Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Abstract

The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R 2 + are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.

Stability and Bifurcation: Discrete Differential Algebraic Planar Model with Square Root Response

“…In recent years, [1,5,8,10,11] due to the following three reasons the discrete-time predator prey models have come to the fore front: First one is, in comparision to continuous time models, the discrete-time models are more suitable to describe the species with non overlapping generations and the second one is discrete models exhibits more complex and rich dynamics. Finally, we get accurate numerical solutions from discrete-time models.…”

Stability, Bifurcation, Chaos : Discrete Prey Predator Model with Step Size

“…In recent years, [1], [2], [8], [9] population dynamics of discrete-time have come to the fore front due to the following reasons: Firstly, they are more appropriate than continuous time systems to define populations with non overlapping generations. Secondly, they yield more rich and complex dynamical behavior than continuous-time systems.…”

“…S is non-hyperbolic. Now, we present the simulations and numerical evidence to understand the dynamical nature of system (2). For example, consider the parameter values = 0.6; = 0.9; = 2.5 h , the interior steady state of the system 2…”

A dynamically consistent nonstandard finite difference scheme for a predator–prey model

“…Similarly, Liu and Xiao [5] presented complex dynamics for a discrete Lotka-Volterra system after implementation of Euler method. For a similar type of investigations related to predator-prey systems the interested reader is referred to [6][7][8][9][10][11][12][13][14][15][16][17]. All these studies reveal that the discrete predator-prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts.…”

“…Suppose that(10),(11) and(17) hold true and L = 0, then the positive steady state (x * , y * ) of model (3) undergoes a Neimark-Sacker bifurcation when the bifurcation parameter s varies in a small neighborhood of s 1 defined in(10). Furthermore, if L < 0, then an attracting invariant closed curve bifurcates from the equilibrium point for s > s 1 , and if L > 0, then a repelling invariant closed curve bifurcates from the equilibrium point for s < s 1 .…”