The paper deals first with the deterministic analysis of stability and bifurcation of a non-linear model prey-predator system and then with the critical analysis of nonequilibrium fluctuation and stability of the model system under random perturbation including the comparative study of both deterministic and stochastic criteria of stability of the system.

We have presented a new axiomatic derivation of Shannon entropy for a discrete probability distribution on the basis of the postulates of additivity and concavity of the entropy function. We have then modified Shannon entropy to take account of observational uncertainty.The modified entropy reduces, in the limiting case, to the form of Shannon differential entropy. As an application, we have derived the expression for classical entropy of statistical mechanics from the quantized form of the entropy.

Abstract. We present axiomatic characterizations of both Boltzmann and Gibbs entropies together with an application.

In this paper, a delayed Holling-Tanner predator-prey model with ratio-dependent functional response is considered. It is proved that the model system is permanent under certain conditions. The local asymptotic stability and the Hopf-bifurcation results are discussed. Qualitative behaviour of the singularity (0, 0) is explored by using a blow up transformation. Global asymptotic stability analysis of the positive equilibrium is carried out. Numerical simulations are presented for the support of our analytical findings.

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The paper deals with the generalization of both Boltzmann entropy and distribution in the light of most-probable interpretation of statistical equilibrium. The statistical analysis of the generalized entropy and distribution leads to some new interesting results of significant physical importance.

The present paper deals with the effect of environmental fluctuation on a prey-predator model with stage structure for prey population. We have studied the stochastic behaviour of the model system around coexisting equilibrium point. Stochastic stability condition in mean square sense is obtained for the stage-structured model with help of a suitable Lyapunov function. Numerical simulations are carried out to substantiate the analytical findings. The main outcomes of mathematical findings are mentioned in conclusion section.