In this paper, we propose a new model of spatio-temporal dynamics concerning the tritrophic reaction-diffusion system by introducing Phytoplankton and Zooplankton. We recall that the phytoplankton and zooplankton species are the basis of the marine food chain. There is prey in each marine tritrophic system. The main objective of this work is to control this species's biomass to ensure the system's sustainability. To achieve this, we determine an optimal control that minimizes the biomass of super predators.
The generalized activator-inhibitor model with cubic nonlinearity, in which the classical Laplacian is replaced by fractional operator has been studied. The fractional operator reflects the nonlocal behavior of superdiffusion. A spatially homogeneous, time independent solution has been found and its linear stability was studied. We have also performed a weakly nonlinear analysis and obtained a system of amplitude equations that are the basis for analysing pattern formation as well as parameter regimes for which various steady-state patterns would exist.
The nonlinear dynamics in generalized activator-inhibitor systems with space fractional derivatives is studied. As an example, the Brusselator model and the reaction–diffusion model with cubic nonlinearity, in which the classical spatial differential operators are replaced by their fractional analogues, are considered. The fractional operator reflects the nonlocal behavior of superdiffusion. The spatially homogeneous, time independent solution has been found for each system. We have also studied its linear stability and determined instability conditions of both Hopf and Turing.