DYNAMICAL ANALYSIS OF POLLUTED PREY-PREDATOR SYSTEM WITH INFECTED PREY

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, a prey-predator model in polluted environment with disease in prey has been proposed and studied. It is assumed that only prey population is prone to disease whereas, both the populations are affected by the pollutant. Boundedness of the solution of the system is discussed. Existence of all possible equilibrium points has been established. Using Routh Hurwitz criterion, local stability of all the possible equilibrium points has been obtained. Also, interior equilibrium point has been proved to be globally asymptotically stable using Lyapunov function. Then time delay has been introduced in the system making the model more realistic. Existence and direction of Hopf bifurcation in the delay model has been established using normal form theory and center manifold theorem. By taking a set of hypothetical and biologically feasible parameters, model has been studied numerically using MATLAB and the effect of pollutant on the system has been deduced.


INTRODUCTION
The relation between the predators and their prey is the building block of ecosystems. Due to its widespread existence and importance, this dynamic relation has always been an important topic of study in ecology as well as mathematical ecology. When a number of prey-predator interactions take place in the environment at different trophic levels, food chains and eventually food webs are formed. There are many factors that influence these food webs such as climate, natural disaster, other food chains etc., due to which species evolve and disperse in order to seek resources for their survival and existence in the ecosystem. So, populations continuously move away from one state to the other state.
There has been huge growth in industry, agriculture etc. which has taken the comfort of mankind to the next level. All these developments including urbanization has helped people attaining a better lifestyle. The byproducts of the processes are not just the products and services that we purchase from the market, but also the waste products that are eliminated into the environment.
This waste is sometimes treated and is less harmful for the environment or sometimes untreated, consumption of which by the organisms could be lethal. The different forms of waste could be organic, inorganic, radioactive etc. In case of radioactive, organisms may suffer from harmful diseases, birth defects or even gene mutation.
The incubation period is defined as the time period between exposure to an infection and appearance of the first symptom. There are very less mechanisms in this world that are instantaneous. For example, human body already contains cancer genes. The symptoms start occurring only when those genes are exposed to the trigger. So, there is a time lag or delay which is termed as the incubation period, after which the effect of infection could be seen. When we consider delay while defining certain mechanism, it becomes more appropriate according to the real life environment, making any study more reasonable. In recent decades many investigators have proposed and analyzed mathematical models to study the effects of toxicants on biological species.
In [14], it is assumed that the toxicant affects both prey and predator population where the infected prey is more vulnerable to be affected by toxicant and predation as compared to the susceptible prey population. In [15], the effect of only disease and the effect of disease as well as toxicant on a plant population has been studied. The problem of ratio-dependent predatorprey model has been studied in [10]. In [9], a prey-predator model has been discussed where prey has logistic growth and the model is modified to include parasitic infection in prey where the infected prey becomes more vulnerable to predation. Four modifications of a predator prey model are developed and analyzed in [11] including parasite infection. In [13], authors have shown that the exposure to the pollutants can lead to immunosuppression and increased disease susceptibility in juvenile salmon. [4] determines the direction of the Hopf bifurcation about the equilibrium using center manifold theorem. Also, a settled modelling approach was proposed to the problem of investigating the effects of a pollutant on an ecological system in [5,6,7,8].
Delay differential equations are widely used in epidemiology and problems related to delay have been studied by various authors [2,3]. A prey-predator model with harvesting and diseased prey, in absence and presence of time delay has been analysed in [16]. [1] talks about the transmission and control of epidemics, where time delay is associated with the infected species.
The chaotic dynamics induced by a disease in an eco-epidemiological prey-predator model with diseased prey and weak Allee in predator has been studied in [12].
Keeping in view the above discussion, in this paper, we have proposed a prey-predator system with combined effect of disease and pollutant in section 2. In our model, we have incorporated disease and pollutant and studied its effect on prey-predator dynamics. After formulating the model, dynamical behavior of the system has been studied in section 3, in which existence of all possible equilibrium point has been obtained and stability, local and global, of the system has been analyzed. Dynamical analysis of the system with time delay has been done in section 4. Section 5 deals with the numerical simulations where a set of hypothetical and biologically feasible parameters has been considered and effect of pollutant on the system has been deduced.
In section 6, the results obtained theoretically and numerically have been discussed.

MATHEMATICAL MODEL
The prey-predator model under the effect of pollutant is considered. The prey population, which is susceptible and infected is denoted by S(t) and I(t) respectively and the predator is denoted by P(t). Also, C(t) is the environmental concentration of the pollutant and U(t) is the concentration of the pollutant in the organisms. The assumptions adopted are: 1. The reproduction in susceptible prey takes place according to constant growth rate. The transmission of disease from infected to susceptible prey takes place by contact. This transmission occurs according to non linear incidence rate of the form λ SI 1+I , where λ SI is the infection force of disease and 1 1+I measures the effect of inhibition from the susceptible prey. This inhibition effect occurs due to behavioral change of susceptible population that includes increase in number or crowding effect of infected prey.
2. The predators attack the susceptible and infected individuals with different rates. The consumption of susceptible prey is according to α 1 S β +S+mI and infected prey is according to α 2 I β +S+mI , which are known as modified Holling type-II functional response.
3. Food and the environment both are the sources of pollutant uptake by the populations. The loss of pollutant from the organisms takes place due to metabolic processing and other causes.
If q is the constant exogenous input rate of the pollutant into the environment, C(t) is the environmental concentration of the pollutant and the natural loss rate of pollutant from environment can be due to biological transformation, hydrolysis, vitalization, microbial degradation, including other processes then the model proposed is as follows: Since we know from (4) and (5) that lim sup t→∞ C(t) ≤ C * and lim sup t→∞ U(t) ≤ U * , thus, using the corollary 1 in [7] in the model we get the limiting system as follows: All the parameters in the above defined systems are assumed to have positive values and are described as follows: Since any infection takes time to get incubated into the organism, so we consider the system defined by equations (6)-(8) with time delay, which is more appropriate according to real environment. Therefore, the system becomes: where, τ ≥ 0 is the time interval for the infection to get incubated into the prey species. Also, , the Banach space of continuous functions mapping the interval [−τ, 0] into R 3 + .

Basic
Properties of the Model. The density of population cannot be negative, so the state space of the system is R 3 + = {(S, I, P) ∈ R 3 : S ≥ 0, I ≥ 0, P ≥ 0}. To support the positivity and boundedness of the system, we start with lemmas given below: Proof: Let (S(t), I(t), P(t)) be any solution of the system defined by equations (6)- (8). We assume that there exists a solution of the system that is at least not positive. Following cases arise: Case 3 ∃t such that P(0) > 0, P(t) = 0, P (t) < 0, S(t) > 0, I(t) > 0, 0 ≤ t <t If case 1 holds then S (t * ) = Λ > 0, that contradicts with S (t * ) < 0.
Since (S(t), I(t), P(t)) was arbitrary, all the solutions of the system are positive ∀ t > 0.
Lemma 2: All the solutions of the system that initiate in the state space R 3 + are uniformly bounded.
Proof: Let (S(t), I(t), P(t)) be any solution of the system with non-negative initial conditions. Consider, W (t) = S(t) + I(t) + P(t), then Since the constant of rate of conversion from prey population to predator population cannot exceed the maximum predation rate constant of predator population to prey population, therefore Thus, all the solutions are bounded. Hence Proved. (b) E 2 = (S, 0, 0) exists uniquely whereS = Λ r 1 U * +d 1 .

DYNAMICAL BEHAVIOR OF
(c) E 3 = (S, 0,P) is a disease free equilibrium and we can see that it always has a unique positive value:S which exists provided that, i.e. disease free equilibrium exists if pollution is under certain level.
(d) E 4 = (Ŝ,Î, 0) is a predator free equilibrium and has a unique positive value: which exists provided that (e) E 5 = (S * , I * , P * ) has a unique existence where S * ∈ 0, Λ d 1 represents a positive root of the equation We now prove the existence of S * . It can be easily verified that h 1 (S) and h 2 (S) are positive for all values of S ∈ 0, Λ d 1 under the following conditions: We have, H(0) = Λ, which is greater than zero. Also, , r 1 , d 1 and U * are all greater than zero. So, H Λ d 1 < 0. Moreover, where,

Stability Analysis.
In this section, stability analysis of all the five equilibrium points is carried out using Routh Hurwitz criterion or Lyapunov function. The Jacobian matrix of the system is given by V (E) = (a i j ) 3X3 and i, j = 1, 2, 3; where Proof: From the Jacobian matrix at E 1 = (0, 0, 0), the eigen values obtained are Theorem 2: The equilibrium point E 2 = (S, 0, 0) of the system is locally asymptotically stable provided that the following conditions are satisfied: The characteristic equation of the Jacobian matrix at E 2 is given by: which are all less than zero provided that hold. Since all the eigen values are negative, therefore E 2 = (S, 0, 0) is locally asymptotically Theorem 3 The equilibrium point E 3 = (S, 0,P) of the system is locally asymptotically stable provided that the following conditions are satisfied: (17) and other two eigen values are the roots of equation Using the given conditions, we get that A > 0 and B > 0. Using Routh Hurwitz criteria, there exist two roots of the polynomial γ 2 + Aγ + B = 0 i.e. the eigen values of V (E) at E 3 with negative real parts. Since all the eigen values are negative, therefore E 3 = (S, 0,P) is locally asymptotically stable.
Theorem 4 Assume that the predator free equilibrium point E 4 = (Ŝ,Î, 0) exists. Then it is locally asymptotically stable provided that the following conditions are satisfied: The characteristic equation of the Jacobian matrix V (E 4 ) is given by: So, using the given condition, and other two eigen values are the roots of equation Theorem 5 Assume that the interior equilibrium point E 5 = (S * , I * , P * ) of the system exists.
Let the following conditions are satisfied: Then, E 5 is locally asymptotically stable.
Proof: The characteristic equation of the Jacobian matrix V (E 5 ) is given by γ 3 +Aγ 2 +Bγ +C = 0 where, where, Using the given conditions, we get that ∆ > 0 =⇒ AB > C. Therefore, by Routh Hurwitz criteria, all the roots of the polynomial γ 3 + Aγ 2 + Bγ + C = 0 have negative real parts. Since all the eigen values are negative, therefore E 5 = (S * , I * , P * ) is locally asymptotically stable.
Proof: Consider the following function: where, C 1 ,C 2 ,C 3 are constants to be determined. It is easy to see that V (S, I, P) ∈ C 1 (R 3 , R) Now, choosing constants as C 1 = 1, which are all positive due to the local stability condition from theorem 5. Then applying the Sylvester's criterion we get that: where, From the condition given in equation (27) we get that q 11 > 0 and q 22 > 0. Then from (28), we obtain that dV dt < 0 is negative definite and hence V is a Lyapunov function with respect to E 5 . So, E 5 is globally asymptotically stable in Ω ∈ IntR 3 + that satisfies the given conditions.

DYNAMICAL ANALYSIS OF SYSTEM WITH DELAY
4.1. Existence of Hopf Bifurcation. In order to study the stability of the system with delay at interior equilibrium point, we begin with linearizing the system defined by equations (9)- (11) at E 5 and obtain the following system: where, The characteristic equation of the system at the interior equilibrium point E 5 = (S * , I * , P * ) is given by: Thus, if C3 holds then the interior equilibrium point E 5 is locally asymptotically stable at τ = 0.
Now, we separate the imaginary and real parts and get =⇒ σ satisfies the following equation: Define a function f as: and ω 3 . Consequently, (31) has three positive roots σ k = √ ω k , k = 1, 2, 3.
Thus, based on this analysis we obtain, if (C3)-(C4) hold, then the system defined by equations (9)-(11) at the interior equilibrium point E 5 is locally asymptotically stable when τ ∈ [0, τ ] and unstable when τ > τ and the system undergoes a Hopf bifrcation at the interior equilibrium point E 5 when τ = τ .

Direction of Hopf Bifurcation.
In the previous section, we obtained certain conditions under which the given system of equations undergoes Hopf bifurcation, with time delay τ = τ being the critical parameter. In this section, by taking into account the normal form theory and the center manifold theorem which were introduced by [12], we will be presenting the formula determining the direction of Hopf bifurcation and will be obtaining conditions for the stability of bifurcating periodic solutions, as well. Since Hopf bifurcation occurs at the critical value τ of τ, there exists a pair of pure imaginary roots ±ισ (τ ) of the characteristic equation (29).
Infact, we can take where, L 1 ,L 2 have already been given above, and δ (θ ) is Dirac delta function.
Next, for ψ ∈ C 1 ([−1, 0], R 3 ), we define the following : Then, the system (34) is equivalent to, Next, for ϕ ∈ C 1 ([0, 1], R 3 ), the adjoint operator A * of A can be defined as, where σ (θ ) = σ (θ , 0), andφ is the complex conjugate of ϕ. It can be verified that the operators A and A * are adjoint operators with respect to this bilinear form. Thus, since ±ισ τ are eigenvalues of A(0), they are the eigenvalues of A * as well.
We need to compute the eigenvectors of A(0) and A * corresponding to the eigenvalues ισ τ and −ισ τ , respectively.
In the remaining part of this section, using the same ideas as in [12], we now compute the coordinates in order to describe the center manifold C 0 at µ = 0. Let x t be the solution of (36) when µ = 0.
Now, on the center manifold C 0 , we have wherez andz are local coordinates for the center manifold C 0 in the direction of q * andq * . We note that W is real if x t is real and we will be considering the real solutions only.

(0)
We can clearly see that in order to determine g 21 , we will have to compute W 20 (θ ) and W 11 (θ ).
•β 2 determines the stability of the bifurcating periodic solutions, for ifβ 2 < 0, the bifurcating periodic solutions will be stable, and ifβ 2 > 0, the bifurcating periodic solutions will be unstable.
•T 2 determines the period of the bifurcating periodic solutions, for ifT 2 > 0, the period increases, and ifT 2 < 0, the period decreases.

NUMERICAL EXAMPLES
We investigate the dynamics of the system numerically. We consider a hypothetical and biologically feasible set of parameters illustrated below:  So, for the above set of data, the system has a globally asymptotically stable interior equilibrium point.  on increasing the value of β , system becomes predator free. Thus, when we take the half saturation constant β ≤ 1, the solution of the system approaches equilibrium point E 3 , which is locally asymptotically stable and due to pollutant, the susceptible population increases and predator population decreases. When 1 < β < 190, the system approaches interior equilibrium point as shown in Figure 1. When we take β ≥ 190, it leads to extinction of the predator population and due to presence of pollutant, prey population decreases. Now take eα 1 = 0.16, eα 2 = 0.16. Figure 5   Here, predator is getting extinct on decreasing conversion rates simultaneously. We observe that when the conversion rates eα 1 , eα 2 ≤ 0.08, system approaches predator free equilibrium E 4 , which is locally asymptotically stable and pollutant leads to decrease in prey population. When 0.08 < eα 1 , eα 2 < 0.16, system approaches interior equilibrium point as shown in Figure 1. When eα 1 , eα 2 ≥ 0.16 the system approaches infective free equilibrium point E 3 which is locally asymptotically stable and due to presence of pollutant susceptible prey increases as predator population decreases.  When we consider, Λ = 500, λ = 0.9103, α 1 = 0.959241, α 2 = 0.56585, β = 30, m = 0.005, e = 0.12, d 1 = 0.0534, d 2 = 00.0010, d 3 = 0.50259, r 1 = 0.937, r 2 = 0.91, r 3 = 0.090001 and U * = 0.0001; we obtain that the periodic solution is locally asymptotically stable when τ < τ 0 = 5.9 (Figure 9(q)) and at τ = τ 0 = 5.9, Hopf Bifurcation occurs (Figure 9(r)). It can be seen from figure 9(s) that if pollution is increased to U * = 0.001 then the periodic solution is stable. Thus, we can say that increase in pollution upto a certain level has stabilizing effect on the system.

CONCLUSION
In this paper, a polluted prey-predator model with disease in prey has been proposed and studied. It is assumed that the pollutant affects both the populations while only prey population is vulnerable to disease. First thing discussed was positivity and boundedness of the solutions of the system. Then we performed stability analysis i.e. local and global stability of the solutions were analyzed. Then we introduced delay in the model to make it more realistic and studied the stability of the delayed system at interior equilibrium point. The existence and direction of Hopf bifurcation was established i.e. Hopf bifurcation occurs at the interior equilibrium point after the delay crosses certain value τ . Further, numerical simulations are carried out in order to investigate that which set of parameters control the dynamic behavior of the system. The parameters chosen were hypothetical and biologically feasible. For the set of data chosen, it is observed that pollutant may not always have negative effect on existence of species, rather it could help the species to survive. On varying the half saturation parameter, if we take half saturation constant below a specific value, disease gets eliminated from the system. The solution of the system in absence of pollutant and in presence of pollutant approaches infective free equilibrium point which is locally asymptotically stable and due to presence of pollutant, the susceptible prey population increases and predator population decreases. If we take half saturation constant above a specific value it leads to extinction of predator population and due to presence of pollutant, the prey population decreases. Varying the conversion rates simultaneously, we get that increasing conversion rates above a specific value, system becomes disease free. The solution of the system in absence of pollutant and in presence of pollutant approaches an infective free equilibrium which is locally asymptotically stable. Also, in this case, due to presence of pollutant, prey population increases and predator population decreases. Decreasing conversion rates below a specific value, predator population extincts and prey population decreases. On decreasing the growth rate constant of the susceptible prey below a specific value, the system approaches predator free equilibrium i.e. it leads to the extinction of predator population. Above that specific value of growth rate constant, the system approaches to interior equilibrium point as shown in figure 1. If we take the infection rate λ and exogenous input rate q of pollutant into the environment below a specific value simultaneously, the pollutant helps in eliminating disease from the prey population and thus increasing the number of healthy prey in the system. Whereas, predator population decreases due to presence of pollutant, which is in sync with real life scenario.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.