# Chaos in a Prey-Predator Model with Infection in

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Chaos in a Prey-Predator Model with Infection in

1 Computational and Mathematical Biology Chaos in a Prey-Predator Model with Infection in Predator - A Parameter Domain Analysis Prasenjit Das, Debasis Mukherjee, Kalyan Das Abstract A prey-predator system with a disease in the predator population is a challenging issue in ecoepidemics which is proposed here and analyzed as a set of three dimensional ordinary nonlinear differential equations. The discrete analogous of the eco-epidemic model is also investigated. We observed that the system converges to a unique equilibrium point for certain prey density threshold level and beyond which stability of the system is disturbed. The results of our study suggest that discrete version of the continuous model exhibits different dynamical complexity including chaos in real situation. 1 Introduction Discrete-time population models can exhibit complex behavior and intriguing dynamics even in the simplest systems in fluctuating environment. A rigorous study of mathematical models on biology shows that discretetime models described by difference equations are more justified than the continuous-time models when the size of the population is rarely small or populations have non-overlapping generations. Further, epidemiological data for infectious diseases is collected in discrete form. Difference equation models give richer dynamics than continuous ones for infection disease model. In recent years diseases in prey-predator systems become most interesting part of research among all mathematical models. Such systems governed mainly by continuoustime models and these studies investigated stability, bifurcation, persistence etc. [3, 4, 7, 8]. However, not much work has been dealt with the stability analysis of discrete eco-epidemic models. There are many published works in mathematical models in ecology and epidemiology with difference equations for two equations [1, 2, 5]. The aim of our current study is to analyze a non-linear system of difference equations with three set of equations. Most of the eco-epidemic models consider the disease among the prey species only as infected prey is more vulnerable to predator except for those developed by Haque et.al [10], Venturino [12] and Issue 3(4), 2014 Haque [9]. In these three papers, the authors studied the spread of disease in predator population. Although parasitic models generally have a tendency to destabilize the system but study by Hilker et al. [11] suggests that predator infection into the system creates paradox of enrichment whereas its removal develops catastrophic effect. Here we establish that destabilization which does not always happen. The disease in predator population may stabilize the oscillatory prey-predator system. Another feature of disease transmission in susceptible predator population allows the prey species to recover. The effect of disease in our model increases the mortality rate of predator. The effect of disease in our model increases the mortality rate of predator which allowing the stability of the system. The present model focuses the complex dynamics of discrete version of continuous eco-epidemic model where predator population is infected. In Section 2 we present our model. Boundedness of the system is studied in Section 3 along with local and global behavior of the continuous-time model. Asymptotic behavior of discrete version is carried out in Section 4. In Section 5 numerical examples and simulations are given and we round up the paper in Section 6. 2 Computational and Mathematical Biology 2 Mathematical Model In the prey-predator system where predator is infected is described by the following equations: ⎡ ⎛ y ⎤ dx x⎞ = x ⎢r ⎜ 1 − ⎟ − ⎥ dt k 1 + x⎦ ⎠ ⎣ ⎝ ⎛ mx ⎞ dy = y⎜ − β I − c⎟ dt 1 + x ⎝ ⎠ where: 3 Qualitative Analysis of Continuous Model In this section we study the dynamical behavior of the continuous-time model. 3.1 Boundedness It has been shown that, all the solutions of system (1) are bounded in a positive octant R+3 . The boundedness of solutions of system (1) is given by the following lemma. Lemma1. Let µ = min(1,c,d) . Then all the solutions of for all positive initial values (x0 , y0 , I 0 ) ∈R+3 as t → ∞ . Issue 3(4), 2014 x̂ = (1) where x, y, I denote respectively, the population of prey species, susceptible predator species and infected predator species. We assume that the infected predators do not predate prey which is very much realistic in real situation. The coefficients r, k, m, β, c and d in model (1) are all positive constants and their ecological interpretation are as follows: r represents the intrinsic growth rate of prey and k denotes the carrying capacity of prey; m, β, c and d represent the uptake constant of the predator, effective transmission rate of disease, death rate of susceptible prey and death rate of infected predator, respectively. Specific example illustrates the above situation. Feline Immunodeficiency Virus (FIV) infects cat population that preys on island bird species. The reduction of a cat population size due to disease can help to recover bird species. Another example is common seal (Phoca Vitulina) is infected by Phocine Distemper Virus (PDV) which preys on fish. The infectious agents like the Avian pox, Newcastle disease, Influenza, Aspergillus’s fumigates etc. affect a wide range of avian species which preys on insects. ⎧ M(r + 1) ⎫ B = ⎨(x, y, I ) ∈R+3 : 0 ≤ x + y + I ≤ ⎬ µ ⎩ ⎭ Stability Analysis of Boundary and Positive Equilibrium Model system (1) has three boundary equilibriums namely (i) E0(0, 0, 0) (ii) E1(k, 0, 0) and (iii) E12 ( x̂, ŷ,0) dI = I( β y − d) dt system (1), will lie in the region: 3.2 c r{(m − c)k − c} . , ŷ = m−c km The local behavior of the model system (1) around each boundary equilibrium points is stated below: Proposition1. (i) E0 is unstable, (ii) E1 is locally stable if mk(1 + k) < c and is unstable otherwise, (iii) E12 is locally stable if ŷ < min{d / β ,r(1 + x̂)2 / k} . It is easy to see that system (1) has a unique positive equilibrium E*(x*, y*, I*) provided either the condition (C1) d/β < r, mx / (1 + x*) > c or the condition (C2) d/β = r, k > 1 is satisfied, where r(k − 1) + r 2 (k − 1)2 − 4rk x = * (d − β r) β 2r , d , β 1 mx * I* = ( − c). β 1 + x* y* = We now state the local stability criterion of E*(x*, y*, I*). Theorem 1: Assume either (C1) or (C2) holds. The equilibrium point of E*(x*, y*, I*) is locally asymptotically stable if a11 < 0 and is unstable otherwise, where x* y * r a11 = − x * + k (1 + x * )2 We now discuss the global stability of the interior equilibrium point E*. The global stability condition is derived using Lyapunov’s direct method. Global stability is given in the following theorem. Theorem 2: The interior equilibrium point E*(x*, y*, I*) of the system (1) is globally asymptotically stable for all positive initial values (x0 , y0 , I 0 ) ∈R+3 if: y* r . > k 1 + x* 3 Computational and Mathematical Biology Remark: When E* is locally asymptotically stable then it may not be globally stable, usually in cases with multiple equilibriums. p3 = p11p22 p33 + p11p23 p32 + p12 p21p33 . Further, p11 = 1 + r − 4 Qualitative Analysis of Discrete Model In this section we investigate the complex dynamics of the discrete version of the continuous model (1). The model is described as follows: x , (1 + x) my p21 = , (1 + x)2 mx p22 = 1 + − β I − c, (1 + x) p23 = β y, p12 = ⎡ ⎛ x ⎞ y ⎤ xn+1 = xn ⎢ 1 + r ⎜ 1 − n ⎟ − n ⎥ k 1 + xn ⎥⎦ ⎝ ⎠ ⎢⎣ ⎛ ⎞ mxn yn+1 = yn ⎜ 1 + − β In − c ⎟ 1 + xn ⎝ ⎠ (2) p32 = β I , I n+1 = I n (1 + β yn − d) We first determine the existence of fixed points of system (2). To determine the fixed points we have to solve the nonlinear system given by ⎡ ⎛ y ⎤ x⎞ x = x ⎢1 + r ⎜ 1 − ⎟ − ⎥ k 1 + x⎦ ⎝ ⎠ ⎣ ⎡ ⎤ mx y = y ⎢1 + − βI − c⎥ 1 + x ⎣ ⎦ I = I[1 + β y − d] By simple computation of the above algebraic system we get three non negative fixed points and the positive fixed point which are as follows: (i) E0(0, 0, 0) (ii) E1(k, 0, 0) and (iii) E12 ( x̂, ŷ,0) where c r{(m − c)k − c} , ŷ = m−c km x̂ = provided m > c and (m – c)k > c, (iv) E(x , y , I ) where x = x ,y = y ,I = I . * * * We study the asymptotic stability of the system of difference equations (2) with the help of Schur-Cohn criterion [6]. That is, asymptotic stability of the system of difference equations can be found with the eigenvalues of Jacobean matrix of system (2). 4.1 Stability Analysis of Fixed Point The characteristic equation of Jacobean matrix of system (2) at the state variable is: λ 3 + p1λ 2 + p2 λ + p3 = 0, p1 = p11 − p22 − p33 , p2 = −(p11p22 + p11p33 + p22 p33 + p23 p32 + p12 p21p33 ), Issue 3(4), 2014 y 2rx − , k (1 + x)2 p33 = 1 + β y − d. We now state the local behavior of boundary fixed points. Proposition2. (i) E0 is unstable, (ii) E1 is locally stable if mk(1 + k) < c and is unstable if mk(1 + k) > c, (iii) E12 is locally stable if: (a) ŷ mŷ r > + , k (1 + x̂)2 (1 + x̂)3 (b) 4 + 2 x̂ŷ mx̂ŷ 2rx̂ + > , and k (1 + x̂)2 (1 + x̂)3 d , β and is unstable otherwise. Using Schur-Cohn criterion we now state the following theorem: (c) ŷ < Theorem 3: Suppose −1 < a11 < 0 and |p2 − p1 p3 |< 1 − p32 then the fixed point E(x , y , I ) is asymptotically stable. 5 Numerical Examples and Simulations In this section we provide comparative study with different numerical scenarios for model systems (1) and (2) using theoretically picked parameter domain values for different situations and explaining their complex dynamical nature. All numerical simulations are generated by using MATLAB 7.4.0. 4 Computational and Mathematical Biology Figure 1a Infected Predator 1.5 1 0.5 0 2 1.5 1.5 1 1 0.5 0.5 0 Susceptible Predator 0 Prey Figure 1(a): When r = k, for the parameter values r = 3/2, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (1) has a unique equilibrium point (1, 1, 1) in the phase portrait which is globally asymptotically stable. Figure 1b 1.4 Infected Predator 1.2 1 0.8 0.6 0.4 0.2 0 2.5 2 1.5 1.5 1 1 0.5 0.5 Susceptible Predator 0 0 Prey Figure 1(b): When r = k, for the same parameter values r = 3/2, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (2) has a unique equilibrium point (1, 1, 1) in the phase portrait which is asymptotically stable. Issue 3(4), 2014 5 Computational and Mathematical Biology Figure 2a Infected Predator 0.2 0.15 0.1 0.05 0 4 3 6 2 4 1 2 0 Susceptible Predator 0 Prey Figure 2(a): When r ≪ k , for the parameter values r = 3/8, k = 5, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (1) has a unique equilibrium point (1, 1, 1) in the phase portrait which is unstable. Figure 2b 0.08 Infected Predator 0.06 0.04 0.02 0 6 6 4 4 2 2 0 Susceptible Predator 0 -2 Prey Figure 2(b): When r ≪ k , for the same parameter values r = 3/8, k = 5, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (2) has a unique equilibrium point (1, 1, 1) in the phase portrait which is also unstable. Issue 3(4), 2014 6 Computational and Mathematical Biology Figure 3a Infected Predator 1.5 1 0.5 0 2 1.5 1.5 1 1 0.5 0.5 0 Susceptible Predator 0 Prey Figure 3(a): If r < k, for the parameter values r = 1, k = 2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (1) has a unique equilibrium point (1, 1, 1) in the phase portrait which is locally stable but not globally. Figure 3b 0.05 Infected Predator 0.04 0.03 0.02 0.01 0 3 2 2 1 1 Susceptible Predator 0 0 -1 Prey Figure 3(b): When r < k, for the same parameter values r = 1, k = 2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (2) has a unique equilibrium point (1, 1, 1) in the phase portrait which is unstable. Issue 3(4), 2014 7 Computational and Mathematical Biology Figure 4a Infected Predator 1.5 1 0.5 0 1.5 3 1 2 0.5 1 0 Susceptible Predator 0 Pery Figure 4(a): When r = k with different values of β, we see that the system (1) is asymptotically stable in the phase portrait for the choice of parametric set of values r = 3/4, k = 3/4, m = 1, β = 1/2, c = 1/4, d = 1/4. Figure 4b Infected Predator 0.4 0.3 0.2 0.1 0 3 3 2 2 1 Susceptible Predator 1 0 0 Prey Figure 4(b): When r = k with different values of β, for the set of parameter values r = 3/4, k = 3/4, m = 1, β = 1/4, c = 1/4, d = 1/4, we see that system (1) has a unique equilibrium point (1, 1, 1) and a11 = 0.This implies that the eigen values of J(E*) are zero and purely imaginary which gives rise to bifurcation. Issue 3(4), 2014 8 Computational and Mathematical Biology Figure 4c Infected Predator 0.2 0.15 0.1 0.05 0 3 3 2 2 1 1 0 Susceptible Predator 0 Prey Figure 4(c): When r = k with different values of β, further the system (1) is unstable for the set of the parameter values r = 3/4, k = 3/4, m = 1, β = 1/8, c = 1/4, d = 1/4. Figure 5a Infected Predator 8 6 4 2 0 8 6 8 6 4 4 2 Susceptible Predator 2 0 0 Prey Figure 5(a): When for the set of the parameter values r = 3/2, k = 7, m = 1, β = 1/4, c = 1/4, d = 1/4, (same values of r with different values of k) we observe that increase in carrying capacity of prey, the system is more destabilized. Issue 3(4), 2014 9 Computational and Mathematical Biology Figure 5b 12 Infected Predator 10 8 6 4 2 0 10 15 5 10 5 0 Susceptible Predator 0 Prey Figure 5(b): When for the set of the parameter values r = 3/2, k = 12, m = 1, β = 1/4, c = 1/4, d = 1/4 (same values of r with different values of k) we observe that increase more in carrying capacity of prey, the system is much more destabilized. Figure 6a Infected Predator 1.5 1 0.5 0 3 1.5 2 1 1 Susceptible Predator 0.5 0 0 Prey Figure 6(a): When r > k for the set of the parameter values r = 2, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (1) has unique interior equilibrium point in the phase portrait which is locally stable. Issue 3(4), 2014 10 Computational and Mathematical Biology Figure 6b 2 Infected Predator 1.5 1 0.5 0 3 1.5 2 1 1 0.5 0 Susceptible Predator 0 Prey Figure 6(b): When r > k for the set of the parameter values r = 2, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (2) has unique interior equilibrium point in the phase portrait which is also stable. Figure 7a 2.5 Infected Predator 2 1.5 1 0.5 0 3 1.5 2 1 1 Susceptible Predator 0.5 0 0 Prey Figure 7(a): When r ≫ k for the set of the parameter values r = 3, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (1) has unique interior equilibrium point in the phase portrait which is stable. Issue 3(4), 2014 11 Computational and Mathematical Biology Figure 7b 3 Infected Predator 2.5 2 1.5 1 0.5 0 4 3 2 1.5 2 1 1 0.5 0 Susceptible Predator 0 Prey Figure 7(b): When r ≫ k for the set of the parameter values r = 3, k = 3/2, m = 1, β = 1/4, c = 1/4, d = 1/4, the system (2) has unique interior equilibrium point in the phase portrait which exhibits chaotic behavior. Parameter Symbol Value (day1) Intrinsic growth rate of prey r 3 ≤r≤3 8 Estimate Carrying capacity of prey K 3 ≤ k ≤ 12 4 Estimate Uptake constant of the predator m m=1 Estimate Effective transmission rate of disease β 1 1 ≤β ≤ 4 2 Estimate Death rate of susceptible prey c c= 1 4 Estimate Death rate of infected d d= 1 4 Estimate Source Table 1: Table showing numerical values of parameters used in the simulations. 6 Conclusion In this paper we have considered a prey-predator system with disease in the predator population which is Issue 3(4), 2014 a challenging problem in eco-epidemics in the modern world real situation. For the continuous model we observed that origin is always unstable. The trivial equilibrium point E1 remains stable as long as predator death rate exceeds a certain threshold value. The planar equilibrium point can be stable if equilibrium level of susceptible predator density must remain below a certain threshold value. For the discrete model, the dynamical behavior of origin and axial fixed point behave in the same way as in continuous model. But for planar fixed point, complex dynamical behavior may not be same as that of continuous model. If the carrying capacity of the environment lies within a certain range then stability of positive fixed point is preserved. When equilibrium level of prey density reached a certain threshold value, then the system converges to a unique equilibrium point. If the prey density at the equilibrium level remains a certain threshold- value and above, then stability of the system is disturbed. Again we have seen that the sensitive parameter, disease transmission rate plays a crucial role to shape the complex dynamical system. In brief, analytical results and numerical 12 Computational and Mathematical Biology findings suggest that discrete version of the continuous model exhibits different dynamics where as the parametric values chosen for obtaining phase portrait do not necessarily have biological meaning or applicable to specific populations. We did our numerical simulations for general cases so as to explore the dynamical behaviors of the two models. However, it would be interesting and challenging to carry out the similar analysis for a given disease in the predator and with given set of data so as to explain the predictive nature of the two types of models as tools for policy guidance and direct implementation in a complex ecosystem with a diseased in predator. Acknowledgements Authors are thankful to the reviewers for their valuable comments and suggestions to improve this paper. Appendix A Simulation Code Program Code used for continuous model: function xdot = Infection(t, y); xdot = [ y(1)*(r*( 1 - (y(1)/k))- y(2)./(1 + y(1))); y(2)*( m * y(1)./(1 + y(1))- beta * y(3)- c); y(3)*(beta * y(2)- d)]; tspan = [0 200]; y0 = [.45; .35; .2]; [t, y] = ode23('Infection', tspan, y0);p = y(:,1); w = y(:,2); f = y(:,3); plot3(p,w,f). For Discrete version of the model standard program code is used. Author Biography Prasenjit Das Baikunthapur High school Kultali, 24 Paraganas (South)-743383, India. [email protected] Debasis Mukherjee Department of Mathematics Vivekananda College Thakurpukur, Kolkata -700 063,India. [email protected] Kalyan Das (Contact Author) Department of Mathematics National Institute of Food Technology Entrepreneurship and Management Plot No. 97, Sector-56, HSIIDC Industrial Estate, Kundli-131028, Haryana, India. [email protected] Issue 3(4), 2014 References 1. Agiza, H.N., Elabbasy, E., EL- Metwally, M. H., & Elasdany, A.A. (2009). 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