Instability Tendencies in A Three Species Ecological Model with Distributed Time Delay

Objectives: The stability analysis of three species ecological model with distributed time delay was investigated. The Hopf bifurcation analysis is discussed using the weight kernel which is also taken as bifurcation parameter. Methods: This study has formulated the mathematical model using system of integro differential equations. The well posed-ness of the model is studied. The linear stability and global stability of the model is discussed. Numerical simulation is carried out using MATLab to identify the kernel weights. Findings: This study has proved that the solutions are positive and bounded in . The co-existing state is identified for the proposed model. The linear stability analysis is studied at the co-existing state and proved that the system is locally asymptotically stable. The global stability analysis is established by constructing suitable Lyapunov’s function and proved that the system is globally asymptotically stable. Hopf bifurcation analysis is studied using numerical simulation. This study identifies the range of kernel strengths, at which the system exhibits the Hopf bifurcation nature. It is concluded that the kernel weights are influential in the dynamics of the model, which leads to the instability tendencies. Novelty: The Hopf bifurcation analysis of distributed time delay model using kernel dynamics is first of its kind. So far, theoretical results have been furnished, and no significant contribution made in this area. The authors elaborated the significance of kernel weights, which are instrumental in the dynamics of the model. The critical values of kernel weights which change the behaviour of the system from stable equilibrium to unstable equilibrium. The Hopf bifurcation analysis is addressed using these weight kernels. Keywords: Prey, Predator, Competitor, Co-existing state, Stability analysis, Hopf bifurcation