1.Introduction: 2. The Model. Key words: Prey, Predator, Seasonality, Stability, Bifurcations, Chaos.

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1 Dynamical behavior of a prey predator model with seasonally varying parameters Sunita Gakkhar, BrhamPal Singh, R K Naji Department of Mathematics I I T Roorkee,47667 INDIA Abstract : A dynamic model based on Leslie - Gower type prey predator model is analyzed. The effects of seasonality on the dynamical behavior of two-dimensional continuous time dynamical system are investigated. The seasonal variations are considered in different parameters. Their variation may be synchronous or asynchronous. The bifurcation diagrams are obtained for different parameters of the model after intensive numerical simulations. The seasonal variations in carrying capacity of the prey due to seasonally varying environment may give rise to complex dynamics of the system. Further, the model with asynchronous seasonal variations in growth rate of two species could exhibit chaotic dynamics for realistic and biologically feasible parametric values. Key words: Prey, Predator, Seasonality, Stability, Bifurcations, Chaos..Introduction: The effects of seasonality on parameters of the interacting biological species have been reported in literature. Without seasonal variations the two-species non-linear system can either have limit cycle or stability. However, due to seasonal variation the behavior of the system becomes complex and can depict chaos [,, 4,5]. The seasonality may affect one or more parameters of the system. The periodic variations in different parameters of real ecological models may not be synchronous. The maximum influence of periodic variation may occur at different times on different parameters. That is, the periodic variations in different parameter are in different phases. Accordingly, the seasonality in two different parameters will be considered simultaneously with different phase angles between them. The model, we study here, is based on a modified version of the Leslie-Gower scheme. The seasonal variations in the carrying capacity of the habitat are considered. In another case Asynchronous seasonal variation is considered in two parameters: the growth rate of the prey and the predator.. The Model Consider a prey-predator system for which it is assumed that the prey X grows logistically in the absence of predation. The predator X consumes the prey according to Holling type-ii functional response. The interaction between species X and its prey X has been modeled by modified Leslie-Gower scheme in which the loss in a predator population is proportional to the reciprocal of per-capita availability of its most favorite food. The state equations are written as follows: ( AX X d X = d T r X X K + BX C X d X d T = r X ( D X + E with X ( 0 0 and X ( 0 0

2 Where X and X represent the population densities at time, r represent the growth rate of prey, K is the carrying capacity in the absence of predation, A is the search rate, /B is the half saturation constant, r describes the growth rate of predator X, which is assumed to be a sexually reproducing species. The square term signifies the fact that mating frequency is directly proportional to the number of males as well as to that of females. The last term in the right-hand side measures the loss of predator population due to rarity of its favorite food X, the constant E normalizes the residual reduction in the predator population X due to severe scarcity of its favorite food X []. In order to reduce the number of parameters in system ( from 8 to 4, we assume the following non-dimensional variable and parameters: t = r T, x = X K, y = A X r, w = B K, w = r A, w = C E A w 4 = D K E. Then system ( takes the nondimensional form ( x ( x y ( + w x y w y ( w d x d t = x, ( d y d t = w x.. Analysis: 4 +. Lemma: In the domain D = {(x,y: 0 < x < x, 0 < y < y}, the system ( is Kolmogorov for w /( + x < w < w /( + x < w, w y /( + x <. (. Theorem The Kolmogorov system ( has bounded solution in the domain D. Proof: Since dx/dt x( x, we have 0 < x < x. Consider n(t= x(t + αy(t α = w /( + x w Or dn / dt x( x + α y ( w w /( + x Or dn / dt + n( t P + / 4, Where max{x + x(-x} = P and max{ α + α ( w w /( + x} = / 4. This shows the bounded ness of n(t implying the bounded ness of y(t in the domain D. There are three non-negative equilibrium points for the Kolmogorov system (: E ( 0,0, E (,0, E = ( x, y ; x = = ( w w w = ; ( x ( w x y = + Note that x is not a feasible equilibrium point of the system with logistic growth for x. Further, x < ( w w w < (4 The equilibrium point E under (4 will be the boundary point of the domain D. The equilibrium points E and E always exist and are non hyperbolic. The Linear stability analysis of the system ( about E is carried out. It is observed that it is locally asymptotically stable under the condition ( w w < x (5 It is observed from ( that in the domain D, condition (4 is not satisfied. Therefore the system will admit a limit cycle in the domain. It is further observed that the system ( admits Hopf bifurcation for ( w w =. ( w w w. (6

3 . Theorem : The System is globally asymptotically stable in the domain D whenever the following is satisfied: ( w w < x. (7 Proof: Consider the following positive definite function: V(X, Y = C[ X x log( + X / x ] ; + D[ Y y log( + Y / y ] x = X + x, x = X + x Here C and D are arbitrarily chosen positive constants. Computing dv/dt and using (, we get dv / dt = CX [ + w x w + w CC XY C = C /( + w x; C = D w y /( + x Further, dv/dt C X [ + w x w + w [ C /( + w x D wm ] XY Choosing arbitrary constants C and D such that C /( + w x = D wm, then dv/dt C X [ + w x w + w It is observed that dv/dt is negative definite provided ( w w < x Therefore V is a Liapunov function. Thus the system ( is globally asymptotically stable under (6. The system may admit a limit cycle in the domain D. 4. The Forced System First we consider the carrying capacity K to be varying periodically due to seasonal variation in the available resources for the prey: K = K + w8 sin ( ΩT (8 T is the periodicity, K w 8 is the amplitude and Ω is the frequency of sinusoidal perturbations in K. Therefore, using z = θ t, and (8, the original system ( can be written in non-dimensional form as: x d t = x x /( + w sin z x y + w ( 8 ( ( x y w y ( w d d y d t = w ( 4 x + d z d t = θ, with z ( 0 = 0. (9 In another case, we consider the intrinsic growth rates r i ( i =, in system ( as periodically varying functions of time due to seasonal variations. For these parameters sinusoidal perturbations are used with the same periodicity T. Therefore, the seasonality is superimposed as follows: r = r ( + ε sin ( Ω T ; r = r ( + ε sin ( ΩT + φ. (0 Here r i ( i =, are the average values of the forced intrinsic growth rates r i ( i =, respectively. The parameters ε i ( i =, represent the degree of seasonality; ri ε i are the magnitude of the perturbations in r i respectively and Ω is the angular frequency of the fluctuations caused by seasonality. Since r i ( i =, are assumed to be positive, therefore 0 ε i. Finally the parameter φ, where 0 φ π, can be interpreted as a difference in phase angle between the seasonality in the intrinsic growth rate of the prey and the predator. Therefore, according to (0, the original system ( can be written in nondimensional form as: d x d t = x ( x ( x y ( + w x + ε xsin ( z d y d t = w z d t = θ y ( w y ( w x + + λ y sin ( z + φ d, with ( 0 = 0 4 z. (

4 5. Results Of Numeric Simulations: Numerical simulations are carried out to study the global dynamics of the preypredator systems (9 and ( with seasonality. The solution of the system with initial conditions in the first octant is obtained numerically for biologically feasible ranges of parametric values. The parametric values are selected in the domain where the unforced system is Kolmogorov. The system being nonlinear and three dimensional, variety of behavior in the solution are expected in contrast to the corresponding twodimensional system without seasonality. The bifurcation diagram provides a summary of essential dynamical behavior of the system. Indeed the points that are plotted will represent either fixed or periodic sinks or other attracting sets including chaos. It shows the birth, evolution, and death of the attracting sets. The term bifurcation refers to significant changes in the set of fixed or periodic points or other sets of interest. The bifurcation diagram for the system (9 are drawn in Fig. for the following choice of parameters: w =.0, w = 0., w = 0.4, =.5, 0.5 < w8 < 0. ( For a fixed value of the key parameter w 8, the local maximum values of prey species x is plotted for a range of t after removing the transient effects. From Fig, it is clear that the twodimensional prey predator model (9 has more complex dynamical behavior than what was expected from a twodimensional prey predator model without seasonal variations. Fig.Bifurcation Diagram for System (9 Number of bifurcation diagrams is obtained for system ( in three different cases φ = 0, π and π respectively.. In each of them, for a fixed value of the key parameter, the local maximum values of prey species x is plotted for a range of t after removing the transient effects. Then increment the key parameter and begin the procedure again. We first assume the critical parameter as w. Fig. shows the bifurcation diagram for φ = 0 as a function of w in the range keeping other parameters fixed as w = 0., w = 0.4, w =.5,.8 w ε = 0.5, λ = 0.,, θ = 0.5 ( It is evident from Fig (a that the dynamics of the system is quite complex. It ( shows existence of limit cycles, periodic doubling leading to chaos. It is evident that the region of chaotic activity is reduced drastically for the case φ = π Narrow regions of chaotic behavior are observed. However, for the φ = π, the chaotic region again becomes denser. The Shifts in windows are observed. 4

5 communities, Bull. Math. Biology, 99, Vol. 55 pp [5] Sabin G.C.W, Summer D. Chaos in a periodically forced predator-prey ecosystem model, Math. Biosciences, 99Vol. pp. 9-, Fig. Bifurcation diagram for (a φ = 0, (b φ = π /, (c φ = π 6. References [] Aziz-Alaoui M.A. Study of a Leslie- Gower-type tri-trophic population model Chaos, Solitons & Fractals, Vol. 4, 00 pp. 75-9, [] Gakkhar S. and Naji R. K. Chaos in seasonally perturbed ratio-dependent prey-predator system Chaos, Solitons & Fractals,, 00, Vol. 5, pp [] Gakkhar S. And Naji R. K. Seasonally perturbed prey-predator system with predator-dependent functional response Chaos, Solitons & Fractals, Vol. 8, 00, pp ,. [4] Rinaldi S., Muratori S. and Kuznetsov Yu. A. Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey