A ratio-dependent predator-prey model with infection in prey population is proposed and analysed. The behaviour of the system near the biological feasible equilibria is observed. The conditions for which no trajectory can reach the origin following any fixed direction or spirally are worked out. We investigate the criteria for which the system will persist. It is observed that the introduction of an infected population in the classical ratio-dependent predator-prey model may act as a biological control to save the population from extinction.

In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors x and y of a commutative ring R are adjacent whenever there exist two nonnegative integers n and m such that x n y m = 0 with x n ̸ = 0 and y m ̸ = 0. This yields an extension of the classical zero divisor graph Γ(R) of R , which will be denoted by Γ(R). First we distinguish when Γ(R) and Γ(R) coincide. Various examples in this context are given. We show that if Γ(R) ̸ = Γ(R) , then Γ(R) must contain a cycle. We also show that if Γ(R) ̸ = Γ(R) and Γ(R) is complemented, then the total quotient ring of R is zero-dimensional. Among other things, the diameter and girth of Γ(R) are also studied.

Abstract. Let R be a commutative ring with zero-divisors Z(R). The extended zero-divisor graph of R, denoted by Γ(R), is the (simple) graph with vertices Z(R) * = Z(R)\{0}, the set of nonzero zero-divisors of R, where two distinct nonzero zero-divisors x and y are adjacent whenever there exist two non-negative integers n and m such that x n y m = 0 with x n = 0 and y m = 0. In this paper, we consider the extended zero-divisor graphs of idealizations R ⋉ M (where M is an R-module). At first, we distinguish when Γ(R ⋉ M ) and the classical zero-divisor graph Γ(R ⋉ M ) coincide. Various examples in this context are given. Among other things, the diameter and the girth of Γ(R ⋉ M ) are also studied.

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The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.