Analysis of plankton-fish system with delay of fear effect

Abstract

We propose a three dimensional system including phytoplankton, zooplankton and fish, which describes the food chain between plankton and fish. We take Holling-II functional response to describe the predation ability of fish and zooplankton. For fear effect of predator exists in reality, so we introduce fear effect of fish into system, besides we think the influence of fear effect is not instantaneous, delay of fear effect has been included. We study the dynamics of plankton-fish system with delay of fear effect. First we study the positivity and boundedness of solution, which mean that all the species exist and their increase is limited by environment, then taking the delay of fear effect as parameter, we discuss Hopf bifurcation, estimate the length of delay to preserve the stability of limit cycle. At last, we give simulations about the influence of delay of fear effect on the dynamics of plankton-fish system and the sensitivity analysis of parameters on the density of plankton and fish without delay. Our results show that delay of fear effect can cause periodic oscillation and chaos for plankton-fish system, uptake rate of fish to zooplankton and death rate of fish have large impact on the density of plankton and fish.