009, India e-mail: banibrat001@yahoo.co.in Rakhi Bhattacharyya has recently submitted her thesis for Ph. D. degree of the University of Calcutta. She has obtained her M. Sc. and M. Phil. degrees from the University of Calcutta. Her research interest focus on stability and bifurcation analysis of dynamical models with application to biological systems. Department of Applied Mathematics, University of Calcutta, 92, A. P. C. Road, olkata - 700 009, India e-mail: rakhi bhattach@yahoo.co.in