# A Stage Structure Prey Predator Model Using Pentagonal Fuzzy Numbers and Functional Response

## Authors

• Vinothini Palani Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India. https://orcid.org/0000-0002-1761-7280
• Kavitha Krishnan Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India. https://orcid.org/0000-0002-1761-7280

## Keywords:

Equilibrium Points, Functional Response, Fuzzy Numbers, Immature Prey, Mature Prey, Prey Predator., Equilibrium Points, Functional Response, Fuzzy Numbers, Immature Prey, Mature Prey, Prey Predator.

## Abstract

In the present study, our work is focused on a prey predator model with a stage structure for the prey. The objective of the study is to find the behavior of the model using parameter values in the presence of Pentagonal fuzzy numbers. The interaction between the species is done by using functional responses, such as the Holling type I reaction for immature prey and the Crowley Martin functional response for mature prey. Prey population categorized as immature and mature prey. The idea of the problem is to construct a fuzzy theoretical method which helps us to create a model to solve this uncertainty problem by using fuzzy parameters and initial values. The mathematical formulation is developed by using a prey predator model in a fuzzy environment. The existence of equilibrium points is carried out. By using the concept of alpha cut the parameters which are used in the prey – predator model is treated as pentagonal fuzzy numbers. The parameters which they have used in the mathematical formulation can be taken as a crisp value by applying the defuzzification method. Here a Robust ranking technique method is utilized. The stability of each equilibrium point is studied by computing the Jacobian matrix and finding the eigenvalues evaluated at each equilibrium point. By utilizing the prey stage structure stability analysis is also discovered. For the dynamical system, numerical simulations are provided using a MATLAB program and also, they show the behavior of the system and determine if it is stable or unstable.

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