International Journal of Advanced Scientific and Technical Research Available online on http://www.rspublication.com/ijst/index.html

Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ An Eco-epidemiological Mathematical model for Predator-Prey systemwith disease infection in prey population

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Alfred Hugo, 2 Goteti V. R. L. Sarma and 3 Nanduri Lakshmi 1 , 2 Department of Mathematics, University of Dodoma, Dodoma, Tanzania 3

Department of Basic Sciences and Humanities, Pragati Engineering College, Surampalem, E.G.Dt., A.P. India

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Abstract: This paper examines an eco-epidemiological model for predator prey system with disease in prey. A model is proposed and analyzed qualitatively using the stability theory of the differential equations. A local study of the model is performed around the disease-free equilibrium while global stability is analyzes using Lyapunov function to estimate the effect of incorporated parameters for predator logistic growth rate and disease infection in prey and species coexistence. The results obtained in numerical simulations are analytically justified to view the reality of the model.

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Key words: Predator-prey system, Eco-epidemiology, disease infection.

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Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ 1

Introduction

Model Formulation

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2

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Eco-epidemiology is a new branch in mathematical biology which considers both ecological and epidemiological issues simultaneously. The first breakthrough in this modern of mathematical ecology was done by Lotka (1924) for a predator-prey competing species. On the other hand, most of the models for the transmission of infectious diseases originated from the classic work of Kermack and McKendrick (1927). Thees models are routinely used now days to understand the spread of the infectious diseases with the goal of determine vaccination policies possibly to eradicate them (Venturino 1994). Bakare et al 2012 analyzed the method of eradicating or reducing the intensity of the disease spread in prey predator ecosystem, by applying vaccination strategies with herd immunity and the vaccination strategies of infectious diseases that can be controlled when there is availability of effective and cheap vaccination. Venturino (2002) considered epidemic aspects in predator-prey models with disease in the predator. While Hethcote et al., (2000), modified a predator-prey model with an SIS disease in the prey where infected prey was more vulnerable to predation, while Mukherjee (2003) studied stability analysis of a stochastic model for prey-predator system with disease in the Prey. Mathematical models are essential tools in order to understand the mechanisms responsible for persistence or extinction of species in natural systems. In ecological models, persistence is in general desired. By contrast, investigations in epidemic models usually aiming at finding the mechanisms that leads to extinction of the parasites or infections (Liu, et al., 2008). In this paper we present an eco-epidemiological model for predator prey system with disease infection in prey to analyze the effect of growth rate of predator and disease infection in prey population.

A mathematical model is proposed and analyzed to study the functional response analysis of the predator toward the susceptible prey as well as infected prey. This dynamics is assumed to follow Michaelis-Menten kinetics Holling type-II predation function (Mukhopadhyay and Bhattacharyya, 2009). The model consists of prey population density denoted by N (t) = S(t) + I(t) and the predator population density denoted by Y (t). In formulating mathematical model we consider the following assumptions (i) In the absence of disease, the prey population grows logistically with intrinsic growth rate r1 and environmental carrying capacity k (ii) With availability of susceptible prey, predator population growth exponentially with intrinsic growth rate r2 . (iii) In the presence of disease, the prey consists of two sub-classes, namely, the susceptible prey S(t) and the infected prey I(t) (iv) The susceptible prey and predator can reproduce. Logistic law is used to model the birth process with the assumption that births should always be positive. The infected

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______________________________ prey is removed with natural death rate e1 , death cause by disease a1 or by predation. However, the infected prey population I contribute with S to population growth towards the carrying capacity k (v) The predator population suffer loss due to death at a constant rate e2 . The predation functional response of the predator towards susceptible as well as infected prey are assumed to follow MichaelisMenten kinetics and is modelled using a Holling typeII functional form with predation coefficient p1 , p2 and half-saturation constant m. Consumed prey is converted into predator with efficiency q

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vi) It is assumed that the disease is spread among the prey population only. Susceptible prey becomes infected when it comes in contact with the infected prey and this contact process is assumed to follow the simple mass action kinetics with β as the rate of conversion Regarding the model assumptions, we formulate a mathematical model that governed by the following system of the differential equations

  dS S+I p1 SY = r1 S 1 − − β SI − dt k m+S (1)

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dI p2 IY = β SI − − (a1 + e1 )I dt m+I dY qp1 SY p2 IY = r2 Y + + − e2 Y dt m+S m+I with the initial conditions:S(0) = S0 , I(0) = I0 , Y (0) = Y0 , p1 , p2 > 0 and 0 < q ≤ 1.

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Model analysis

We qualitatively analyze the model equations (1) to get insights into its dynamical features which will give better understanding of the effect of incorporating exponential growth rate of predator population.

3.1

Boundedness of the model

The boundedness of the system equations (1) implies that the system is biologically valid and well behaved. Theorem 3.1 All solutions of the system (1) are uniformly bounded

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Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ Proof 3.2 Assume W denote the total populations in the specific model, that is W =S+I +Y

(2)

this gives dW dS dI dY = + + dt dt dt dt Now, substituting the model equations (1) into (3)and simplify we get

(3)

dW ≤ r1 S − (a1 + e1 )I − (e2 − r2 )Y dt on simplifying we get dW ˆ 1 + 1) − hW ≤ k(r dt

(4)

where kˆ = max{S(0), k} and h =min{1, (a1 + e1 ), (e2 − r2 )}. The equation (4) can be written as dW ˆ 1 + 1). − hW ≤ k(r dt

(5)

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Solving (5) and substituting the initial conditions we get

as t → ∞ we have

kˆ (r1 + 1)(1 − e−ht ) h

(6)

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W ≤

kˆ (r1 + 1) h which is implies that the solution is bounded for W ≤

0≤W ≤

kˆ (r1 + 1). h

Therefore, all solutions of the model (1) in <3+ are confined in the region ( ) ˆ k Γ = (S, I, Y ) ∈ <3+ W ≤ (r1 + 1) + ε . h

3.2

Positivity of solutions

For model (1) to be epidemiologically meaningful and well posed, we need to prove that all solutions of system with positive initial data will remain positive for all times. This will be established by the following theorem Theorem 3.3 Let S(0) > 0, I(0) > 0, Y (0) > 0 this implies that S(t), I(t) and Y (t) of system (1) are all positive for ∀t ≥ 0.

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Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ Proof 3.4 To prove theorem (3.3), we use all equations of the model (1). From the 1st equation, we obtain the inequality expression   dS S ≤ r1 S 1 − (7) dt k which gives S≤

e−r1 t

kS(0) {k − S(0)} + S(0)

(8)

As t → ∞ we obtain 0 ≤ S ≤ k. Hence all feasible solution of system (1) is feasible in region Γ = {S, I, Y } . Similar proofs can be established for the positivity of the other solutions.

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Equilibria and stability analysis

4.1

Equilibrium points

The system of differential equation (1)has the following equilibrium points by setting

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dS dI dY = = =0 dt dt dt The model equations (1) has the following equilibrium points (i) A trial equilibrium point ET (S ∗ , I ∗ , Y ∗ ) = (0, 0, 0),

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(ii) The axial equilibrium point EA (S ∗ , I ∗ , Y ∗ ) = (k, 0, 0), (iii) The boundary equilibrium point where predator population dies out,   (βk−a −e )r a +e 1 1 1 that is Y ∗ = 0 is E(S ∗ , I ∗ , Y ∗ ) = 1 β 1 , (r1 +βk)β ,0 (iii) The equilibrium point where the disease eventually disappears from prey population, that is I ∗ = 0 is   (e2 −r2 )m (r1 mq(r2 m−e2 m+kr2 +kqp1 −ke2 ) ∗ ∗ ∗ E(S , I , Y ) = (r2 +qp1 −e2 , 0, ) (r2 +qp1 −e2 )2 k (iv) Endemic equilibrium point of the model equation (1) are Y∗ =−

(β S − a1 − e1 ) (−r1 m2 − β km2 − mβ kS − r1 km − r1 kS + r1 S 2 ) r1 p2 m + r1 p2 S + β kp2 m + β kp2 S + p1 kβ S − p1 ka1 − p1 ke1

−r1 p2 km − r1 p2 kS + r1 p2 Sm + r1 p2 S 2 + p1 mkβ S − p1 mka1 − p1 mke1 I =− r1 p2 m + r1 p2 S + β kp2 m + β kp2 S + p1 kβ S − p1 ka1 − p1 ke1 ∗ and S is the positive root for the equation ∗

r2 r1 m2 + r2 β km2 + r2 mβ kS + r2 r1 km + AA + e2 r1 S 2 =0 −r1 m2 − β km2 − mβ kS − r1 km − r1 kS + r1 S 2

(9)

where AA = r2 r1 kS − r2 r1 S 2 + r1 p2 km + r1 p2 kS − r1 p2 Sm − r1 p2 S 2 − p1 mkβ S + p1 mka1 + p1 mke1 − e2 r1 m2 − e2 β km2 − e2 mβ kS − e2 r1 km − e2 r1 kS

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Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ 4.2

Local stability analysis

Here, we study the local stability and existence criteria of the different equilibrium points by computing the Jacobian matrix of the system of equations (1). From the model equation (1) we form jacobian matrix as   p1 Y p1 SY p1 S aa − β L − m+S + (m+S) − r1kS − β S − m+S 2    p2 Y p2 LY p2 L  βL β S − m+L + (m+L) − m+L J = (10) 2 − a1 − e1    qp1 Y qp1 SY p2 Y p2 LY − (m+S) − (m+L) r2 + bb 2 2 m+S m+L where aa = r1 1 −

S+L k




−

r1 S k

and bb =

qp1 S m+S

+

p2 L m+L

− e2

Now, we check the stability around the disease free equilibrium points as follows:The eigenvalues for trivial equilibrium point ET (S ∗ , I ∗ , Y ∗ ) = (0, 0, 0) are given by λ1 = r1 , λ2 = −a1 − e1 and λ3 = r2 − e2 then, ET (S ∗ , I ∗ , Y ∗ ) = (0, 0, 0) is unstable since λ1 > 0.

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The axial equilibrium EA (S ∗ , I ∗ , Y ∗ ) = (k, 0, 0) will be stable if all its eigenvalues λ < 0 1 , and will exist if that is λ1 = −r1 , λ2 = −(a1 + e1 − βk) and λ3 = − (e2 −r2 )(m+k)−kqp m+k m + k 6= 0.

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With the absence of the disease in prey population, the model (1) is stable due to existence of all negative eigenvalues. A That is λ1 = −1/2 kqp1 (r2 +qp 1 −e2 ) λ2 = −1/2(p21 r1 q 2 k 3 − 2p1 r1 qk 3 e2 − 2r1 qp1 r2 m + 2r1 qp1 e2 m − r1 qp1 kr2 − B λ3 = −(−2βmr2 e2 + 2a1p1 qr2 − 2a1p1 qe2 + 2e1 p1 qr2 + βmr22 + βme22 + a1p21 q 2 − 2a1r2 e2 + e1 p21 q 2 − 2e1 r2 e2 + a1r22 + a1e22 + βmp1 qr2 − 2e1 p1 qe2 + p2 r1 qkr2 m − p2 r1 qke2 m + p2 r1 qk 2 r2 + p2 r1 q 2 k 2 p1 − p2 r1 qk 2 e2 − βmp1 qe2 + e1 r22 + e1 e22 )/(2p1 qr2 + p21 q 2 − 2p1 qe2 + r22 − 2r2 e2 + e22 ) where A = p1 2 r1 q 2 k 3 − 2 p1 r1 qk 3 e2 − 2 r1 qp1 r2 m + 2 r1 qp1 e2 m − r1 qp1 kr2 − r1 q 2 p1 2 k + r1 qp1 ke2 + p1 r1 qk 2 r2 m − p1 r1 qk 2 e2 m +√2 p1 r1 qk 3 r2 + r1 k 2 e2 2 m + r1 k 3 r2 2 + r1 k 3 e2 2 − 2 r1 k 3 r2 e2 + r1 k 2 r2 2 m − 2 r1 k 2 r2 e2 m − A1 and A1 = r1 2 q 2 p1 2 k 2 e2 2 + 6 p1 3 r1 2 q 3 k 3 e2 m + r1 2 k 4 e2 4 m2 − 16 r2 3 k 4 qp1 r1 e2 + 2 r1 2 k 5 e2 4 m − 2 r1 2 qp1 k 4 r2 3 + p1 4 r1 2 q 4 k 6 + r1 2 k 4 r2 4 m2 − 8 r1 2 k 5 e2 3 mr2 +r1 2 k 6 r2 4 +12 r1 2 k 5 e2 2 mr2 2 +4 p1 r1 2 qk 6 r2 3 +4 e2 4 k 4 qp1 r1 −6 r1 2 q 2 p1 2 k 4 r2 2 +4 r1 2 q 2 p1 2 r2 2 m2 − 18 p1 r1 2 qk 5 e2 r2 2 m+6 p1 3 r1 2 q 3 k 4 e2 −2 p1 2 r1 2 q 2 k 4 r2 m2 e2 +18 p1 r1 2 qk 5 e2 2 r2 m+6 p1 r1 2 qk 4 r2 m2 e2 2 − 6 p1 r1 2 qk 4 r2 2 m2 e2 −8 r1 2 q 2 p1 2 r2 mke2 +8 r1 2 q 2 p1 2 r2 m2 k 2 e2 −12 r1 2 qp1 r2 m2 k 2 e2 2 −6 r1 2 qp1 k 4 r2 e2 2 + 18 r1 2 qp1 r2 2 mk 3 e2 −18 r1 2 qp1 r2 mk 3 e2 2 +2 p1 3 r1 2 q 3 k 5 r2 m+12 p1 2 r1 2 q 2 k 4 e2 r2 +12 r1 2 qp1 r2 2 m2 k 2 e2 − 12 p1 2 r1 2 q 2 k 3 e2 2 m + 6 r1 2 qp1 k 4 r2 2 e2 + r1 2 q 4 p1 4 k 2 − 6 p1 r1 2 qk 5 e2 3 m + 24 k 3 q 2 p1 2 r1 r2 e2 2 m − 8 k 3 q 3 p1 3 r1 r2 me2 − 24 k 3 q 2 p1 2 r1 r2 2 me2 − 12 p1 r1 2 qk 6 e2 r2 2 +

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Issue 4 volume 5, Sep.-Oct. 2014 ISSN 2249-9954

______________________________ 24 r2 2 k 3 qp1 r1 e2 2 m−12 r1 2 q 2 p1 2 r2 2 mk 3 −4 r1 2 qp1 r2 3 m2 k 2 −4 r1 2 q 2 p1 2 e2 2 m2 k 2 +4 r1 2 q 2 p1 2 e2 2 mk− 4 r1 2 q 3 p1 3 e2 mk−24 k 4 q 3 p1 3 r1 r2 e2 +6 r1 2 qp1 e2 3 mk 3 +4 r1 2 qp1 e2 3 m2 k 2 +4 k 3 q 3 p1 3 r1 e2 2 m+ 4 k 3 q 3 p1 3 r1 r2 2 m+8 k 3 q 2 p1 2 r1 r2 3 m−8 k 3 q 2 p1 2 r1 e2 3 m−6 p1 3 r1 2 q 3 k 3 r2 m+36 k 4 q 2 p1 2 r1 r2 e2 2 − 36 k 4 q 2 p1 2 r1 r2 2 e2 −16 r2 k 4 qp1 r1 e2 3 +4 r2 4 k 3 qp1 r1 m+4 p1 3 r1 2 q 3 k 6 r2 −16 r2 3 k 3 qp1 r1 me2 + 6 p1 2 r1 2 q 2 k 5 r2 2 m+6 p1 2 r1 2 q 2 k 5 e2 2 m+r1 2 q 2 p1 2 k 2 r2 2 +6 p1 2 r1 2 q 2 k 6 r2 2 −12 p1 2 r1 2 q 2 k 6 r2 e2 − 2 p1 3 r1 2 q 3 k 5 e2 m + 6 p1 2 r1 2 q 2 k 6 e2 2

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B = r1 q 2 p21 k + r1 qp1 ke2 + p1 r1 qk 2 r2 m − p1 r1 qk 2 e2 m + 2p1 r1 qk 3 r2 + r1 k 2 e22 m + r1 k 3 r22 + r1 k 3 e22 − 2r1 k 3 r2 e2 + r1 k 2 r22 m − 2r1 k 2 r2 e2 m + (−12p21 r12 q 2 k 5 r2 e2 m +24p21 r12 q 2 k 3 e2 r2 m−2r12 qp1 k 4 r23 −6r12 qp1 k 4 r2 e22 +6r12 qp1 k 4 r22 e2 −12p21 r12 q 2 k 3 e22 m+12p21 r12 q 2 k 4 e2 r2 − 6p21 r12 q 2 k 4 e22 − 2r12 q 3 p31 k 2 e2 − 6p1 r12 qk 5 e32 m − 12p1 r12 qk 6 e2 r22 − 4p1 r12 qk 6 e32 + 12p1 r12 qk 6 e22 r2 − 18p1 r12 qk 5 e2 r22 m + r12 q 2 p21 k 2 e22 + 2r12 qp1 k 4 e32 +18p1 r12 qk 5 e22 r2 m+4r12 q 2 p21 r22 m2 −6r12 q 2 p21 k 4 r22 −2p21 r12 q 2 k 4 r2 m2 e2 +6p1 r12 qk 4 r2 m2 e22 +6p1 r12 qk 5 r23 m+ p21 r12 q 2 k 4 r22 m2 −6p1 r12 qk 4 r22 m2 e2 +2p1 r12 qk 4 r23 m2 +p21 r12 q 2 k 4 e22 m2 −2p1 r12 qk 4 e32 m2 −8r12 q 2 p21 r2 m2 e2 + 4r12 q 2 p21 r22 mk + 4r12 q 3 p31 r2 mk − 8r12 q 2 p21 r2 mke2 + 4e42 k 4 qp1 r1 + 4e42 k 3 qp1 r1 m − 4r12 q 2 p21 r22 m2 k 2 + 8r12 q 2 p21 r2 m2 k 2 e2 −12r12 q 2 p21 r22 mk 3 −12r12 qp1 r2 m2 k 2 e22 +4p1 r12 qk 6 r23 −6r12 qp1 r23 mk 3 −18r12 qp1 r2 mk 3 e22 + 18r12 qp1 r22 mk 3 e2 −4r12 qp1 r23 m2 k 2 +r12 k 4 e42 m2 +2r12 k 5 e42 m+6r12 k 6 r22 e22 −4r12 k 6 r23 e2 +12r12 k 5 e22 mr22 − 8r12 k 5 e32 mr2 + 6r12 k 4 e22 m2 r22 − 4r12 k 4 e32 m2 r2 + r12 k 6 r24 + 2r12 k 5 r24 m − 4r12 k 6 e32 r2 − 8r12 k 5 r23 e2 m + r12 k 6 e42 +12r12 qp1 r22 m2 k 2 e2 −2p41 r12 q 4 k 4 +4r12 q 2 p21 e22 m2 −4r12 q 3 p31 e2 mk+4r12 q 2 p21 e22 mk−4r12 q 2 p21 e22 m2 k 2 + 4r12 qp1 e32 m2 k 2 +6r12 qp1 e32 mk 3 −4r12 k 4 r23 m2 e2 +r12 k 4 r24 m2 +p41 r12 q 4 k 6 +12k 4 q 3 p31 r1 r22 −24k 4 q 3 p31 r1 r2 e2 + 12k 4 q 2 p21 r1 r23 −12k 4 q 2 p21 r1 e32 +8k 3 q 2 p21 r1 r23 m+4k 3 q 3 p31 r1 r22 m−8k 3 q 3 p31 r1 r2 me2 +24k 3 q 2 p21 r1 r2 e22 m+ 4k 3 q 3 p31 r1 e22 m − 6p31 r12 q 3 k 3 r2 m − 4p31 r12 q 3 k 6 e2 − 6p31 r12 q 3 k 4 r2 + 6p31 r12 q 3 k 4 e2 − 8k 3 q 2 p21 r1 e32 m − 36k 4 q 2 p21 r1 r22 e2 +36k 4 q 2 p21 r1 r2 e22 −24k 3 q 2 p21 r1 r22 me2 +12k 4 q 3 p31 r1 e22 +4k 4 q 4 p41 r1 r2 −4k 4 q 4 p41 r1 e2 + 24r22 k 3 qp1 r1 e22 m+4r24 k 4 qp1 r1 −16r2 k 4 qp1 r1 e32 +4r24 k 3 qp1 r1 m−16r2 k 3 qp1 r1 e32 m+6p31 r12 q 3 k 3 e2 m− 16r23 k 4 qp1 r1 e2 +24r22 k 4 qp1 r1 e22 −16r23 k 3 qp1 r1 me2 +2p31 r12 q 3 k 5 r2 m−2p31 r12 q 3 k 5 e2 m+4p31 r12 q 3 k 6 r2 + 6p21 r12 q 2 k 5 e22 m + r12 q 2 p21 k 2 r22 + 2r12 q 3 p31 k 2 r2 − 2r12 q 2 p21 k 2 r2 e2 + 6p21 r12 q 2 k 6 r22 + 6p21 r12 q 2 k 6 e22 − 12p21 r12 q 2 k 6 r2 e2 + 6p21 r12 q 2 k 5 r22 m + r12 q 4 p41 k 2 )( 1/2))/k/q/p1 /(r2 + qp1 − e2 ) With the absence of predator population, we have the following eigenvalues λ1 = −1/2(r1 β 3 k − r1 β 2 e1 + 2e1 r1 − βkr1 − r1 β 2 a1 + 2a1 r1 − (4e21 r12 + 4a21 r12 − 2r12 β 5 ke1 + 6r12 β 3 ke1 − 2r12 β 5 ka1 + 6r12 β 3 ka1 + r12 β 6 k 2 − 4r12 β 2 e21 + r12 β 4 e21 − 2r12 β 4 k 2 + β 2 k 2 r12 + 8e1 r12 a1 + 2r12 β 4 e1 a1 − 8r12 β 2 e1 a1 − 4e1 r12 βk − 4r12 β 2 a21 + r12 β 4 a21 − 4 βkr12 a1 − 4 β 4 k 2 r1 e1 + 4 β 3 ke21 r1 − 4 β 4 k 2 r1 a1 + 8 β 3 ka1 e1 r1 + 4 β 3 ka21 r1 )( 1/2))/ β/k λ2 = −1/2(r1 β 3 k − r1 β 2 e1 + 2e1 r1 − βkr1 − r1 β 2 a1 + 2a1 r1 + (4e21 r12 + 4a21 r12 − 2r12 β 5 ke1 + 6r12 β 3 ke1 − 2r12 β 5 ka1 + 6r12 β 3 ka1 + r12 β 6 k 2 − 4r12 β 2 e21 + r12 β 4 e21 − 2r12 β 4 k 2 + β 2 k 2 r12 + 8e1 r12 a1 + 2r12 β 4 e1 a1 − 8r12 β 2 e1 a1 − 4e1 r12 βk − 4r12 β 2 a21 + r12 β 4 a21 − 4 βkr12 a1 − 4 β 4 k 2 r1 e1 + 4 β 3 ke21 r1 − 4 β 4 k 2 r1 a1 + 8 β 3 ka1 e1 r1 + 4 β 3 ka21 r1 )( 1/2)) βk and λ3 = (r2 a1 r1 β 2 k − 2r2 a1 r1 βe1 + r2 e1 m βk + r2 e1 r1 β 2 k + r2 β 3 mr1 k − r2 β 2 mr1 a1 − r2 β 2 mr1 e1 − e2 a1 mr1 + e2 r1 βa21 − e2 e1 mr1 + e2 r1 βe21 − e2 βm2 r1 − e2 β 2 m2 k − p2 r1 βa21 − p2 r1 βe21 + qp1 a1 mr1 + qp1 a1 m βk + qp1 a1 r1 β 2 k − qp1 r1 βa21 − 2qp1 a1 r1 βe1 + qp1 e1 mr1 +

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______________________________ qp1 e1 m βk + qp1 e1 r1 β 2 k − qp1 r1 βe21 − e2 a1 m βk − e2 a1 r1 β 2 k + 2e2 a1 r1 βe1 − e2 e1 r1 β 2 k − e2 β 3 mr1 k + e2 β 2 mr1 a1 − e2 e1 m βk + r2 a1 m βk + r2 a1 mr1 − r2 r1 βa21 + r2 e1 mr1 − r2 r1 βe21 + r2 βm2 r1 + r2 β 2 m2 k + e2 β 2 mr1 e1 + p2 r1 β 2 a1 k + p2 r1 β 2 e1 k + p2 r1 β 3 km − 2p2 r1 βa1 e1 − p2 r1 β 2 a1 m − p2 r1 β 2 e1 m)/(a1 mr1 + a1 m βk + a1 r1 β 2 k − r1 βa21 − 2a1 r1 βe1 + e1 mr1 + e1 m βk + e1 r1 β 2 k − r1 βe21 + βm2 r1 + β 2 m2 k + β 3 mr1 k − β 2 mr1 a1 − β 2 mr1 e1 ) The local stability of coexistence is given by the polynomial equation λ3 + B1 λ2 − B2 λ − B3 = 0

(11)

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where B1 = (m+L)B11 2 k(m+S)2 and B11 = 2 p1 Y km2 I − 2 kβ S 3 mI + 4 ke2 m2 IS − 2 kr2 mI 2 S + 2 kp2 Y m2 S − 2 kp2 SI 2 m − 2 kqp1 Sm2 I +3 kβ SI 2 m2 −3 kβ S 2 m2 I +2 ke2 mS 2 I +2 ke2 mI 2 S +4 ke1 m2 IS +2 ke1 mIS 2 + ka1 m4 + ke1 m4 − r1 km4 + 2 r1 Sm4 + 4 r1 m3 S 2 + 2 r1 S 3 m2 + 2 r1 S 3 I 2 + r1 Im4 + 2 r1 I 2 m3 + r1 I 3 m2 +r1 I 3 S 2 −kβ Sm4 −2 kβ S 2 m3 −kβ S 3 m2 −kβ S 3 I 2 +kp2 Y m3 +2 ka1 m3 S+ka1 m2 S 2 + 2 ka1 m3 I + ka1 I 2 m2 + ka1 I 2 S 2 + 2 ke1 m3 S + ke1 m2 S 2 + 2 ke1 m3 I + ke1 I 2 m2 + ke1 I 2 S 2 − 2 r1 km3 I − r1 km2 I 2 − 2 r1 km3 S − r1 kS 2 m2 − r1 kS 2 I 2 + 6 r1 Sm3 I + 6 r1 Sm2 I 2 + 9 r1 m2 S 2 I + 6 r1 mS 2 I 2 + 4 r1 S 3 mI + 2 r1 I 3 mS + β Ikm4 + 2 β I 2 km3 + β I 3 km2 + β I 3 kS 2 + p1 Y km3 + kp2 Y mS 2 + 4 ka1 m2 IS + 2 ka1 mIS 2 + 2 ka1 I 2 mS + 2 ke1 I 2 mS − 4 r1 km2 SI − 2 r1 kmSI 2 − 2 r1 kS 2 mI +2 β I 3 kmS +p1 Y kmI 2 −2 p2 IkSm2 −p2 Ikm3 −p2 ImkS 2 +ke2 m2 S 2 +ke2 m2 I 2 + 2 ke2 m3 I + 2 ke2 m3 S − kp2 I 2 m2 − 2 kr2 m3 I − 2 kr2 m3 S − kr2 S 2 I 2 − kp2 S 2 I 2 − kr2 m2 I 2 − kr2 m2 S 2 + ke2 S 2 I 2 + ke2 m4 − kr2 m4 − 2 kqp1 S 2 mI − kqp1 SI 2 m − kqp1 S 2 I 2 − kqp1 S 2 m2 − 2 3 2 2 2 1 m kβ+4 ISr2 m ke1 −Iqp1 S β km kqp1 Sm3 − 2 kr2 mS 2 I − 4 kr2 m2 IS, B2 = BB+2 ISqp1 m ke1 +ISqp (m+I)2 k(m+S)2 where BB = 2 IS 3 e2 mkβ + I 3 S 2 r2 β k + 5 S 2 p2 I 2 r1 m − 6 Se2 m3 r1 I − 6 e1 m3 r1 IS − p2 Y m3 β Ik − 2 a1 m3 β IkS−a1 m3 p1 Y k−4 ISr2 m2 r1 k−9 S 2 e2 m2 r1 I+2 e1 mIr1 kS 2 −2 Se2 m4 r1 +I 2 S 2 qp1 ka1 − 4 S 2 e2 m3 r1 − 2 e1 mI 2 β kS 2 + e1 I 2 r1 km2 + S 2 qp1 I 3 r1 − β Sm4 r1 k − 2 β S 2 m3 r1 k + S 2 p2 I 3 r1 + 2 ISp2 m2 ka1 + 2 ISqp1 m2 ka1 + 2 ISp2 m2 ke1 + 4 β S 2 Ir1 m3 + 8 β S 3 Ir1 m2 + I 2 S 2 r2 ka1 − 2 I 2 Sr2 mr1 k +4 ISr2 m2 ka1 −I 2 S 2 e2 ka1 +I 2 S 2 e2 r1 k −I 2 S 2 qp1 r1 k +I 2 S 2 r2 ke1 +Sqp1 I 3 r1 m+ 2 β S 2 m4 r1 + 4 β S 3 m3 r1 + 2 β S 4 m2 r1 + 2 β S 4 I 2 r1 + a1 m4 r1 k − 2 a1 m4 r1 S − 4 a1 m3 r1 S 2 − 2 a1 m2 r1 S 3 −a1 m4 r1 I−2 a1 m3 I 2 r1 −2 a1 I 2 r1 S 3 −a1 I 3 r1 m2 −a1 I 3 r1 S 2 +e1 m4 r1 k−2 e1 m4 r1 S− 4 e1 m3 r1 S 2 − 2 e1 m2 r1 S 3 − e1 m4 r1 I − 2 e1 m3 I 2 r1 − 2 e1 I 2 r1 S 3 − e1 I 3 r1 m2 − e1 I 3 r1 S 2 + 2 β S 2 I 2 r1 m2 + 4 β S 3 I 2 r1 m − β S 3 m2 r1 k + β Sm3 p1 Y k − 2 β Sm3 Ir1 k − 4 β S 2 m2 Ir1 k − 2 β S 3 mIr1 k + 4 β S 4 mIr1 + 2 β Sm2 Ip1 Y k − β SI 2 r1 km2 − 2 β S 2 I 2 r1 km − β S 3 I 2 r1 k + β SI 2 p1 Y km + p2 Y m3 r1 k + 2 p2 Y m2 r1 kS + p2 Y mr1 kS 2 − 2 p2 Y m3 r1 S − 4 p2 Y m2 r1 S 2 − 2 p2 Y mr1 S 3 − p2 Y m3 r1 I − 2 p2 Y m2 r1 IS − p2 Y mr1 IS 2 − 2 p2 Y m2 β IkS − p2 Y mβ IkS 2 − p2 Y 2 m2 p1 k+2 a1 m3 r1 kS+a1 m2 r1 kS 2 −6 a1 m3 r1 IS−9 a1 m2 r1 IS 2 −a1 m4 β Ik−a1 m2 β IkS 2 + 2 a1 m3 Ir1 k + 4 a1 m2 Ir1 kS + 2 a1 mIr1 kS 2 − 4 a1 mIr1 S 3 − 6 a1 m2 I 2 r1 S − 6 a1 mI 2 r1 S 2 − 2 a1 m3 I 2 β k − 4 a1 m2 I 2 β kS − 2 a1 mI 2 β kS 2 − 2 a1 m2 Ip1 Y k + a1 I 2 r1 km2 + 2 a1 I 2 r1 kmS + a1 I 2 r1 kS 2 −2 a1 I 3 r1 mS −a1 I 3 β km2 −2 a1 I 3 β kmS −a1 I 3 β kS 2 −a1 I 2 p1 Y km+2 e1 m3 r1 kS + e1 m2 r1 kS 2 −9 e1 m2 r1 IS 2 −e1 m4 β Ik−2 e1 m3 β IkS −e1 m2 β IkS 2 −e1 m3 p1 Y k+2 e1 m3 Ir1 k+ 4 e1 m2 Ir1 kS − 4 e1 mIr1 S 3 − 6 e1 m2 I 2 r1 S − 6 e1 mI 2 r1 S 2 − 2 e1 m3 I 2 β k − 4 e1 m2 I 2 β kS − 2 e1 m2 Ip1 Y k+2 e1 I 2 r1 kmS+e1 I 2 r1 kS 2 −2 e1 I 3 r1 mS−e1 I 3 β km2 −2 e1 I 3 β kmS−e1 I 3 β kS 2 − e1 I 2 p1 Y km+r2 m2 r1 I 3 +r2 m4 ka1 +r2 m4 ke1 −r2 m4 r1 k +r2 m4 r1 I +2 r2 m3 r1 I 2 −2 e2 m3 ka1 I − e2 m4 kβ I + r2 m4 kβ I + 2 r2 m3 ke1 I + p2 I 2 m3 r1 + p2 I 3 m2 r1 + r2 m2 ka1 I 2 + r2 m2 ke1 I 2 −

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r2 m2 r1 kI 2 + r2 m2 β I 3 k + r2 m3 kp2 Y + 2 r2 m3 ka1 I − 2 r2 m3 r1 kI + p2 Im3 ka1 + p2 Im3 ke1 − p2 Im3 r1 k + p2 I 2 m2 ka1 − p2 I 2 m2 r1 k + p2 I 3 kβ m2 + p2 I 2 m3 kβ + p2 I 2 m2 ke1 − e2 m2 r1 I 3 − e2 m4 ka1 −e2 m4 ke1 +e2 m4 r1 k−e2 m4 r1 I −2 e2 m3 r1 I 2 −2 e2 m3 ke1 I −e2 m2 ka1 I 2 −e2 m2 ke1 I 2 + e2 m2 r1 kI 2 −e2 m2 β I 3 k −e2 m3 kp2 Y +2 e2 m3 r1 kI +2 r2 m3 kβ I 2 −2 e2 m3 kβ I 2 +2 Sr2 mI 3 r1 + 4 S 2 r2 m3 r1 +2 S 2 qp1 m3 r1 +6 S 2 r2 mr1 I 2 +4 S 2 p2 Ir1 m2 +2 Sr2 m4 r1 +9 S 2 r2 m2 r1 I +S 2 r2 I 3 r1 + 4 S 2 qp1 mr1 I 2 +5 S 2 qp1 m2 r1 I−6 S 2 e2 mr1 I 2 −S 2 e2 I 3 r1 +6 Sr2 m2 r1 I 2 +6 Sr2 m3 r1 I+2 Sp2 Im3 r1 + 2 Sp2 I 3 mr1 +4 Sp2 I 2 m2 r1 +Sqp1 m3 r1 I+2 Sqp1 m2 r1 I 2 −2 Se2 mI 3 r1 −6 Se2 m2 r1 I 2 +S 3 e2 m2 kβ− 2 Se2 m2 kp2 Y −S 2 e2 kp2 Y m+S 2 r2 m2 ka1 +2 r2 m2 S 3 r1 −2 e2 m2 S 3 r1 +r2 m3 p1 Y k+2 qp1 S 3 m2 r1 − S 3 qp1 m2 kβ−2 S 2 r2 m3 kβ+S 2 r2 m2 ke1 −S 2 r2 m2 r1 k−S 2 qp1 m3 kβ+S 2 qp1 m2 ke1 +S 2 qp1 m2 ka1 + S 2 r2 kp2 Y m−S 2 qp1 m2 r1 k+S 2 qp1 kp2 Y m−S 2 e2 m2 ke1 −e2 m3 p1 Y k−S 3 r2 m2 kβ−Sqp1 m3 r1 k+ Sqp1 kp2 Y m2 −2 Se2 m3 ke1 +2 Se2 m3 r1 k−S 2 e2 m2 ka1 +S 2 e2 m2 r1 k+2 S 2 e2 m3 kβ−2 Se2 m3 ka1 + Se2 m4 kβ −Sr2 m4 kβ +2 Sr2 m3 ke1 +2 Sr2 m2 kp2 Y +Sqp1 m3 ke1 +2 Sr2 m3 ka1 −2 Sr2 m3 r1 k + Sqp1 m3 ka1 + 2 I 2 p2 r1 S 3 + 2 I 2 r2 S 3 r1 − 2 I 2 e2 S 3 r1 + I 2 e2 S 3 β k − I 2 p2 kβ S 3 + I 2 p2 kβ Sm2 − I 2 p2 mβ kS 2 −I 2 qp1 S 3 β k+3 I 2 r2 m2 β kS+2 I 2 qp1 S 3 r1 +2 I 2 qp1 Sβ km2 −I 2 r2 S 3 β k+I 2 qp1 S 2 β km− 3 I 2 e2 m2 β kS−I 2 S 2 e2 ke1 +I 2 Sqp1 ka1 m+I 2 S 2 qp1 ke1 +I 2 r2 p1 Y km−I 2 e2 p1 Y km+I 2 p2 p1 Y km− I 2 S 2 r2 r1 k+I 2 S 2 p2 ka1 +I 2 S 2 p2 ke1 −I 2 S 2 p2 r1 k+2 I 2 Sp2 mke1 −2 I 2 Sp2 mr1 k+I 2 Sqp1 ke1 m− I 2 Sqp1 r1 km − 2 I 2 Se2 mka1 + 2 I 2 Se2 mr1 k + 2 I 2 Sr2 mke1 − 2 I 2 Se2 mke1 + 2 I 2 Sr2 mka1 + 2 I 2 Sp2 mka1 +I 3 S 2 qp1 β k+I 3 S 2 p2 β k−I 3 S 2 e2 β k+2 I 3 Sp2 mβ k+I 3 Sqp1 β km−2 I 3 Se2 mβ k+ 2 I 3 Sr2 mβ k+2 Imp2 r1 S 3 −Imp2 kβ S 3 −Ip2 kβ Sm3 −2 Ip2 m2 β kS 2 +4 Ir2 mS 3 r1 +IS 2 p2 ka1 m− 2 IS 2 e2 mka1 − 3 IS 2 r2 m2 kβ − 4 Ie2 mS 3 r1 + 2 Ir2 m2 p1 Y k − 2 Ie2 m2 p1 Y k + Ip2 p1 Y km2 + 4 Iqp1 S 3 mr1 − 2 IS 3 r2 mkβ − 2 IS 3 qp1 mkβ + 2 IS 2 r2 mka1 + 2 IS 2 r2 mke1 − 2 IS 2 r2 mr1 k + IS 2 p2 ke1 m+2 IS 2 qp1 mka1 +2 IS 2 qp1 mke1 −2 IS 2 qp1 mr1 k −2 ISp2 m2 r1 k −2 ISqp1 m2 r1 k − 4 ISe2 m2 ka1 − 4 ISe2 m2 ke1 + 4 ISe2 m2 r1 k − 2 IS 2 e2 mke1 + 2 IS 2 e2 mr1 k − IS 2 p2 r1 km + 3 IS 2 e2 m2 kβ and B3 = (m+I)2CC k(m+S)2 where CC = r2 m4 a1 Ir1 + 4 r2 m3 a1 r1 S 2 + e2 m4 e1 r1 k + e2 m4 a1 r1 k + I 3 S 2 p2 a1 β k − 2 e2 m3 e1 I 2 r1 + r2 m2 e1 I 3 r1 +r2 p2 Y 2 m2 p1 k+p2 I 3 m2 e1 r1 +IS 2 r2 p2 Y mr1 +p2 I 2 m3 e1 r1 −e2 m4 a1 r1 I−e2 m3 p2 Y r1 I+ r2 m2 a1 I 3 r1 + 2 r2 m3 a1 I 2 r1 + I 2 S 2 p2 a1 mβ k + 4 I 2 S 2 qp1 me1 r1 + 2 r2 m3 e1 I 2 r1 − I 3 e2 m2 a1 β k − S 2 r2 m2 a1 r1 k+2 S 2 e2 m4 β r1 +2 I 2 Se2 ma1 r1 k−4 IS 2 r2 m3 β r1 +2 S 3 qp1 p2 Y mr1 +2 ISr2 m3 e1 β k− 4 IS 4 qp1 mβ r1 +4 IS 4 e2 mβ r1 +I 3 Sqp1 a1 r1 m+9 IS 2 r2 m2 a1 r1 +2 Sr2 m3 p2 Y r1 −2 S 4 r2 m2 β r1 + 2 S 4 e2 m2 β r1 −2 S 4 qp1 β r1 m2 −2 I 2 Sp2 ma1 r1 k−I 2 p2 m2 e1 r1 k−6 I 2 S 2 e2 ma1 r1 −I 2 S 2 qp1 a1 r1 k− I 2 S 2 r2 a1 r1 k +6 ISr2 m3 e1 r1 −4 IS 3 p2 m2 β r1 +e2 m3 p2 Y r1 k +S 3 qp1 β r1 km2 −2 S 3 qp1 β r1 m3 − 4 S 2 e2 m2 p2 Y r1 +S 2 e2 m2 a1 r1 k +4 S 2 r2 m2 p2 Y r1 +S 2 e2 m2 e1 r1 k +I 3 S 2 r2 e1 r1 +2 I 3 Sp2 ma1 r1 + 2 I 3 Sp2 me1 r1 +2 I 3 Sr2 ma1 r1 +ISqp1 e1 r1 m3 +5 IS 2 qp1 e1 r1 m2 −IS 2 e2 p2 Y mr1 +ISqp1 a1 r1 m3 + 2 ISqp1 p2 Y m2 r1 −2 I 2 S 3 qp1 mβ r1 −6 I 2 Se2 m2 a1 r1 −6 I 2 Se2 m2 e1 r1 +4 I 2 Sp2 m2 a1 r1 +2 I 2 Sqp1 m2 e1 r1 − 4 I 2 S 3 r2 mβ r1 +4 I 2 S 3 e2 mβ r1 +6 I 2 Sr2 m2 e1 r1 −I 3 S 2 e2 e1 r1 +4 S 3 e2 m3 β r1 +2 S 2 qp1 p2 Y m2 r1 − 2 S 2 e2 m3 β r1 k−2 S 2 r2 m4 β r1 −2 Sr2 m2 p2 Y r1 k−e2 p2 Y 2 m2 p1 k−r2 m3 p2 Y r1 k−2 S 3 e2 p2 Y mr1 + 2 S 3 r2 p2 Y mr1 −e2 m3 e1 p1 Y k −S 3 e2 m2 β r1 k −S 2 r2 p2 Y mr1 k −Sqp1 e1 r1 km3 −S 2 qp1 e1 r1 km2 + S 2 qp1 β r1 km3 −2 e2 m3 a1 I 2 r1 +S 2 e2 p2 Y mr1 k+IS 2 qp1 p2 Y mβ k−4 IS 3 e2 me1 r1 −4 IS 3 e2 ma1 r1 + IS 2 r2 m2 e1 β k+IS 2 r2 m2 a1 β k−Ie2 m4 a1 β k−2 ISqp1 m2 a1 r1 k−2 ISr2 m2 β p1 Y k+2 IS 3 p2 ma1 r1 + 4 IS 3 qp1 a1 r1 m−2 ISe2 m3 e1 β k+Ir2 m4 e1 β k+2 ISr2 m3 a1 β k−I 2 r2 m2 e1 r1 k−I 2 e2 a1 p1 Y km−

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I 2 p2 m2 a1 r1 k + I 2 r2 e1 p1 Y km + I 2 p2 m3 e1 β k − I 2 r2 m2 a1 r1 k + 2 I 2 S 3 p2 a1 r1 − 2 I 2 S 4 p2 β r1 + 2 I 2 S 3 r2 e1 r1 +2 I 2 S 3 qp1 a1 r1 −2 I 2 S 4 qp1 β r1 −2 I 2 S 3 e2 a1 r1 −4 I 2 Se2 m2 a1 β k−I 2 Se2 m2 β r1 k+ I 2 e2 m2 e1 r1 k + I 2 e2 m2 a1 r1 k − I 2 e2 e1 p1 Y km + I 2 S 2 e2 e1 r1 k + I 2 S 2 e2 a1 r1 k + 2 I 2 S 3 r2 a1 r1 − I 2 S 2 r2 e1 r1 k+2 I 2 S 3 qp1 e1 r1 +2 I 2 r2 m3 e1 β k+2 I 2 r2 m3 a1 β k+I 2 p2 e1 p1 Y km−2 I 2 e2 m3 a1 β k+ I 2 Sp2 m2 β r1 k+2 I 2 Sp2 m2 a1 β k−I 2 Sr2 β p1 Y km−I 2 Sp2 β p1 Y km−2 I 2 Sp2 me1 r1 k−4 I 2 Se2 m2 e1 β k+ I 2 Sr2 m2 β r1 k+I 2 SY mqp1 p2 r1 −2 I 2 Sr2 ma1 r1 k+I 2 p2 a1 p1 Y km−2 I 2 e2 m3 e1 β k+2 I 2 Sqp1 m2 e1 β k+ I 2 Se2 β p1 Y km+2 I 2 Se2 me1 r1 k+I 2 S 3 qp1 β r1 k+4 I 2 Sr2 m2 a1 β k+4 I 2 Sr2 m2 e1 β k+2 I 2 S 2 r2 ma1 β k+ 2 I 2 S 2 r2 me1 β k+2 I 2 S 2 r2 mβ r1 k+2 I 2 S 2 p2 mβ r1 k+I 2 S 2 p2 e1 mβ k−2 I 2 S 2 e2 mβ r1 k−2 I 2 S 2 e2 me1 β k− 2 I 2 S 2 e2 ma1 β k+2 I 2 S 4 e2 β r1 +I 2 SY mqp1 p2 kβ−2 I 2 Sr2 me1 r1 k−I 2 Sqp1 e1 r1 km−2 I 2 S 4 r2 β r1 + 2 I 2 S 2 qp1 ma1 β k−I 2 Sqp1 a1 r1 km+2 I 2 Sp2 m2 e1 β k+2 I 2 S 2 qp1 me1 β k−I 2 S 2 p2 e1 r1 k+I 2 S 2 qp1 β r1 km− I 2 S 2 qp1 e1 r1 k − I 2 S 2 p2 a1 r1 k − I 2 S 3 e2 β r1 k + I 2 S 3 r2 β r1 k + I 2 S 3 p2 β r1 k + 2 I 2 Sqp1 m2 a1 β k − 2 I 2 S 3 e2 e1 r1 + 2 I 2 S 3 p2 e1 r1 + I 3 r2 m2 e1 β k − I 3 e2 m2 e1 β k + I 3 S 2 r2 e1 β k + I 3 S 2 r2 a1 β k + 2 I 3 Sp2 ma1 β k+I 3 r2 m2 a1 β k+2 I 3 Sp2 me1 β k+2 I 3 Sr2 ma1 β k+I 3 p2 m2 e1 β k+I 3 p2 m2 a1 β k− 2 I 3 Se2 ma1 β k−I 3 S 2 e2 a1 β k−2 I 3 Se2 me1 β k+I 3 Sqp1 a1 β km+I 3 Sqp1 e1 β km+2 I 3 Sr2 me1 β k+ I 3 S 2 p2 e1 β k + I 3 S 2 qp1 a1 β k + I 3 S 2 qp1 e1 β k + p2 I 3 m2 a1 r1 + 6 I 2 S 2 r2 ma1 r1 − 2 I 2 S 2 p2 m2 β r1 − 4 I 2 S 3 p2 β mr1 +6 I 2 Sr2 m2 a1 r1 +4 I 2 Sp2 m2 e1 r1 +4 I 2 S 2 qp1 ma1 r1 −4 IS 3 qp1 β r1 m2 −9 IS 2 e2 m2 a1 r1 − 9 IS 2 e2 m2 e1 r1 +4 IS 2 e2 m3 β r1 −2 IS 2 p2 m3 β r1 +4 IS 2 p2 m2 a1 r1 +2 ISr2 m2 p2 Y r1 +8 IS 3 e2 m2 β r1 + 4 IS 2 p2 m2 e1 r1 +5 IS 2 qp1 a1 r1 m2 +IS 2 qp1 p2 Y mr1 +9 IS 2 r2 m2 e1 r1 −2 ISe2 m2 p2 Y r1 −8 IS 3 r2 m2 β r1 − 6 ISe2 m3 a1 r1 − 6 ISe2 m3 e1 r1 + 2 ISp2 m3 a1 r1 + 2 ISp2 m3 e1 r1 + 6 ISr2 m3 a1 r1 − r2 m4 a1 r1 k − 2 I 3 Se2 ma1 r1 − 2 I 3 Se2 me1 r1 + I 3 S 2 p2 e1 r1 + I 3 S 2 qp1 e1 r1 + I 3 S 2 p2 a1 r1 + I 3 S 2 qp1 a1 r1 + I 3 S 2 r2 a1 r1 +2 I 3 Sr2 me1 r1 +I 3 Sqp1 e1 r1 m+2 I 2 Sqp1 m2 a1 r1 +6 I 2 S 2 r2 me1 r1 +2 I 2 S 2 e2 m2 β r1 − 2 I 2 S 2 r2 m2 β r1 −6 I 2 S 2 e2 e1 r1 m+5 I 2 S 2 p2 e1 mr1 +5 I 2 S 2 p2 a1 mr1 −Sqp1 a1 r1 km3 −I 3 S 2 e2 a1 r1 − e2 a1 p1 Y km3 −4 S 3 r2 m3 β r1 −S 2 qp1 a1 r1 km2 −Sqp1 p2 Y m2 r1 k−2 Se2 m3 p2 Y r1 −I 3 S 2 e2 e1 β k− S 2 qp1 p2 Y mr1 k + r2 m4 e1 r1 I + r2 m3 p2 Y r1 I + 2 S 2 r2 m3 β r1 k − S 2 r2 m2 e1 r1 k − e2 m4 e1 r1 I − e2 m2 a1 I 3 r1 − e2 m2 e1 I 3 r1 + 4 IS 3 r2 ma1 r1 − 2 Ie2 a1 p1 Y km2 − Ip2 m3 a1 r1 k − 2 Ir2 m3 a1 r1 k − Ip2 m3 e1 r1 k + 2 Ie2 m3 e1 r1 k + Ir2 m4 a1 β k − IS 2 e2 m2 a1 β k + 2 IS 3 qp1 mβ r1 k − 4 IS 4 r2 mβ r1 − 2 IS 4 p2 β mr1 −2 ISe2 m3 a1 β k+4 ISe2 m2 a1 r1 k+4 ISe2 m2 e1 r1 k−2 Ie2 m2 e1 p1 Y k−Ie2 m4 e1 β k− 2 IS 2 qp1 me1 r1 k+4 IS 3 r2 me1 r1 +4 IS 3 qp1 e1 r1 m+2 IS 3 p2 me1 r1 −IS 2 e2 m2 e1 β k+IS 2 qp1 e1 β km2 + 2 Ie2 m3 a1 r1 k−2 Ir2 m3 e1 r1 k+Ip2 e1 p1 Y km2 −Ie2 m3 p2 Y β k−2 ISp2 m2 a1 r1 k+2 ISY m2 qp1 p2 kβ+ 2 ISr2 m2 p2 Y β k + Ip2 a1 m2 p1 Y k + 2 Ir2 m2 a1 p1 Y k − 2 ISqp1 m2 e1 r1 k + 2 ISe2 m2 β p1 Y k + 2 Ir2 m2 e1 p1 Y k+Ir2 m3 p2 Y β k+2 IS 3 r2 mβ r1 k−2 IS 3 e2 mβ r1 k−2 ISe2 m3 β r1 k−4 ISr2 m2 e1 r1 k− 2 IS 2 r2 me1 r1 k+IS 2 r2 p2 Y mβ k−4 IS 2 e2 m2 β r1 k+4 IS 2 r2 m2 β r1 k+ISqp1 e1 β km3 +2 IS 2 e2 e1 r1 km− IS 2 p2 e1 mr1 k−IS 2 p2 a1 mr1 k−2 IS 2 r2 ma1 r1 k+2 IS 2 p2 m2 β r1 k+IS 3 p2 β mr1 k−2 ISe2 m2 p2 Y β k− IS 2 Y mp2 kβ p1 + 2 IS 2 e2 ma1 r1 k + 2 IS 2 qp1 m2 β r1 k − 2 ISY m2 p2 kβ p1 − 4 ISr2 m2 a1 r1 k + ISp2 m3 β r1 k+ISqp1 a1 β km3 −2 ISp2 m2 e1 r1 k−IS 2 e2 p2 Y mβ k−2 IS 2 qp1 ma1 r1 k+IS 2 qp1 a1 β km2 + 2 ISr2 m3 β r1 k +r2 m3 e1 p1 Y k +r2 m3 a1 p1 Y k +2 Se2 m2 p2 Y r1 k −Se2 m4 β r1 k +Se2 m3 β p1 Y k + I 2 p2 m3 a1 β k + I 2 r2 a1 p1 Y km − r2 m4 e1 r1 k + p2 I 2 m3 a1 r1 − 2 e2 m4 e1 r1 S − 2 e2 m2 e1 r1 S 3 − 4 e2 m3 a1 r1 S 2 + 2 r2 m4 a1 r1 S − 4 e2 m3 e1 r1 S 2 + 2 r2 m2 e1 r1 S 3 + 4 r2 m3 e1 r1 S 2 − 2 e2 m4 a1 r1 S − 2 e2 m2 a1 r1 S 3 +2 r2 m2 a1 r1 S 3 +2 qp1 S 3 a1 r1 m2 +2 r2 m4 e1 r1 S+2 qp1 S 2 a1 r1 m3 +2 qp1 S 2 e1 r1 m3 + 2 qp1 S 3 e1 r1 m2 +S 3 r2 m2 β r1 k −Sr2 m3 β p1 Y k +Sr2 m4 β r1 k +2 Se2 m3 a1 r1 k +2 Se2 m3 e1 r1 k − 2 Sr2 m3 a1 r1 k − 2 Sr2 m3 e1 r1 k Using the Routh-Hurwitz criteria, the coexistence equilibrium point will be stable if the equation (11) will obey B1 > 0, B2 > 0, B3 > 0; B1 B2 > B3 . Otherwise the coexistence

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4.3

Global stability Analysis

We perform global stability analysis of the system (1) around the positive equilibrium point of the coexistence E(S ∗ , I ∗ , Y ∗ ). We consider the following theorem on the Lyapunov function U . Theorem 4.1 Let 1 1 1 U = (S − S ∗ )2 + µ1 (I − I ∗ )2 + µ2 (Y − Y ∗ )2 2 2 2

(12)

0

where µ1 , µ2 > 0 to be chosen carefully such that U (E) = 0 then E(S ∗ , I ∗ , Y ∗ ) and U = (S, I, Y ) > 0, ∀S, I, Y | {E} The time derivative of U is

dU dt

(13)

≤ 0, ∀S, I, Y ∈ τ + then dU = 0, ∀S, I, Y ∈ τ + dt

(14)

ST

implies that E ∗ of the system is Lyapunov stable and

near E ∗ is global stable. Proof 4.2

(15)

IJ

dU < 0, ∀S, I, Y ∈ τ + dt

dU dS dI dY = (S − S ∗ ) + µ1 (I − I ∗ ) + µ2 (Y − Y ∗ ) dt dt dt dt substituting the model equations (1) and simplifying we get

(16)

    dU S+I p1 Y ∗ 2 = − (S − S ) r1 −1 +βI + dt k m+S   p2 Y ∗ 2 − µ1 (I − I ) + (a1 + e1 ) − β S m+I   qp1 S p2 I ∗ 2 − µ2 (Y − Y ) e2 − r2 − − (17) m+S m+I 0

Thus it is possible to set µ1 , µ2 > 0 such that U ≤ 0 and endemic positive equilibrium point is globally stable. Therefore, the parameters k, q and m play important roles in controlling the stability aspects of the system.

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Numerical simulation

In order to illustrate some of the analytical results of the study, we present the numerical simulations of the model (1) using Rung-Kutta iteration scheme with a set of reasonable parameter values given in Table 1. These parameter values are mainly hypothetical. They are chosen following realistic ecological observations.

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Table 1: Table for parameter values for the model Parameter symbol Parameter value r1 300.2 k 8 β 30.2 p1 0.4 p2 0.8 m 0.5 a1 0.1 e1 0.04 e2 0.8 r2 0.02 q 0.25

60

50

Suceptible prey Infected prey Predator

Predator

40

30

20

10

0 0

10

20

30

40

50

60

70

80

90

100

time

Figure 1: Total population around the disease free parameter values in table 1

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14

Susceptible Prey

12

10

8

6

4

2

0 0

5

10

15

20

25

30

35

40

time

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Figure 2: Prey population around disease free parameter values in table 1

50 45 40

Infected prey

35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

40

45

50

time

Figure 3: Infected Prey population around disease free parameter values in table 1

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60

Predator

50

40

30

20

10

0

0

10

20

30

40

50

60

70

time

Figure 4: Predator population around disease free parameter values in table 1

70

50

40

ST

Predator population

60

30

20

0

0

IJ

10

10

20

30

40

50

60

Infected prey population

Figure 5: variation of predator and infected prey population around disease free parameter values in table 1

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infected prey population

50

40

30

20

10

0

0

5

10

15

20

25

30

susceptible prey population

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Figure 6: Variation of susceptible prey population against infected prey around disease free parameter values in table 1

70

predator population

60

50

40

30

20

10

0

0

2

4

6

8

10

12

14

16

18

20

susceptible prey population

Figure 7: Variation of susceptible prey population against infected prey around disease free parameter values in table 1

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Predator

50 40 30 20 10 0 60 50

60

40

50 30

40 30

20

20

10 0

10 0

infected prey

susceptible prey

Figure 8: variation of predator population agaist susceptible prey around disease free parameter values in table 1

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It is observed that in Figure 1, the distribution of population with time is shown in all classes with the predation rate of p1 and p2 . It is observed that susceptible populations decreases with time due to the disease infection. The infected classes increases with time and eventually attain its equilibrium point. Figure 2 shows the variation of the susceptible prey population as the result of the disease infection in the population. The population oscillates at high amplitude at the beginning and then decreases to low amplitude until the steady state attained. The Figure 3 shows the variation of the infected prey with high oscillation amplitude compared with Figure (2). It is observed that the population increases with an increase of the disease, and eventually reached the steady state.Figure 4 shows how the predator population vary with time depending on the interaction with prey and infected prey populations. The sharply decrease of the population occurs as the result of high number of infected prey that are easily captured by the predator. The high death rate of the predator also contribute in decline of the population.

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References 1. Alfred Hugo., Estomih S. Massawe., Oluwole Daniel Makinde., (2012). An eco-epidemiological mathematical model with treatment and disease infection in both prey and predator population, Journal of Ecology and the Natural Environment Vol. 4(10), pp. 266-279 2. Anderson R., May R., (1986). The Invasion, Persistence and Spread of InfectiousDiseases within Animal and Plant-Communities. Philosophical Transactions of The Royal Society of London Series B-Biological Sciences 314, pp. 533570. 3. Arino O., El abdllaoui A., Mikram J., Chattopadhyay J., (2004). Infection in prey population may act as a biological control in ratio-dependent predator-prey models. Nonlinearity 17, pp. 11011116. 4. Apreutesei N.C., (2009) An optimal control problem for a prey -predator system with a general functional response. ELSEVIER , Applied Mathematics Letters 22 (2009) 1062 -1065

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5. Bakare E.A., Adekunle Y. A., Nwagwo A., (2012). Mathematical Analysis of the control of the spread of the infectious disease in a prey-predator. International Journal of computer and Organization Trends Volume 2Issue 1. ISSN: 2249-2593. 6. Beltrami E, Carroll TO (1994). Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol. 32:857-863.

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7. Blumstein D (2003) Flight-initiation distance in birds is dependent on intruder starting distance. J Wildlife Manage 64: 852-857. 8. Han L., Ma Z., Hethcote H. W., (2001). Four eco-epidemiological models (SI and SIR types) with standard and mass action incidences. Math. Comp. Modelling 34 849. 9. Hethcote, H.W., Wang, W., Han, L. and Ma, Z. (2004), A predatorprey model with infected prey 66, 259268. 10. Hethcote H. W., (2000). The mathematics of infectious diseases. SIAM Review 42, 599653. 11. Raid Kamel N, Kawa Ahmed H, (2012), The dynamics of prey-predator model with disease in prey. Math comput. sci.2, no. 4, 1052-1072, ISSN: 1927-5307 12. Kermack W. O., McKendrick A. G., (1927). The Evolution of Virulence in Sterilizing Pathogens. Pro-Royal Soc. London, A., 115, 700. 13. Lotka A. J. (1924). Elements of Physical Biology (Baltimore Williams and Wilkins Co., Inc.). 14. Liu R., Duvvuri V. R. S. K., Wu J., (2008). Spread pattern formation of h5n1avian influenza and its implications for control strategies. Mathematical Modelling of Natural Phenomena.3, 161179.

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______________________________ 15. Makinde O. D., Okosun., (2011). Impact of Chemo-therapy on Optimal Control of Malaria Disease with Infected Immigrants. ELSEVIER, BioSystems 104 (2011) 3241. 16. Mukherjee D., (2003). Stability Analysis of a Stochastic Model for Prey-Predator System with Disease in the Prey, Vivekananda College, Nonlinear Analysis: Modelling and Control, 2003, Vol. 8, No. 2, 8392. 17. Mukhopadhyay, B., Bhattacharyya, R., (2009). Role of predator switching in an ecoepidemiological model with disease in the prey. ELSEVIER- Ecological Modelling 220, pp. 931939. 18. Venturino E (1994). The influence of diseases on Lotka-Volterra systems. Rocky Mount. J. Math. 24:381-402. 19. Venturino E (1995). Epidemics in predatorprey models: Disease among the prey. In: Arino O, Axelrod D, Kimmel M, Langlais M (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. I: Theory of Epidemics. Wuertz Publishing Ltd, Winnipeg, Canada, pp. 381-393.

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20. Venturino E (2002). Epidemics in predator-prey model with disease in the predators. IMA J. Math. Appl. Med. Biol. 19:185.

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21. Joel Z., Michael Robert De Weese, (2011). How should prey animals respond to uncertain threats?. arXiv:1104.3805v1 [q-bio.PE]

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