In this paper we have analyzed the stability and Hopf-bifurcation behaviors of a multi-delayed two-species competitive system affected by toxic substances with imprecise biological parameters. We have exercised a method to handle these imprecise biological parameters by using parametric form of interval numbers. We have studied the feasibility of various equilibrium points and their stability. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation due to variation in the delay parameters which lead to Hopf-bifurcation. Numerical simulations with a hypothetical set of data have been done to support the analytical findings.

A statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.

In this article, a two prey - one predator model has been studied where two prey species are competitive in nature and also uses toxic substances for own existence. Biologically well posedness of the model system has been shown through positivity and boundedness of solutions. Existence criterion and stability analysis of the non-negative equilibrium points have been discussed. The sufficient conditions for existence of Hopf bifurcation and stability switches induced by delay are investigated. The direction and the stability criteria of the bifurcating periodic solutions are determined with the help of the normal form theory and the center manifold theorem. Numerical simulations are performed to illustrate the theoretical analysis results.