Role of fear factor in a two-prey one-predator model: comparison between crisp and fuzzy environment

Abstract

A prey–predator model with fear effect on two prey species in presence of predator and the intraspecific competition between two prey species has been considered first time in crisp and fuzzy environment. To incorporate the natural imprecision present in the environmental factors, some parameters are considered as fuzzy numbers. The positivity, boundedness and the permanence of the solutions of the fuzzy system have been established here. The existence and local stability of the equilibrium points have also been discussed. In this article, we have mainly focussed on our study to examine the effect of fear on the stability of the system in crisp as well as fuzzy environment. For this reason, Hopf bifurcation analysis has been done with respect to fear factor in the crisp as well as fuzzy environment. These results are also verified numerically from computer simulation of the system.

Keywords:

• Prey–predator model
• fear effect
• fuzzy number
• signed distance method
• stability analysis
• Hopf-bifurcation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

In this article, the studies of Ritwick Banerjee is funded by Indian Institute of Engineering Science and Technology, Shibpur, under institute fellowship.

Notes on contributors

Soumya Das

Soumya Das has completed his BSc degree from Calcutta University in 2014 and MSc degree from Calcutta University in 2016. He has qualified Gate exam in 2018. He is pursuing PhD degree from the Department of Mathematics, IIEST Shibpur, Howrah. His area of research interest is Mathematical Biology.

Suvankar Biswas

Suvankar Biswas has graduated from A. P. C. College under West Bengal State University in 2011. He has completed his post graduation from Indian Institute of Engineering Science and Technology, Shibpur (Formerly, Bengal Engineering and Science University, Shibpur.) in 2013. He has been awarded by Arun Chandra Mitra Memorial Medal for securing highest marks among the candidates of the of the Master of Science examination, 2013 conducted under the Faculty of Basic and Applied Sciences, Bengal Engineering and Science University, Shibpur. He has been awarded by Prof. S. C. Dasgupta Gold Medal for securing highest marks among the candidates of the Master of Science in Applied Mathematics examination, 2013 conducted under the Faculty of Basic and Applied Sciences, Bengal Engineering and Science University, Shibpur. He has completed his Ph.D. on “Studies on Differential and Integral Equations in Fuzzy Environment” from Indian Institute of Engineering Science and Technology, Shibpur in 2018. Now he is working as an Assistant Professor at Indira Gandhi National Open University,New Delhi – 110068, India. His area of research interest are Numerical Analysis, Differential equation, Integral Equation, Integro-Differential Equation, Fuzzy Mathematics, Mathematical Biology.

Ritwick Banerjee

Ritwik Banerjee has completed his BSc degree from Calcutta University in 2014 and MSc degree from Calcutta University in 2016. Now, he is a research scholar in the Department of Mathematics, IIEST Shibpur, Howrah. His area of research interest is Mathematical Biology.

Pritha Das

Pritha Das has completed her BSc and MSc degrees from Jadavpur University in 1987 and 1989, respectively. She has completed her MSc degree with specialisation in Bio-Mathematics. Also, she has completed PhD degree from Jadavpur University in 1996. Now, she is working as a professor in the Department of Mathematics, IIEST Shibpur, Howrah. Her area of research interest are Mathematical Biology, Ecological Modeling, Neural Networks, Nonlinear Data Analysis, Nonlinear Dynamics. She has many published papers in many reputed journals.