Anuraj Singh

EFFECTS OF DELAY AND SEASONALITY ON TOXIN PRODUCING PHYTOPLANKTON–ZOOPLANKTON SYSTEM

A dynamical model for toxin producing phytoplankton and zooplankton has been formulated and analyzed. Due to gestation of prey, a discrete time delay is incorporated in the predator dynamics. The stability of the delay model is discussed and Hopf bifurcation to a periodic orbit is established. Stability and direction of bifurcating periodic orbits are investigated using normal form theory and center manifold arguments. Global existence of periodic orbits is also established. To substantiate analytical findings, numerical simulations are performed. The system shows rich dynamic behavior including chaos and limit cycles. The influence of seasonality in intrinsic growth parameter of the phytoplankton population is also investigated. Seasonality leads to complexity in the system.

Controlling chaos in a food chain model

In this paper, different mechanisms are used to suppress the chaos in a food chain model. The control is applied to the chaotic system so as the controlled system admits a stable attractor which may be an equilibrium point or a limit cycle. The bounded feedback is used to achieve the stabilization of unstable fixed point of the uncontrolled chaotic system. Delayed feedback control is used to control the chaos to periodic orbits. Numerical results substantiate the analytical findings.

Stabilization of Prey Predator Model via Feedback Control

In this paper, the effect of feedback linearization in Leslie–Gower type prey-predator model with Holling-type IV functional response is investigated. It is shown that the closed loop system may be stabilized using either approximate or exact linear approach. The former approach uses a linear control variable to provide a feedback linearization law whereas in latter approach, state space coordinates are suitably changed. Using this feedback control, a complex non-linear system is reduced to a linear controlled system that yields a globally asymptotically stable equilibrium point. Finally Analytical findings are validated through numerical simulations.

The dynamics of a delayed predator-prey model with state dependent feedback control

A delayed prey‐predator model with state‐dependent impulses is investigated. The sufficient conditions of existence and stability of semi‐trivial solution and positive period‐1 solution are obtained by using the Poincare map and analogue of the Poincare Criterion. The qualitative analysis shows that the positive period‐one solution bifurcates from the semi‐trivial solution through a fold bifurcation. The complex dynamics including chaos is obtained and numerical simulations substantiate the analytical results.

Synchronization of Chaos in a Food Chain

The three species ratio‐dependent food chain (Holling‐Tanner type) model is investigated by Gakkhar and Naji is known to have chaotic behavior for a choice of parameters. An attempt has been made to synchronize the chaos in the model using bidirectional coupling. Numerical simulations are presented to demonstrate the effectiveness and feasibility of the analytical results. Numerical results show that for higher value of coupling strength, chaotic synchronization is achieved. Chaos can be controlled to achieve stable synchronization in natural systems.