Modeling and analysis of inhibitory effect in plankton-fish model: Application to the hypertrophic Swarzedzkie Lake in Western Poland

The hypertrophic Swarzedzkie Lake, Poland, is characterized by high species diversity, abundance and biomass of both phytoplankton and zooplankton. A relationship has been observed only with size groups and/or living forms of a few taxonomical groups of phytoplankton and a negative effect is found in zooplankton growth rate due to these groups of phytoplankton. This paper is devoted to mathematical study of such influences of toxic grouped phytoplankton on zooplankton and fish populations. The tri-trophic reaction-diffusion model incorporates Holling type III and Monod-Haldane functional response for three state variables namely, phytoplankton, zooplankton and fish. In order to investigate the dynamics of model, we have analyzed local and global stability of both nonspatial and spatial model systems. We have proved Hopf-bifurcation for non-spatial model and discussed maximum sustainable yield as well as optimal harvesting policy for maximizing economic gain. Different conditions for Turing instability have been obtained for spatial model. Numerical simulation is carried out for both temporal and spatial models. The temporal model is numerically analyzed with the help of phase portraits, time evaluation and bifurcation diagrams. Very beautiful Turing patterns are observed for the diffusion induced model. Influences of inhibitory effect are numerically investigated in both zooplankton and fish populations. This mathematical study suggests that inhibitory effect is able to destabilize homogeneous steady state and also able to produce chaos in plankton-fish system. (C) 2018 Elsevier Ltd. All rights reserved.