This paper deals with a construction and an analysis of harvested predator-prey model with ratio-dependent response function and prey refuge. The harvesting is applied on both of predator and prey because they have a commercial value, while the prey refuge is applied in accordance with the fact that prey has a refuge instinct enabling to reduce the possibility of prey catching rate. According to the analysis, there are four equilibrium points which are stable under certain conditions, namely the prey extinction, the predator extinction, and two coexistence points. Finally, numerical solutions are presented not only to illustrate each equilibrium points but also to illustrate the effects of prey refuge.

No abstract

No abstract

No abstract

A distance vertex irregular total k-labeling of a simple undirected graph G = G(V, E), is a function f : V (G) ∪ E(G) −→ {1, 2, . . . , k} such that for every pair vertices u, v ∈ V (G) and u ̸ = v, the weights of u and v are distinct. The weight of vertex v ∈ V (G) is defined to be the sum of the label of vertices in neighborhood of v and the label of all incident edges to v. The total distance vertex irregularity strength of G (denoted by tdis(G)) is the minimum of k for which G has a distance vertex irregular total k-labeling. In this paper, we present several results of the total distance vertex irregularity strength of some corona product graphs.

Let H= (T,S), be a finite simple graph, T(H)= T and S(H)= S, respectively, are the sets of vertices and edges on H. Let σ:T∪S→1,2,· · · ,k, be a total k-labeling on H and wσ(x), be a weight of x∈T while using σ labeling, which is evaluated based on the total number of all vertices labels in the neighborhood x and its incident edges. If every x∈T has a different weight, then σ is a distance vertex irregular total k-labeling (DVITL). Total distance vertex irregularity strength of H (tdis(H) is defined as the least k for which H has a DVITL. Our research investigates the DVITL of the path (Pr) and cycle (Cr) graphs. We establish a lower bound and then calculate the precise value of tdis(Pr) and tdis(Cr).

Zadeh have introduced fuzzy set in 1965 and Atanassov have introduced intuitionistic fuzzy set in 1986 in theirs paper. Now, many of researcher connecting intuitionistic fuzzy set with algebraic structure. We interested to combine some concepts over intuitionistic fuzzy set, module of a ring, t-norm, t-conorm, and intuitionistic anti fuzzy. In this paper, we discusses about intuitionistic anti fuzzy module t-norm and t-conorm (IAFMTC) and their properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. We have investigate all properties of IAFMTC.

Forest with two components 2-regular graph Join product graph A graph is called super edge-magic if there exists a bijection ∶ () ∪ () ⟶ {1, 2, ⋯ , | ()| + | ()|}, where (()) = {1, 2, ⋯ , | ()|}, such that () + () + () is a constant for every edge ∈ (). Such a case, is called a super edge magic labeling of. A bipartite graph with partite sets and is called consecutively super edge-magic if there exists a super edge-magic labeling with the property that () = {1, 2, ⋯ , | |} and () = {| | + 1, | | + 2, ⋯ , | ()|}. The super edge-magic deficiency of a graph , denoted by (), is either the minimum nonnegative integer such that ∪ 1 is super edge-magic or +∞ if there exists no such. The consecutively super edge-magic deficiency of a bipartite graph , denoted by (), is either the minimum nonnegative integer such that ∪ 1 is consecutively super edge-magic or +∞ if there exists no such. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph 2 3 ∪ and join product of 1, ∪ with an isolated vertex.