![]() ![]() It is investigated that system has unique positive steady state. A planar cubic dynamical system governed by nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction is studied. The mathematical framework of rate equations enables us to discuss steady-states, stability and oscillatory behavior of a chemical reaction. ![]() The resonance bifurcation in the discrete-time map indicates that both species coincide till order 4 in stable periodic cycles near some critical parametric values.Ĭhemical reactions reveal all types of exotic behavior, that is, multistability, oscillation, chaos, or multistationarity. The results in this manuscript reveal that the dynamics of the discrete-time model in both single-parameter and two-parameter spaces are inherently rich and complex. The numerical simulation gives a wide range of periodic cycles including codim-1 bifurcation and resonance curves in the system. The different bifurcation curves of fixed points are drawn which validate the analytical findings. Using the critical normal form theorem and bifurcation theory, normal form coefficients are calculated for each bifurcation. The model undergoes fold bifurcation, flip bifurcation, Neimark–Sacker bifurcation and resonances 1:2, 1:3, 1:4 at different fixed points. Here, codim-1 and codim-2 bifurcation including multiple and generic bifurcations in the discrete model are explored. The local stability conditions of all the fixed points in the system are determined. This paper investigates bifurcations analysis and resonances in a discrete-time prey-predator model analytically and numerically as well. Finally, numerical simulations are presented to confirm the theoretical and analytical findings. The maximum Lya-punov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. The fractal dimensions of the proposed model are calculated. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. ![]() By contrast, chaotic attractors ensure chaos. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark-Sacker bifurcation around the interior equilibrium point. Moreover, bounded-ness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. Discretization is conducted by applying a piecewise constant argument method of differential equations. The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator prey model accompanied by a Holling type III functional response. Thus, this counter-intuitive phenomenon (positive effect on predator biomass) referred to as hydra effect is detected in our discrete-time system. However, we have identified a situation for which the unstable equilibrium predator biomass decreases while mean predator density increases under predator mortality. With prey harvesting, the mean predator density increases when the system exhibits nonequilibrium dynamics. The system also exhibits complicated dynamics including multistability and Neimark–Sacker bifurcation when prey (or predator) harvesting rate is varied. When the system is subjected to prey (or predator) harvesting, it stabilizes into equilibrium state. Further, we introduce prey and predator harvesting to the system. Therefore, the paradox of enrichment is prominent in our discrete-time system. The sufficient increase in the nutrient supply to the prey species may have negative effect in form of decrease in mean predator stock which leads to extinction of predator. The basins of attraction for these multistabilities show complicated structures. Multistability of different kinds: periodic–periodic and periodic–chaotic are also revealed. The transversality condition for the Neimark–Sacker bifurcation at the bifurcation point is derived using a different approach compared to the existing ones. We use bifurcation theory along with numerical examples to show the existence of Neimark–Sacker bifurcation in the system. The system undergoes a Neimark–Sacker bifurcation leading to complex behaviors including quasiperiodicity, periodic windows, period-bubbling, and chaos. First, we analyze the system by varying carrying capacity of the prey species. We investigate a discrete-time system derived from the continuous-time Rosenzweig–MacArthur (RM) model using the forward Euler scheme with unit integral step size. ![]()
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