An Eco-Epidemiological Predator-Prey Model where Predators Distinguish Between Susceptible and Infected Prey

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1 An Eco-Epidemiological Predator-Prey Model were Predators Distinguis Between Susceptible and Infected Prey David Greenalg 1, Qamar J. A. Kan 2 and Josep S. Pettigrew 1 1 Department of Matematics and Statistics, University of Stratclyde, Glasgow, U.K. 2 Department of Matematics and Statistics, College of Science, Sultan Qaboos University, Muscat, Sultanate of Oman Abstract A predator-prey model wit disease amongst te prey and ratio-dependent functional response for bot infected and susceptible prey is proposed and its features analysed. Tis wor is based on previous matematical models to analyse te important ecosystem of te Salton Sea in Soutern California and New Mexico were birds particularly pelicans prey on fis particularly tilapia. Te dynamics of te system around eac of te ecologically meaningful equilibria are presented. Natural disease control is considered before studying te impact of te disease in te absence of predators, te interaction of predators and ealty prey and te disease effects on predators in te absence of ealty prey. Our teoretical results are confirmed by numerical simulation. Keywords: Epidemiology, ecology, differential equations, equilibrium, stability, Hopf bifurcation. 1 Introduction Te Salton Sea in te desert of Soutern California, New Mexico is an important eco-epidemiological system were birds particularly pelicans prey on fis particularly tilapia. Over te past few years tere ave been large-scale bird mortalities at te sea wit tousands of pelicans dying from avian botulism. Te vibrio class of bacteria is very common in salt water fis and tose infected wit vibrio may ave salt water present in teir tissue. As te fis struggle in teir deat process tey tend to rise to te surface of te sea for oxygen. Wen tey do tey become very attractive to fis-eating birds, specifically te 1

2 pelicans. Wen pelicans and oter fis-eating birds eat vibrio infected fis alive a large number become infected wit botulism and die off. In tis paper we sall study a predator-prey model were disease spreads amongst te prey wit different ratio-dependent functional responses for bot infected and susceptible prey. Tis wor is based upon previous matematical models of te Salton Sea. Tere is a large literature already existing on te subject of ecological systems under te influence of epidemiological factors. Anderson and May [1] studied five prey-predator, parasite interactions were parasites infect eiter te prey or te predator. Hadeler and Freedman [2] described a predator-prey model were te prey is infected by a parasite and in turn infects te predator wit tat parasite. Gulland [3] described a number of possible situations in wic predators may be affected by various diseases. Haque and Venturino [4] proposed a predator-prey model wit an epidemic spreading only among te predators by contact between a well individual and an infected one. Tere are two types of prey-predator interaction models in te literature: te first is were prey eaten per predator per unit time is a function of te prey density and in te second type called predator-dependent functional response tis is a function of bot prey and predator. Ratio-dependent functional response is a subtype of predator-dependent functional response and assumes tat te prey eaten per unit time is a function of te ratio of prey to predator. Ratio-dependent predator-prey models are more appropriate for predator-prey interaction were predation involves serious searcing, for example, predator animals searcing for prey in Kuang [5]. It is also a fact tat ratio-dependent models are more flexible and versatile. Various field study and laboratory experiments support ratio-dependent predator-prey models, for example [6], [7] and [8]. Cattopadyay and Bairagi [9], and Sarar et al. [10] studied tilapia and pelican interaction models for te Salton Sea. In bot models te autors assumed tat due to infection, tilapia become wea and pelicans interact only wit infected tilapia. Later using te same viewpoint Greenalg and Haque [11] described a ratio-dependent predator-prey interaction model were susceptibles experienced no predation. Tis is unliely to be realistic for most species. Cattopadyay et al. [12] modified te model discussed in [9] by introducing interaction of pelicans wit susceptible fis and assumed tat feeding on infected fis increases te deat rate of te pelicans. Here we modify te ratio-dependent model of [11] and consider tat te susceptible and te infected populations are exposed to te predator to varying degrees. However, te predator preys preferentially on te most numerous prey type. 2 Te Matematical Model We consider ere a predator-prey model, in wic an epidemic spreads in te prey. We are tining of predators being pelicans and prey tilapia. We assume tat te disease is transmitted by direct contact among te prey. In te presence 2

3 of vibrio infection te prey population is divided into two classes, namely susceptible tilapia, denoted by St, and infected tilapia, denoted by It. Terefore at time t te total tilapia population is X t = S t + I t. We assume tat only susceptible tilapia breed and tat, in te absence of infection, population growt is given by a logistic function wit carrying capacity. Bot susceptible and infected tilapia are subject to predation by te pelicans. Te pelicans preferentially eat infected tilapia because te tilapia become wea due to infection and rise to te surface of te sea for oxygen. As tey are relatively easy to catc, tese infected tilapia become more attractive to pelicans. Wit te above assumptions te basic equations of te model are ds dt = rs 1 S + I di dt = λsi cy I my + I γi, dy dt = δy 1 Y I + S Here te parameters are as follows: r λ γ p and c δ m λsi. py S my + S, te species growt rate of tilapia in te breeding subpopulation, te disease transmission coefficient, te per capita deat rate of infected prey, searc rate of te pelicans towards susceptible and infected tilapia, respectively, te per capita growt rate of te pelicans, a constant relating to te density dependent mortality of te predator population, a strictly positive constant. In order to reduce te number of parameters, we define τ = λt, = r λ, p 1 = p λ, γ 1 = γ λ, δ 1 = δ λ and c 1 = c λ. Ten coosing te new parameters and renaming τ as t, we find tat te system is ds dt = S 1 S + I SI p 1Y S my + S, di dt = SI c 1Y I my + I γ 1I, 2.1 dy dt = δ 1Y 1 Y I + S It is straigtforward to sow tat te solutions to tese equations are always positive.. 3

4 We assume tat all te parameters in te model are strictly positive and tat S0 0, I0 0 and Y 0 0. As 0 S my + S 1, 0 < p 1Y S my + S p 1Y. Hence it is natural to interpret tis item as zero wen S = Y = 0. Similarly it is natural to interpret te term c 1 Y I my + I as zero wen Y = I = 0. If I0 = S0 = 0 and Y 0 0, ten 2.1iii is interpreted as implying tat Y t = 0 for all t. It is straigtforward to sow tat te solution to tese equations are always positive and S0 > 0 implies tat St > 0 for all t, I0 > 0 implies tat It > 0 for all t and Y 0 > 0 and at least one of S0 and I0 are strictly positive implies tat Y t > 0 for all t. Leslie [13, 14] introduced te following matematical model for a predatorprey system were X is te prey and Y is te predator Y X dx dt = bxx pxy, dy dt = r 2 a 2Y Y. X is te Leslie-Gower term wic measures te loss in te predator population due to te relative scarcity of te prey. px corresponds to te per capita rate at wic predators consume prey and, r 2, b and a 2 are constants. Tus our model is based on te Leslie-Gower model. Tere are several previous papers concerning Leslie-Gower eco-epidemiological models wit ratio-dependent and prey-dependent functional response. Here we sall list some of tem and differences between our model and teirs. Te study of eco-epidemiology was started in 1999 by Cattopadyay and Arino [15] wo studied te following system of differential equations: ds = rs + I 1 S + I dt di = bsi γiy ci, dt dy dt = εγi + ηεγ 1S dy, bsi ηγ 1 SY, were γi and ηγ 1 S are te predator response functions, is te carrying capacity of te environment for prey and r, b, c and d are constants. Tis differs from our model in tat firstly bot infected and susceptible prey contribute to prey growt and secondly te predator equation is different. Haque and Cattopadyay [16] and later Jin and Haque [17] study an eco-epidemiological 4

5 model for te Salton Sea wit disease in te prey. Tis model differs from ours in tat te predator consumes only infected prey not bot types of prey and also te predator response function is not ratio-dependent. Also te disease transmission term is a generalisation of ours to λi p S q were p and q are positive integers and again te predator equation is different. Arino et al. [18] discuss a ratio-dependent predator-prey model wit disease in te prey. Tis as some similarities wit our model but also some differences. Firstly in te logistic term for te growt of te susceptible prey population only deats of susceptible prey are taen into account wereas in our model tere are deats of bot susceptible and infected prey. Secondly in te functional response terms for bot susceptible and infected prey te denominators in te model of Arino et al. are of te form using our notation my + S + I, wereas in our model tey are my +S and my +I respectively. Tirdly tere are differences in te predator growt equation wic as predator growt terms corresponding to consumption of susceptible and infected prey and a linear deat term. Kundu and Cattopadyay [19] study a model for a predator-prey system wit disease amongst te prey wic was similar to ours but te predator response to infected prey is simply a function of te number of infected prey and te predator growt equation is not a Leslie-Gower one. Greenalg and Haque [11] study a ratio-dependent predator-prey model wit disease in te prey but tis differs from te current model as follows: Firstly te growt function of te susceptible prey is purely logistic, depending only on te susceptible prey, wereas in te current model te infected prey also indirectly affects te susceptible prey growt rate. Secondly in [11] te predator preys only on infected prey wereas in te current model te predator preys on bot susceptible and infected prey. Tirdly te predator carrying capacity depends only on te infected prey wereas in te current model it depends on te total number of prey. Xaio and Cen [20] also study a ratio-dependent predator-prey model wit disease in te prey but in teir model te predators do not consume susceptible prey and te predator growt equation is different and is not of Leslie-Gower type. Pal and Samanta [21] study a predator-prey model incorporating a prey refuge wit disease in te prey. Te paper extends te model of Xaio and Cen [20]. In a later paper, Pal and Samanta [22] tey modify tis model in an arguably more realistic fasion by taing te prey disease transmission term to be frequency dependent asi S+I rater tan density dependent asi. Tus in te frequency dependent model te per capita disease contact rate is constant. Wang and Feng [23] also extend te model of Xaio and Cen [20] including stage structure into te predator population and a time delay due to te gestation of te predator into te system. However again in all tree models unlie our current model te predators do not consume infected prey and te predator growt equation is not te same as ours. Raman and Caravarty [24] discuss a predator-prey model wit disease in te prey in wic te functional response to susceptible and infected prey are 5

6 c 1SY a+s+iy and c 2IY a+s+iy respectively were a, c 1 and c 2 are constants. Tis is a modification of a model studied by Cosner et al. [25]. Tis model differs from ours in tat bot te per capita predation rates of susceptible and infected prey ave different functional forms and yet again te predator growt equation is not of Leslie-Gower type. Tere are many oter papers wic consider eco-epidemiological models wit disease in te prey or te predator, for example Venturino [26, 27], Haque and Venturino [4, 28], Cattopadyay and Bairagi [9], Xaio and Cen [29, 30], Cattopadyay, Srinivasu and Bairagi [12] and Muopadyay and Battacarya [31] amongst oters but we believe tat te models wic we ave surveyed above are te most relevant to our study. 3 Boundedness Results We ave tree results on te boundedness of te system 2.1. First of all, we consider te prey. Proposition 1 Te prey are always bounded above. Proof. If S0 = 0 ten te result is trivial. If S0 > 0 ten St > 0 for all t. On adding equations 2.1i and 2.1ii, we get ds dt + di dt S 1 S + I. It follows ten tat lim sup t St + It. Proposition 2 Te number of predators Y is always bounded above. Proof. If Y 0 = 0 te result is obvious. If Y 0 > 0, ten using equation 2.1iii we see tat dy dt < 0 if Y I + S > 1. Suppose tat lim sup t Y >. i.e. lim sup t Y > +ɛ for some ɛ > 0. Hence Y +ɛ > 0 in infinitely many disjoint intervals t 0, t 1, t 2, t 3, t 4, t 5,... and outside tese intervals Y +ɛ 0. As lim sup t I + S witout loss of generality we can assume tat in tese intervals +ɛ > I+S so dy dt < 0 in tese intervals. If one suc interval is t 2r, t 2r+1, r an integer, ten for t t 2r, t 2r+1 Y t Y t 2r = + ɛ. Hence lim sup t Y t +ɛ. As ɛ > 0 is arbitrary we deduce tat lim sup t Y t. Proposition 3 Te trajectories of system 2.1 are bounded. 6

7 Proof. Define te function l = S + I + Y and tae its time derivative along te solution of 2.1 dl dt = ds dt + di dt + dy dt. Now dl dt + ql = S S S + I p 1Y S my + S c 1Y I my + I γ 1 I + δ 1 Y δ 1Y 2 + qs + qi + qy, I + S S + I = q + S + q + δ 1 Y γ 1 qi S p 1Y S my + S c 1Y I my + I δ 1Y 2 I + S, were q is a positive constant. For q < γ 1 given ɛ > 0 tere exists t 0 suc tat for t t 0 dl dt + ql m + ɛ were m = q + + q + δ 1. Hence so d dt leqt m + ɛe qt, lt lt 0 e qt t0 + m + ɛ q Letting t, ten letting ɛ 0 independently of te initial conditions. 4 Te Equilibria lim sup lt m t q We now examine te equilibria of te system 2.1. teorem: 1 e qt t0. We ave te following Teorem 4 Te ecologically meaningful possible equilibria of system 2.1 are i Ē0 = 0, 0, 0, were all populations are extinct, wic always exists. ii Ē1 =, 0, 0 were tere is only susceptible prey wic always exists. iii Ē2 = Y 2, 0, Y 2 were Y 2 = p1 m+. In tis equilibrium tere are only susceptible prey and predators. Tis equilibrium always exists if > p1 m+. 7

8 iv Ē3 = γ 1, r1 γ1 +, 0. Tis equilibrium as only infected and susceptible prey and no predators. It is possible if > γ 1. v Up to two co-existence equilibria Ē4, Ē 5. Tese are given by Ēi = x i ȳ, x i ȳ, ȳ, for i = 4, 5 were x i is te it root of te cubic equation Here Define fx = a 0 x 3 + a 1 x 2 + a 2 x + a 3 = [ [ ]] in 0, min, m + p1 were a For p1 a 0 = + γ 1, a 1 = p 1 + c 1 2 γ 1, a 2 = 2 m 2 m p 1 m c 1 [m + ] γ 1 mm + γ 1 2, a 3 = m + m c 1 γ 1 m p 1m. m + ȳ = c 1 + γ 1 m + x i, for i = 4, 5. m + x i x i = 4 27 a a2 1a a3 1a a 1a 2 a 3 + a 2 3. m +, tere are no feasible co-existence equilibria. b For p1 < m + and m > c 1 + γ 1 m + p1m m+, te cubic equation 4.1 as exactly one real root in te given interval. c Suppose tat p1 < m + and m < c 1 + γ 1 m + p1m m+. For a 2 1 > 3a 0 a 2, a 1 < 0, a 2 > 0 and < 0, te cubic equation 4.1 as two real positive turning points at α 1, β 1 α 1 < β 1, te roots of 3a 0 x 2 + 2a 1 x + a 2 = 0. If [ α 1 < min, m + p ] 1, ten tere are two strictly positive real roots of 4.1 in te interval. If eiter i a 2 1 3a 0 a 2, ii a 1 0, iii a 2 0, iv > 0 or v a 2 1 > 3a 0 a 2, a 1 < 0 a 2 > 0, 0 and [ α 1 min, m + p ] 1, 8

9 ten tere are no strictly positive real roots of 4.1 in te interval. If a 2 1 > 3a 0 a 2, a 1 < 0, a 2 > 0, = 0 and [ α 1 < min, m + p ] 1 ten 4.1 as one strictly positive repeated real root in te interval. d Suppose tat p1 < m + and m = c 1 + γ 1 m + p1m m+. For a 2 1 > 3a 0 a 2, a 1 < 0, a 2 > 0 and < 0 if [ α 1 < min, m + p ] 1 ten tere [ is exactly one ] strictly positive real root of 4.1 in te interval 0, min, m + p1. If eiter i a 2 1 3a 0 a 2, ii a 1 0, iii a 2 0 or iv 0 ten tere are no strictly positive real roots of 4.1 in tis interval. Ecological Interpretation of Tese Conditions. p m+. Te condition for te existence of Ē 2 is > p1 m+, equivalently r > Tis means tat bot r, te per capita growt rate of te tilapia and, te density-dependent mortality constant are relatively large compared wit p, te searc rate of te pelicans toward infected tilapia. Te condition for te existence of Ē 3 is λ > γ. Tis means tat te product of te carrying capacity of te environment for prey and te disease transmission coefficient λ exceeds γ te per capita deat rate of infected prey. Provided tat > p1 m+, te condition in Teorem 4vb for te existence of one or more co-existence equilibria is p 1 λ > 1 c + γm m + r m so te weigted sum of te searc rate c of te pelicans towards infected tilapia and te per capita deat rate of infected tilapia γ is less tan te disease transmission coefficient λ multiplied by te carrying capacity of te environment for prey,, is less tan a term tat is related to ow strongly te first condition is satisfied. > p 1 m + Proof. From equations 2.1 we see tat 0,0,0 is always a solution. Moreover for any feasible solution 9

10 1 Eiter S = 0 or 1 S +I I p1y my +S = 0, 2 Eiter I = 0 or S c1y my +I γ 1 = 0, 3 Eiter Y = 0 or I + S = Y. i If S = 0 ten I = 0 and Y = 0. ii and iii If S 0 and I = 0, eiter ii Y = 0 wic implies tat S =, leading to te equilibrium Ē1 or iii Y 0 wic implies tat Hence so Y = = S p 1 m + + p 1Y my + S and S = Y. = Y + p 1 m +,, S = p 1 m + Tis equilibrium Ē2 will be feasible if m + > p 1.. iv If S 0, I 0 and Y = 0 ten S I = γ 1 I so S = γ 1 wic implies tat 1 S + I I = 0, I 1 + Tus we ave Ē3 wic is feasible if > γ 1. = 1 γ 1 I = γ 1 +. v Finally for a co-existence equilibrium we ave S 0, I 0, Y 0. Hence 1 S + I I p 1Y my + S = 0, 4.2 S c 1Y my + I γ 1 = 0, I + S = Y. Write x Y = I x Y = = c 1 Y my + x Y + γ 1, c 1 m + x + γ 1, = c 1 + γ 1 m + x m + x., 10

11 So Y = c 1 + γ 1 m + γ 1 x x m + x. 4.3 Now from equation 4.2 we deduce tat 1 Y x Y p 1 m + x = 0. Tus Y r1 m + x p 1 = m + x x + For a feasible equilibrium we need 0 < x < and x < also c 1 + γ 1 m + γ 1 x x m + x = r1 m + x p 1 m + x i.e.. m + p1 x + c 1 + γ 1 m + γ 1 x m + x x + + x m + x m + x p 1 = Expanding tis expression we can see tat it is te cubic 4.1 described earlier. a 0 x 3 + a 1 x 2 + a 2 x + a 3 = 0 a Clearly for p1 m + tere are no feasible co-existence equilibria. b For m > c 1 + γ 1 m + p1m r, a 1m+ 3 > 0. Using 4.4, f = c 1 + γ 1 m + γ 1 m + < 0. If x = m + p1 < deduce tat f m + p 1 note tat m + > p1, again using 4.4 we = c 1 + γ 1 m + γ 1 x p 1 x + < 0. [ ] Terefore f min, m + p1 < 0. Hence te cubic equation 4.1 [ [ as one or tree real roots in te interval 0, min, m + p1 ]]. But [ [ ] ] it as one real root in min, m + p1,. Terefore it as exactly [ [ ]] one real root in te interval 0, min, m + p1 in tis case. 11

12 c For p1 < m + and m < c 1 + γ 1 m + p1m r, a 1m+ 3 < 0 and as above [ ] f min, m + p1 < 0. Hence te cubic equation 4.1 as eiter zero or two positive real roots in te interval. Te derivative of fx given by 4.1 is gx = 3a 0 x 2 + 2a 1 x + a 2. Te turning points are te roots α 1, β 1 of gx = 0 and = fα 1 fβ 1 [32]. If a 2 1 > 3a 0 a 2, a 1 < 0, a 2 > 0 and < 0, te cubic equation 4.1 as two real turning points at α 1, β 1 were 0 α 1 < β 1. If [ α 1 < min, m + p ] 1, ten by considering te sape of fx we see tat tere are two strictly positive real roots of 4.1 in te interval. If a 2 1 > 3a 0 a 2, a 1 < 0, a 2 > 0, = 0 and [ α 1 < min, m + p ] 1, ten fx as one repeated strictly positive real root in te interval. If eiter a 2 1 3a 0 a 2, or a 1 0 or a 2 0 or > 0 ten fx as no strictly positive real roots in te interval. 4.1 Te Case p 1 = 0 We can gain furter insigt into wen wat number of equilibria are possible by considering te special case p 1 = 0. Wen p 1 = 0 we need 0 < x < for a feasible equilibrium. Te cubic 4.4 reduces to x = m + infeasible or x m + x + c 1 + γ 1 m + γ 1 x x + = Tis is a quadratic were lx = b 0 x 2 + b 1 x + b 2 = b 0 = γ 1 +, b 1 = c 1 + γ 1 m + γ 1 + m, b 2 = c 1 + γ 1 m m. i Note tat l > 0. Hence if c 1 + γ 1 m < m tere is exactly one root of lx = 0 in [0, ]. 12

13 ii If c 1 + γ 1 m > max[m, ] ten b 1 > 0, b 2 > 0, so lx = 0 as no real roots in [0, ]. iii To see tat it is possible to ave two feasible co-existence equilibria it is sufficient to sow tat it is possible to ave two positive real roots of lx = 0 in [0, ]. Coose γ 1 < and c 1 = γ 1 m + ɛ were ɛ > 0 is very small. Coose very large so tat b 1 = [m + ɛ] + γ 1 + m < 0. Ten if is large enoug and ɛ is small enoug b 1 < 0, b 2 > 0 and b 2 1 4b 0 b 2 > 0. So te quadratic lx = b 0 x 2 + b 1 x + b 2 as two positive real roots and is minimised at x = b 1 2b 0 > 0. But recalling tat is very large and ɛ is very small b 1 γ 1 2b 0 2 γ 1 + < as 2γ 1 + > γ 1. Terefore te quadratic as two real roots in [0, ]. Hence it is possible for tere to be eiter no, one or two co-existence equilibria. Tis concludes our equilibrium analysis of te predator-prey model wit disease. In te next section we sall loo at te stability of tese equilibria. 5 Stability At a general equilibrium S, I, Y te stability matrix or Jacobian of te system is 2r1S r1i I mp1y 2 my +S 2 S S p 1S 2 my +S 2 J = I S γ 1 c1my 2 c 1I 2 my +I 2 my +I 2 δ 1Y 2 I+S 2 δ 1Y 2 I+S 2 δ 1 1 2Y I+S. 5.1 Dynamics of te System Around Ē0 Te stability matrix is not well-defined at te equilibrium Ē0. To sow tat Ē0 is unstable it is sufficient to sow tat not all trajectories starting in a small ball of radius ɛ > 0 approac Ē0. Consider a trajectory wit Y 0 = 0 and S 0 > 0 ten Y t = 0 and St > 0 for all t. 13

14 Hence if S and I are small enoug. approac Ē0 wic is unstable. 1 ds S dt = 1 S + I I > 2 Hence S S 0 e t 2. So tis trajectory cannot 5.2 Dynamics of te System Around Ē1 Te stability matrix is also not well defined at te equilibrium Ē1. To sow tat Ē1 is unstable suppose tat Ē1 is locally asymptotically stable LAS. Now consider a trajectory wit Y 0 > 0 and eiter I 0 > 0, or S 0 > 0, ence eiter It > 0 for all t or St > 0 for all t. As tis trajectory approaces te equilibrium Ē1 dy dt δ 1Y. Hence if tis trajectory approaces Ē1 tere is a t 0 suc tat for t t 0 Also Y t 0 > 0 as Y 0 > 0. So dy dt δ 1Y 2. Y t e δ 1 t t 0 2 for t t 0 wic is a contradiction. Terefore Ē1 is unstable. 5.3 Dynamics of te System Around Ē2 At te equilibrium Ē2 one eigenvalue is J 22 = p1 m+ remaining eigenvalues satisfy te caracteristic equation r det 1 + p1m+2 m+ ω p12 2 m+ 2 = 0. δ 1 δ 1 ω Tis is equivalent to were and ω 2 + a 1 ω + a 2 = 0 a 1 = δ 1 + p 1m + 2 m + 2, a 2 = δ 1 Y 2 > 0. Hence tis equilibrium is stable if γ 1 c1 m. Te γ 1 + c 1 m + p 1 m + >

15 and Te equilibrium is unstable if eiter or δ 1 + > p 1m + 2 m > γ 1 + c 1 m + p 1 m p 1 m + 2 m + 2 > δ We ave already discussed te biological significance of te first of tese stability conditions 5.1. Te second condition can be rewritten as 2 p 1 m + > δ + r. So tis condition will be satisfied if p, te searc rate of te pelicans towards susceptible tilapia is ig, and, te density-dependent mortality rate of te pelicans is small compared wit te sum of te per capita birt rates of te pelicans and te tilapia. 5.4 Dynamics of te System Around Ē3 At te equilibrium Ē3 = S 3, I 3, 0 te Jacobian is J 3 = r1s3 + 1 S 3 p 1 I 3 0 c δ 1 One eigenvalue is clearly δ 1 > 0 so tis equilibrium is unstable. Te oter two eigenvalues ave negative real parts. 5.5 Dynamics of te System Around te Co-existence Equilibria Ē4 and Ē5 At te equilibria Ē4 and Ē5 = S, Ī, Ȳ te Jacobian matrix is J 4 = = r1 S p1ȳ + mȳ + S 2 Ī r1 S S 2 p1 S mȳ + S 2 c 1 Ȳ Ī c1ī2 mȳ +Ī2 mȳ δ +Ī2 1 δ 1 δ 1 a 11 + a 12 Ȳ ω a 11 S a 12 S Ī A ω Āx δ 1 δ 1 δ 1 ω

16 were a 11 = S, a 12 = p 1 S mȳ + S 2 and A = c 1 Ȳ Ī mȳ + Ī2. Te caracteristic equation associated wit te possible equilibria Ē4 and Ē5 of tis model is δ 1 ωa ω a 11 + a 12 Ȳ ω + δ1 Ax a 11 + S δ1 Ia 12 S +a 12 S δ 1 A ω + δ1 Ax a 11 + a 12 Ȳ ω δ 1 + ωa 11 + SI = 0. Tis can be expressed as a cubic equation were e 1 = δ 1 + a 11 a 12 Ȳ A, ω 3 + e 1 ω 2 + e 2 ω + e 3 = 0, 5.6 e 2 = δ 1 a 11 a 12 Ȳ A Aa 11 a 12 Ȳ + a 12 Sδ1 + δ 1 Ax + a 11 + SĪ, e 3 = δ 1 Aa 11 a 12 Ȳ δ 1 Ax a 11 + S + δ 1 Īa 12 S δ 1 A Sa 12 + δ 1 Ax a 11 a 12 Ȳ + δ 1 a 11 + SĪ. Te Rout-Hurwitz conditions for te caracteristic equation to ave only roots wit strictly negative real parts and ence te co-existence equilibrium to be LAS are e 1 > 0, e 3 > 0 and e 1 e 2 > e 3. If we consider te special case were p 1 = c 1 = 0 ten tis implies tat e 1 = δ 1 + a 11, e 2 = δ 1 a 11 + a 11 + SĪ, e 3 = δ 1 a 11 + SĪ, in tis case. So te co-existence equilibrium is LAS wen it exists. Note tat equation 4.1 and ence x does not depend on δ 1. Hence it is natural to express e 1, e 2 and e 3 in terms of δ 1 and see wat appens as δ 1 varies. In terms of δ 1 e 1 = δ 1 + ξ 1, were e 2 = δ 1 ξ 2 + ξ 3, e 3 = δ 1 ξ 4, ξ 1 = a 11 a 12 Ȳ A, ξ 2 = a 11 a 12 Ȳ S A 1 x, ξ 2 > ξ 1, ξ 3 = a 11 + SĪ Aa 11 a 12 Ȳ, ξ 4 = Aa 11 a 12 Ȳ A Sx + Ia 12 S 16 a 12 SA AȲ x a 12 + a 11 + SĪ.

17 If ξ 4 < 0, te equilibrium is unstable for all δ 1. Tis situation is rater complicated to analyse so we consider te different possibilities depending on te signs of ξ 1, ξ 2, ξ 3 and ξ 4. Examining te Rout- Hurwitz conditions e 1 > 0, e 3 > 0 and e 1 e 2 > e 3 in terms of δ 1, te condition e 1 > 0 is given by a straigt line of slope one intercept ξ 1, te condition e 3 > 0 by a straigt line troug te origin and in general apart from degenerate parameter values e 1 e 2 > e 3 by a parabola. Looing at tese conditions grapically it is straigtforward to sow tat: i If ξ 4 < 0 ten any co-existence equilibrium is unstable for all δ 1 ; ii If ξ 1 > 0, ξ 3 > 0, ξ 4 > 0 ten any co-existence equilibrium is stable in a range suc as 0, δ 0 1, δ 1 1, including always stable as a possibility; iii If ξ 1 0, ξ 2 > 0, ξ 4 > 0 ten a co-existence equilibrium is stable in a range suc as δ 0 1, ; iv If ξ 1 0, ξ 2 < 0, ξ 1 ξ 2 > ξ 3, ξ 4 > 0 ten a co-existence equilibrium is always unstable; v If ξ 1 0, ξ 2 < 0, ξ 1 ξ 2 < ξ 3, ξ 4 > 0 ten a co-existence equilibrium is eiter a always unstable, or b stable in a range suc as δ 1 0, δ 1 1 and unstable in 0, δ 1 0 δ 1 1, Very Large, m + > p 1 Tis case is covered by Teorem 4vb so tere is a unique co-existence equilibrium wit susceptible prey, infected prey and predators present. From 4.4 te limiting value of x as is te unique root of x m + x m + x p 1 c 1 + γ 1 m + γ 1 x m + x x [ [ in te interval 0, min, m + p1 ]]. Hence a 11 = S = c 1 + γ 1 m + γ 1 x m + x 0 as. So as te limiting values of ξ 1, ξ 2, ξ 3 and ξ 4 are ξ 1 a 12 Ȳ A < 0, ξ 2 [a 12 Ȳ S + A = [a 12Ī + A x ] ξ 3 SĪ + Aa 12Ȳ > 0, ξ 4 Ax S + Ī Sa SĪ. ] 1 x, < 0,

18 To determine te sign of te limiting value of ξ 4 as we note tat Ī > Ax But tis will always be true as if and only if > c 1x Ȳ mȳ + Ī2, if and only if Ȳ > c 1 x m + x 2. Ȳ = Ȳ x + x Ȳ > Ȳ x > c 1 m + x > c 1 x m + x 2 using 4.3. So in tis case lim ξ 4 > 0 and by cases iv and v above in general apart from degenerate parameter values te co-existence equilibrium is eiter always unstable, or stable in a range suc as δ1, 0 δ1 1 and unstable in 0, δ0 1 δ1, Te Inequality ξ 4 0 Motivated by te example discussed above we may as weter it is always te case tat ξ 4 0. However tis is not always te case as te following counterexample sows: Counterexample 5 It is possible to ave a co-existence equilibrium wit ξ 4 < 0. Proof. Coose p 1 = 0, m = c 1 +γ 1 m+ɛ, were ɛ is very small. Also coose very large in comparison wit m. Ten again we are in case vb of Teorem 4 and tere is a unique co-existence wit susceptible prey, infected prey and predators present. As a 3 is proportional to ɛ it is very small, so x is very small and we ave x 0, Ȳ c 1 + γ 1 m 5.7 m by 4.3. As a 12 = 0 ξ 4 = a 11 Ī A + S Ī Ax = r S 1 Ī A + S Ī Ax c 1 Ȳ 1 = SĪ = SĪ 1 mȳ + Ī2 c 1 Ȳ m + x 2,, + SĪ 1 + SĪ 1 c 1 x Ȳ mȳ + Ī2, c 1 x Ȳ m + x 2 Substituting in te approximate values of x and Ȳ from 5.7 we deduce tat ξ 4 SĪ r1 c c 1 + γ 1 mm Hence ξ 4 is negative if is large in comparison wit m. Terefore it is possible to ave a co-existence equilibrium wit ξ 4 <

19 6 Hopf Bifurcation Around te Possible Co-existing Equilibria Ē4 and Ē5 By considering te diagrams giving te stability of te co-existence equilibria for various values of ξ 1, ξ 2, ξ 3 and ξ 4 as δ 1 increases, we see tat at points were te stability canges eiter from stable to unstable or from unstable to stable it is te constraint e 1 e 2 > e 3 tat canges rater tan e 1 > 0 or e 3 > 0. To formalise tis as a teorem: Teorem 6 Recall tat δ 1 is te ratio of δ, te per capita growt rate of te pelicans, to λ, te prey disease transmission coefficient. Suppose tat ξ 4 0. As δ 1 increases te co-existence equilibrium can cange stability only as δ 1 passes troug a positive root of te quadratic equation in δ 1 e 1 δ 1 e 2 δ 1 e 3 δ 1 = Proof. To prove tis algebraically note tat if ξ 4 < 0 ten a co-existence equilibrium is always unstable so we can assume tat ξ 4 > 0 and te constraint e 3 > 0 is always satisfied. i If ξ 1 0 ten te constraint e 1 > 0 is always satisfied so as δ 1 increases te stability of te co-existence equilibrium can cange only as δ 1 passes troug one of te at most two possible roots of 6.1. ii If ξ 1 < 0 ten for δ 1 < ξ 1 te co-existence equilibrium is unstable. At δ 1 = ξ 1, e 1 δ 1 e 2 δ 1 = 0 < e 3 δ 1 so te co-existence equilibrium is still unstable. For δ 1 ξ 1 te coexistence equilibrium will again cange stability only as δ 1 passes troug one of te at most two possible positive roots of 6.1 for δ 1. We investigate te Hopf bifurcation for te system 2.1, taing δ 1 as te bifurcation parameter. Suppose tat ωδ 1 is an eigenvalue of te stability matrix at a co-existence equilibrium Ē4 or Ē5. Ten Hopf bifurcation will occur if tere exists a δ 1 = δ 1 suc tat i e 1 δ 1 e 2 δ 1 = e 3 δ 1 wit e 1 δ 1, e 2 δ 1, e 3 δ 1 > 0. ii d dδ 1 Reωδ 1 δ1= δ 1 0. Clearly 5.6 as two purely imaginary roots if and only if e 1 e 2 = e 3 for some value of δ 1 say δ 1 = δ 1 and e 2 δ 1 > 0. Tere are at most two positive values of δ 1 at wic we ave Hopf bifurcation. For δ 1 = δ 1 we ave ω 2 + e 2 ω + e 1 = 0 19

20 wic as tree roots ω 1 = i e 2, ω 2 = i e 2, ω 3 = e 1. Tus in a neigbourood of δ 1 te caracteristic equation 5.5 cannot ave real positive roots. In suc a neigbourood te roots ave te form ω 1 δ 1 = uδ 1 + ivδ 1, ω 2 δ 1 = uδ 1 ivδ 1, ω 3 δ 1 = φδ 1 were φ δ 1 > 0. To apply te Hopf Bifurcation Teorem we ave to verify te transversality condition d dω Re 0, = 1, 2. dt dδ 1 δ 1= δ 1 Substituting ω 1 δ 1 = uδ 1 + ivδ 1 into 5.6, equating real and imaginary parts and calculating te derivative, we get were Rδ 1 u δ 1 Sδ 1 v δ 1 + T δ 1 = 0, Sδ 1 u δ 1 + Rδ 1 v δ 1 + Uδ 1 = 0, Rδ 1 = 3uδ e 1 δ 1 uδ 1 + e 2 δ 1 3vδ 1 2, Sδ 1 = 6uδ 1 vδ 1 + 2e 1 δ 1 vδ 1, T δ 1 = u 2 δ 1 e 1δ 1 + e 2δ 1 uδ 1 + e 3δ 1 e 1δ 1 vδ 1 2, Uδ 1 = 2uδ 1 vδ 1 e 1δ 1 + e 2δ 1 vδ At δ 1 = δ 1, u δ 1 = 0 and v δ 1 0. If S = 0 ten e 1 δ 1 = 0 wic implies tat e 3 δ 1 = 0 wic is a contradiction. Hence S 0. If SU + RT 0 at δ 1 = δ 1, ten d dt Re dω dδ 1 Now from equation 6.2 at δ = δ 1, δ 1= δ 1 SU + RT = R 2 + S 2 SU + RT = 2e 2 e 1 e 2 + e 2 e 1 e 3, = 2e 2 d dδ 1 e 1 e 2 e δ1= δ 1 Suppose tat ξ 2 0. Write Qδ = e 1 e 2 e 3. Ten te equation Qδ 1 = 0 is te equation of a parabola. At δ 1, e 1 e 2 = e 3 and so e 2 δ 1 > 0. Te tangent to te parabola dq dδ 1 δ1= δ 1 will be eiter increasing or decreasing except at te vertex. Hence tere will be a Hopf bifurcation at δ 1 = δ 1, unless δ 1 is a repeated root of Qδ 1 = 0. 20

21 In te degenerate case ξ 2 = 0 we must ave ξ 3 > 0 and ξ 4 > 0 for a stable equilibrium to be possible. If ξ 3 ξ 4 ten Qδ 1 is a line wit non-zero slope and a root at δ 1 = δ 1 = ξ 1ξ 3 ξ 4 ξ 3. Hence if ξ 3, ξ 4 and ξ1ξ3 ξ 4 ξ 3 are all strictly positive, tere is a Hopf bifurcation. If ξ 3 = ξ 4 ten Qδ 1 is a constant, SU + RT = 0 and we cannot conclude anyting. Tis completes our matematical analysis of te general predator-prey disease system. In te next sections we sall loo at some special cases of tis system. First of all we sall examine te system wen tere is no infection in te prey and te predator interacts wit te susceptible prey only. Next we establis a condition for disease to die out in te full system wit susceptible and infected prey and predators. Ten we sall loo at te system wit disease in te prey but no predators. Following tis we loo at disease effects on predators in te absence of ealty prey. Next in Section 11 we sall verify some of our teoretical results using numerical simulation wit realistic parameter values. 7 Interaction of Healty Prey wit te Predator in te Absence of Infected Prey Putting I 0 in te system 2.1, we get ds dt = S dy dt = δ 1Y 1 S 1 Y. S p 1Y S my + S, As before te term p1y S my +S is interpreted as zero at S = Y = 0. Hence tere are tree equilibria, te origin Ẽ0 = 0, 0, Ẽ 1 =, 0 and Ẽ 2 = S, Ỹ were S = Ỹ and Ỹ = feasible only for > p1 m+. i Beaviour of te System around Ẽ0 p1 m+. Te last equilibrium is Te stability matrix is not well-defined at te trivial equilibrium Ẽ0. However, it is straigtforward to sow tat tis equilibrium is unstable as described earlier in te paper. ii Beaviour of te System around Ẽ1 Te stability matrix of te system is given by r1 p 1. 0 δ 1 21

22 Te caracteristic equation as one positive root so Ẽ1 will be unstable. iii Beaviour of te System around Ẽ2 Te stability matrix of te system is given by p1 r1 S m+ 2 p12 m+ 2 δ 1 δ 1. Te equilibrium Ẽ2 is LAS if δ 1 + > p1m+2 m+ and unstable if δ < p 1m+2 m+. Note tat 2 i If S0 = 0 ten St = Y t = 0 for all t so te system approaces Ẽ0. ii If S0 > 0 but Y 0 = 0 ten Y t = 0 for all t and St as t so te system approaces Ẽ1. It is tempting to conjecture: Conjecture 7 If S0 > 0 and Y 0 > 0 and δ 1 + > p1m+2 m+ 2 system approaces Ẽ2 for large times, i.e. St S and Y t Ỹ. ten te 8 Natural Disease Control We may wis to naturally eliminate disease from te system. We note tat te infected prey will be wiped out from te ecosystem if γ, te per capita deat rate of te infected prey, exceeds λ, were is te carrying capacity of te system. Under tis condition te wole infected prey population disappears and so te disease will also vanis. Proposition 8 If λ < γ ten disease will disappear from te ecosystem. Proof. Coose ɛ > 0 suc tat + ɛλ < γ. By te proof of Proposition 1 t 0 suc tat S + ɛ for t t 0. Ten for t t 0 = I λs, Hence I 0 as t. di dt cy my + I γ Iλ + ɛ γ. Te possible equilibria are Ē0 and Ē1, bot of wic are always unstable, Ē 2 wic exists if > p1 m+, Ē 4 and Ē5. If i > p 1 m +, 22

23 ii γ 1 + c 1 m + p 1 m + > and iii δ 1 + > p 1 m + 2 m + 2 ten we conjecture tat Ē2 is globally asymptotically stable GAS. Conjecture 9 If i-iii above are satisfied ten te equilibrium Ē2 is GAS. 9 Impact of Disease on Prey in te Absence of Predators Putting Y 0 in te system 2.1, we get 1 S + I ds dt = S di dt = SI γ 1I, SI, 9.1 were S denotes ealty prey and I denotes infected prey. Tere are tree ecologically meaningful equilibria for tis system: Ê 0 = 0, 0, Ê 1 =, 0 and Ê 3 = S, Ī, were S = γ 1 and Ī = r1 γ1 +. Ê 3 is a feasible equilibrium if and only if > γ 1. It is easily sown tat te equilibrium Ê0 is unstable and te equilibrium Ê 1 is LAS if < γ 1 and unstable if > γ 1. If = γ 1, Ê 1 is neutrally stable. Te stability matrix at Ê3 is given by S S S J =. Ī S γ1 Tis leads to te caracteristic equation ω 2 + Hω + H + SĪ = 0, were H = S. 9.2 Bot roots of te caracteristic equation 9.2 ave negative real parts so Ê3 = S, Ī is LAS wen it exists. It is tempting to conjecture tat watever te initial conditions te system tends to Ê1 if γ 1 and to Ê3 if > γ 1. Conjecture 10 Watever te initial conditions of te system 9.1 were te predators are absent tends to Ê1 if γ 1 and to Ê3 if > γ 1. 23

24 10 Disease Effects on Predators in te Absence of Healty Prey It is straigtforward to sow tat if tere are no ealty prey ten bot te infected prey and predators die out. Teorem 11 If tere are no ealty prey ten bot te infected prey and te predators die out so te system approaces E 0. Proof. On simplifying te system 2.1, by letting S 0, we get di dt = c 1Y I my + I γ 1I, dy dt = δ 1Y 1 Y. I Clearly It 0 as t. Hence given ɛ > 0 t 0 > 0 suc tat for t t 0, I ɛ. For t t 0 1 dy Y dt δ 1 1 Y ɛ so if Y 2ɛ, dy dt < δ 1. Hence tere is t 1 t 0 suc tat for t t 1, Y 2ɛ so I 0, Y 0 as t. 11 Numerical Wor We explored our teoretical result wit numerical simulations. We too most of our base parameter values from te study of an eco-epidemiological model of pelicans at ris in te Salton Sea by Cattopadyay et al. [12], r = 3.0 per day, = 45.0, λ = per day, γ = 0.24 per day, c = 0.05 per day. Additionally we tae δ = 0.09 per day and p = 0. For our first simulation we too parameters as above except m = 1.0 and = 0.2. Wit tis set of parameter values te equilibria are Ē 0 = 0.0,0.0,0.0, Ē 1 = 45.0,0.0,0.0, Ē 2 = 45.0,0.0,225.0 and Ē3 = 40.0,4.587,0.0. Note tat p 1 = 0, c 1 + γ 1 m > m and m >. So in 4.6 b 1 > 0 and b 2 > 0, so tere is no co-existence equilibrium. As te conditions 5.1 and 5.2 are satisfied Ē2 is LAS and te oter equilibria are unstable. We performed a variety of simulations and eac time watever te starting value te system tended to Ē 2 as time became large. Figure 1 sows a typical simulation wit starting values S, I, Y = 60,40,300. For te second simulation we too te same parameters except = 75, = 0.04 and m = 5.0. Wit tese parameter values we find tat Ē0 = 0.0,0.0,0.0, Ē 1 = 75.0,0.0,0.0, Ē2 = 75.0,0.0,1,875.0 and Ē3 = 40.0,30.435,0.0 are possible equilibria. For tese parameters 24

25 population size System approaces disease-free equilibrium susceptible infected predator time days Figure 1: Simulation wit r = 3 per day, =45, λ = per day, γ = 0.24 per day, c = 0.05 per day, δ = 0.09 per day, = 0.2, p = 0 and m = 1.0. Starting values S = 60, I = 40, Y = 300. System tends to Ē2. 25

26 population size System approaces co-existence equilibrium susceptible infected time days Figure 2: Simulation wit parameter values as in Figure 1 except tat = 75, = Starting values S = 50, I = 15, Y = 1,400. System tends to unique co-existence equilibrium Ē4. Susceptible and infected prey. γ 1 + c 1 m + p 1 m + < so te above four equilibria including Ē2 are all unstable. p 1 = 0 and c 1 + γ 1 m < m so tere is a unique co-existence equilibrium. We calculate te unique root of equation 4.1 to be approximately x = in tis case and te corresponding co-existence equilibrium to be S, Ī, Ȳ = 41.7,29.0,1,766.0 woring only to tree significant figures. We performed simulations wit a range of starting values and found tat watever te initial value te system seemed to tend to te unique co-existence equilibrium. Figures 2 and 3 sow a typical simulation wit starting values S, I, Y = 50,15,1,400. We ave presented te simulation on two graps, te first representing te tilipia and te second te pelicans. To sow tat tere could be two co-existence equilibria we too te parameter values to be te same as in Figure 1 except tat c = per day, = 50 and m = 5.0. In tis situation p =0, γ 1 <, c 1 +γ 1 m > m and b 1 < 0. Also te quadratic lx given by 4.6 as two real roots, x 1 = and x 2 = to four significant figures in [0, ]. Tere are two corresponding co-existence equilibria Ē4 = , , and Ē5 = , , To examine te existence of limit cycles arising by Hopf bifurcation we too 26

27 population size System approaces co-existence equilibrium predator time days Figure 3: Simulation wit parameter values as in Figure 1 except tat = 75, = Starting values S = 50, I = 15, Y = 1,400. System tends to unique co-existence equilibrium Ē4. Predators. 27

28 population size Existence of Hopf Bifurcation: Limit cycles S and I susceptible infected time days Figure 4: Simulation wit parameter values and initial values as detailed in te text δ = 0.09/day sowing te existence of stable limit cycles about te unique co-existence equilibrium Ē4. Susceptible and infected prey. r = 3.0/day, = 2,000,000, λ = 0.006/day, γ = 0.24/day, p = 0, c = 0.05/day, = 0.04 and m = 5.0. Again we ave p 1 = 0 and c 1 + γ 1 m < m so tere is a unique co-existence equilibrium wic is independent of δ. Wit tese parameters we found ξ 1 = , ξ 2 = , ξ 3 = 20, and ξ 4 = 20, Examining te Rout-Hurwitz conditions we find tat we expect te co-existence equilibrium to be unstable for δ [0, δ 0 and stable for δ > δ 0 were δ 0 = to eigt significant figures. Numerical simulations confirmed tese results and sowed tat in te region [0, δ 0 stable limit cycles existed. Figures 4 and 5 sow te stable limit cycles for δ = 0.09/day in te susceptible and infected prey Figure 4 and te predator Figure 5 and Figures 6 and 7 wit δ = 100/day sow tat te numbers of susceptible and infected prey Figure 6 and te number of predators Figure 7 tend to teir steady-state equilibrium values Ē4 = , , 13, In all of tese figures te initial values were S0 = 17.28, I0 = and Y 0 = 13, For te parameter values for wic we were able to sow te existence of Hopf bifurcation te approac of te solution to te equilibrium value was very slow compared wit te relatively fast transient oscillations. Hence it is not possible to sow bot on te same timescale and Figures 6 and 7 sow te envelope of te solutions over a very long time wic clearly sows tat wit tis value of 28

29 population size #10 4 Existence of Hopf Bifurcation: Limit cycles Y predator time days Figure 5: Simulation wit parameter values and initial values as detailed in te text δ = 0.09/day sowing te existence of stable limit cycles about te unique co-existence equilibrium Ē4. Predators. 29

30 population size Existence of Hopf Bifurcation: Stability S and I susceptible infected time days #10 5 Figure 6: Simulation wit parameter values and initial values as detailed in te text δ = 100.0/day sowing te system approaces te unique co-existence equilibrium Ē 4. Susceptible and infected prey. 30

31 population size #10 4 Existence of Hopf Bifurcation: Stability Y predator time days #10 5 Figure 7: Simulation wit parameter values and initial values as detailed in te text δ = 100.0/day sowing te system approaces te unique co-existence equilibrium Ē 4. Predators. 31

32 0.12 Bifurcation Diagram of I* against c I * c /day Figure 8: Bifurcation diagram of I against c. Oter parameters are as detailed in te text. A solid line indicates stability, a dased line instability. δ te solutions approac te unique co-existence equilibrium value. To investigate te bifurcation structure and stability wen multiple endemic equilibria exist we too c as a bifurcation parameter. We really need a four dimensional bifurcation diagram but to understand te situation in a simple way we plot only te bifurcation diagram of I against c. Hence te equilibria Ē 0, Ē1 and Ē2 are all represented by I = 0 on tis bifurcation diagram. We too as parameters r = 3.0/day, = 45, λ = 0.006/day, γ = 0.24/day, p =0, δ =0.09/day, = 50.0 and m =5.0. Wit tese parameters te unstable equilibrium Ē3 is represented by a constant line at I = Te susceptible prey and predator equilibrium Ē2 is stable for c < 0.15/day and unstable if c > 0.15/day. For c < 0.15/day tere is a unique co-existence equilibrium wic is stable. At c = 0.15/day an additional stable co-existence equilibrium bifurcates in a forward direction from Ē2. For 0.15/day < c < c 0 were c 0 is approximately /day to eleven significant figures tere are two co-existence equilibria but at c = c 0 tey coalesce and disappear and for c > c 0 tere do not appear to be any co-existence equilibria. Figure 8 sows te bifurcation diagram in te region /day < c < /day. 32

33 12 Summary and Discussion In tis paper we ave proposed and analysed a tree-species eco-epidemiological model for te pelican-tilapia system in te Salton Sea. We modified te ratiodependent model of Greenalg and Haque [11] and considered tat te susceptible and infected populations of tilapia are exposed to te predator to varying degrees for a more biologically realistic model. We sowed tat te system is bounded and found six equilibria. Te first Ē0 was were all populations are extinct wic is always unstable. Te second Ē1 was were tere is only susceptible prey wic is also unstable. Te tird Ē2 was were tere is no infected prey wic exists if > p1 m+. Conditions 5.1 to 5.4 for stability and instability of tis equilibrium were derived. Te fourt equilibrium Ē3 was were tere are no predators. Tis is possible if > γ 1 and is unstable. Te fift and sixt equilibria Ē4 and Ē5 were co-existence equilibria were bot susceptible and infected prey co-existed wit predators. We sowed tat for certain parameter values tere could be eiter zero, one or two co-existence equilibria. Numerical simulations were used to confirm te analytical results. In addition we carried out teoretical analysis of Hopf bifurcation around te co-existence equilibria. Next we moved on to some important special cases of te model were eiter te susceptible or infected prey or te predator was not present. If infection was not present in te prey ten tere were found to be tree equilibria. Te first, Ẽ 0, is trivial wit neiter susceptible prey or predators present and tis is found to be unstable. Tere was also an equilibrium Ẽ1 wit te predator absent, wic was also found to be unstable. Tere was a unique equilibrium Ẽ 2 wit bot predator and susceptible prey present, and conditions for stability of tis equilibrium were found. If te predators are not present ten tere are tree equilibria present, Ê 0, wit neiter infected nor susceptible prey present, wic is always unstable, Ê 1 wit only susceptible prey present, wic is LAS if < γ 1 and unstable if > γ 1. If > γ 1 tere is an additional equilibrium Ê3 wit bot susceptible and infected prey present wic is always LAS. For te model wit susceptible prey not present we sowed tat bot te infected prey and te predators became extinct. Te results are more interesting tan tose of te original model by Greenalg and Haque [11] because in our current model te predator can consume susceptible prey as well as infected prey and bot infected and susceptible prey can contribute to te carrying capacity. Tus our model is more general and realistic tan te original model studied in [11]. As te model is more complex tan te original model te potential beaviour is correspondingly more ric, diverse and interesting. For example an extra potential equilibrium Ē3 were tere are no predators just susceptible and infected prey is possible in our model but not possible in te model studied in [11]. Additionally wilst te model of [11] always ad a unique co-existence equilibrium in te current model tere can be two co-existence equilibria simultaneously. We ave already discussed several oter models of predator-prey systems 33

34 wit disease in te prey and pointed out te differences between tem and our model. In particular none of tese models uses a Leslie-Gower type equation for te predator growt. Cattopadyay and Arino [15] sow tat in teir model te prey population ultimately tends to its carrying capacity and tere are four equilibria: one were bot te prey and te predator population ave died out lie Ē0, one were te prey population is at its carrying capacity and bot te disease and te predators ave died out lie Ē1, one wit bot susceptible and infected prey and no predators lie Ē3 and a unique co-existence equilibrium lie Ē4 and Ē 5. Tey discuss stability of te equilibria and sow tat limit cycles can arise from te stable co-existence equilibrium by Hopf bifurcation and discuss conditions for subcriticality and supercriticality. Te models discussed by Xaio and Cen [20], Haque and Cattopadyay [16], Jin and Haque [17] and Pal and Samanta [21, 22] ave only te four possible equilibria discussed above. Wang and Feng [23] discuss only te equilibrium wic corresponds to Ē1 and te unique co-existence equilibrium: tey do not establis weter oter equilibria exist. All of tese papers discuss conditions of stability of te equilibria and sow tat limit cycles can arise from te unique co-existence equilibrium by Hopf bifurcation. Te models of Arino et al. [18] and Kundu and Cattopadyay [19] ave all of te above equilibria and additionally an equilibrium wit no infected prey only susceptible prey and predators. So also as te model of Raman and Caravarty [24]. Arino et al. give only a set of sufficient conditions for te co-existence equilibrium to exist and be unique, tey do not discuss parameter values wic do not satisfy tis sufficient set of conditions. In te model of Kundu and Cattopadyay [19] te general existence and uniqueness of te co-existence equilibrium is not discussed. Moreover Raman and Caravarty give two separate conditions for te uniqueness of eac of i te equilibrium wit only susceptible prey and predators and ii te co-existence equilibrium. Tey do not rule out multiple equilibria in eiter case if tese conditions are not satisfied but on te oter and tey do not sow explicitly tat multiple equilibria of eiter of tese types is possible. Tey also discuss stability of possible equilibria and weter limit cycles can arise from te co-existence equilibrium by Hopf bifurcation. Tus our model differs from most of te oter models surveyed above in tat te only one wit a Leslie-Gower predator growt equation is Greenalg and Haque [11]. Our results differ from te results of most of tese models in tat in te majority of tem te equilibrium corresponding to Ē2 does not exist and te co-existence equilibrium Ē4 or Ē5 is unique. Te model of Raman and Caravarty [24] does not eiter sow uniqueness of te co-existence equilibrium or demonstrate non-uniqueness. Tus our model is te first of tese to explicitly demonstrate non-uniqueness of te co-existence equilibrium. Acnowledgements We are grateful to te referees for teir elpful comments. We are also grateful to Sultan Qaboos University, Oman for financial support troug a 34