Nonlinear Dyn DOI 10.1007/s11071-014-1784-4 ORIGINAL PAPER Stability and bionomic analysis of fuzzy parameter based prey–predator harvesting model using UFM D. Pal · G. S. Mahapatra · G. P. Samanta Received: 5 April 2014 / Accepted: 22 October 2014 © Springer Science+Business Media Dordrecht 2014 Abstract In this paper, a novel concept of fuzzy prey– predator model is introduced by considering the impre- cise nature of the biological parameters. We consider the imprecise biological parameters as a form of trian- gular fuzzy number in nature. These imprecise parame- ters first transform to the corresponding intervals and then using interval mathematics the related differen- tial equation is converted to two differential equations. Then using utility function method, the converted dif- ferential equations is changed to a single differential equation. The possibility of existence of both biological and bionomic equilibrium is presented. We obtain the conditions of local and global stability under imprecise- ness. We also study the optimal harvesting policy and derive the optimal solution under imprecise biological parameters. Lastly, numerical examples are presented to the support of our proposed approach. D. Pal · G. P. Samanta Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India e-mail:pal.debkumar@gmail.com G. P. Samanta e-mail:g_p_samanta@yahoo.co.uk G. S. Mahapatra (B ) Department of Mathematics, National Institute of Technology Puducherry, Karaikal 609605, India e-mail:g_s_mahapatra@yahoo.com Keywords Prey–predator · Stability · Bio-economic · Optimal harvesting · Fuzzy set 1 Introduction Research in the area of theoretical ecology initiated by Lotka [1] and Volterra [2]. Exploitation of biological resources such as fisheries and forestries is going to be a very alarming situation for the human being. For the development of human society, it is undoubtedly essen- tial problem how to utilize the biological resources. In the last decades, ordinary differential equations is used to present this type of problem and provide some mathematical answer and explanations [3–17]. How- ever, it is notable that the biological parameters used in the differential equations are not always fixed. In real world, almost every community is imbedded in periodically varying environments. Temperature vari- ations strongly influence the reproduction rate of bac- teria during the day and tide cycles regulate migra- tion rates of numerous species in aquatic and terres- trial ecosystems; again light intensity controls photo- synthesis during the seasons. It is therefore quite nat- ural to try to identify role of the impreciseness of the biological parameters in the behavior of population communities. In fact, many parameters may oscillate simultaneously with the periodically varying environ- ments in real-world ecosystems. They are also vary- ing due to both nature and human society, such as fire, earthquake, climate warming, financial crisis. There- 123