Extension and dynamical phases in random walkers depositing and following chemical trails

Active walker models have proved to be extremely effective in understanding the evolution of a large class systems biology like ant trail formation and pedestrian trails. We propose simple model random which modifies its local environment that turn influences motion at {\em later} time. perform direct numerical simulations discrete lattice with actively depositing chemical attracts trajectory also evaporates method look structural transitions using radius gyration for finite time walks. The extension over definite time-window shows non-monotonic change deposition rate characteristic coil-globule transition. At certain regions parameter space evaporation rates, extensions re-entrant behavior. dynamics, characterised by mean-squared displacement, deviation from diffusive scaling intermediate scales, returning behavior asymptotically. A mean field theory captures variation asymptotic diffusivity.