Analysis of a plankton–fish model with external toxicity and nonlinear harvesting.

In this paper, we consider a three species plankton–fish system that incorporates external toxicity and nonlinear harvesting. We consider that the growth of species are affected directly or indirectly by an external toxic substance and the feeding of the predator on the affected prey is considered as Holling type II functional response. All the possible biological feasible equilibrium points are determined analytically as well as numerically and performed stability analysis around these equilibrium points. It is shown that the system undergoes for Hopf bifurcation when the growth rate of prey passes some threshold value. Furthermore, Pontryagin's maximum principle has been applied to obtain optimal control of harvesting to maximize the benefit as well as the conservation of the ecosystem. We perform numerical simulations to justify and illustrate our analytical results. Some numerical tools such as phase portraits, time evaluation and bifurcation diagrams are presented to ensure the complex dynamics in the system. Period doubling cascade route to chaos is examined and validated through the numerical calculation of Lyapunov exponents and sensitivity analysis. [ABSTRACT FROM AUTHOR]