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In this paper, we investigate the system undergoes flip and Neimark Sacker bifurcation in the interior of R+2 by using the center manifold theorem and bifurcation theory. The dynamics of this discrete time predator-pre model is investigated in the closed first quadrant of R+2 .

Let $G$ be a connected graph and $d(\mu,\omega)$ be the distance between any two vertices of $G$ . The diameter of $G$ is denoted by $diam(G)$ and is equal to $\max \{d(\mu,\omega); \\ \mu,\omega \in G\}$ . The radio labeling (RL) for the graph $G$ is an injective function $\digamma:V(G)\rightarrow N\cup \{0\}$ such that for any pair of vertices $\mu $ and $\omega \,\,|\digamma (\mu)-\digamma (\omega)|\geq diam(G)-d(\mu,\omega)+1$ . The span of radio labeling is the largest number in $\digamma (V)$ . The radio number of $G$ , denoted by $rn(G)$ is the minimum span over all radio labeling of $G$ . In this paper, we determine radio number for the generalized Petersen graphs, $P(n,2)$ , $n=4k+2$ . Further the lower bound of radio number for $P(n,2)$ when $n=4k$ is determined.

The study of complex graphics due to their charm and complexity has become a significant area of research nowadays. The aim of this paper is to visualize antifractals like tricorns and multicorns in S-iteration orbit with s-convexity. Several beautiful aesthetic patterns are presented for antipolynomial z̅m+c of complex polynomial zm + c, for m ≥ 2 to explore the geometry of antifractals.

International audience; Web conversational services are exposed to several threats in which the social context between communicating participants is manipulated. Cybercrimes based on identity misrepresentation to obtain sensitive information are on the rise. Various scams and frauds are conducted by distributing malicious content, viruses, and spam over established communication sessions. In order to maintain overall security and enhance privacy, methods of estimating trustworthiness and reputation should be built into Web calling services. In this paper, we propose "TrustCall" a reputation-based trust model for real-time Web conversational services. In our approach, the reputation of a caller is evaluated using Authenticity Trust and Behavioral Trust. Authenticity Trust describes the legitimacy of a caller by collecting recommendations from other members of the network, whereas Behavioral Trust determines a caller's popularity based on its communication behavior. Other contributions include a threat taxonomy for Web calling services, including social threats, that directly target users. A set of experiments are conducted in order to prove the feasibility and effectiveness of our model