Nonlinear Dyn DOI 10.1007/s11071-016-3326-8 ORIGINAL PAPER Dynamic behaviour of a reaction–diffusion predator–prey model with both refuge and harvesting Lakshmi Narayan Guin · Sattwika Acharya Received: 23 September 2016 / Accepted: 27 December 2016 © Springer Science+Business Media Dordrecht 2017 Abstract An appropriate mathematical structure to describe the population dynamics is given by the par- tial differential equations of reaction–diffusion type. The spatiotemporal dynamics and bifurcations of a ratio-dependent Holling type II predator–prey model system with both the effect of linear prey harvesting and constant proportion of prey refuge are investigated. The existence of all ecologically feasible equilibria for the non-spatial model is determined, and the dynam- ical classifications of these equilibria are developed. The model system representing initial boundary value problem under study is subjected to zero flux boundary conditions. The conditions of diffusion-driven instabil- ity and the Turing bifurcation region in two parameter space are explored. The consequences of spatial pat- tern analysis in two-dimensional domain by means of numerical simulations reveal that the typical dynamics of population density variation is the formation of iso- lated groups, i.e. spotted or stripe-like patterns or coex- istence of both the patterns or labyrinthine patterns and so on. The results around the unique interior feasible equilibrium solution indicate that the effect of refuge and harvesting plays a significant role on the control of spatial pattern formation of the species. Finally, the L. N. Guin (B ) · S. Acharya Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal 731 235, India e-mail: guin_ln@yahoo.com S. Acharya e-mail: sattwikacharya@gmail.com paper ends with a comprehensive discussion of biolog- ical implications of our findings. Keywords Interacting populations · Prey refuge · Prey harvesting · Stability and instability · Self- and cross-diffusion · Reaction–diffusion predator–prey model · Spatiotemporal pattern formation Mathematics Subject Classification 34A12 · 35B36 · 35G31 · 35K57 · 35K61 · 37B25 · 65N06 · 70K50 · 91B76 · 93B18 · 93D20 1 Introduction The word “ecology” was first coined in 1866 by Ernest Hackel, which refers to the “totality pattern of relation- ship between organism and its environment” and an ecosystem consisting of biotic society and the abiotic features on which the individuals depend. Ecological systems are open in which the interaction between the module parts is nonlinear and the significant variety of dynamical behaviour by several interacting populations has inspired an immense attention in the development of mathematical models for a number of ecological sys- tems. In the field of ecological modelling, the simplest form of ecological model for predator–prey dynamics is a two-species model, where a single predator predates upon a single prey. Studies of ecological models are enlightening in accepting predator–prey interactions in these systems as an outcome, and predator–prey mod- 123