Numerical investigation of fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

The Assessment of Phytoplankton Dynamics in Two Reservoirs in Southern Africa with Special Reference to Water Abstraction for Inter-Basin Transfers and Potable Water Production

Toxic phytoplankton in the aquatic ecosystems are dynamic, affecting water quality. It remains unclear as to how possible toxic phytoplankton assemblages vary vertically and temporally in Swakoppoort and Von Bach dams, located in a dry subtropical desert region in central Namibia. The following variables were analyzed: pH, Secchi depths, turbidity, water temperature, total phosphorus, orthophosphate, chlorophyll-a, phytoplankton cells, and water depths. Cyanobacteria dominated the phytoplankton community in the autumn, winter and spring (dry) and summer (wet) seasons, at all the depth ranges in both dams. Microcystis dominated the vertical and temporal dynamics, followed by Dolichospermum. In the dry seasons, higher cyanobacteria cell numbers were observed in comparison to the rainy season in both dams. Spring blooms of cyanobacteria were evident in the Von Bach Dam while autumn and spring cyanobacteria blooms were observed in the Swakoppoort Dam. In the Swakoppoort Dam, the preferable depth ranges for toxic cyanobacteria species were at 5 to 10 m while in the Von Bach Dam at 0 to 5 m range. The findings of the current study indicate that the traditional selective withdrawal of water in the two dams should be performed with vertical and temporal dynamics of possible toxic cyanobacteria accounted for to aid the abstraction of water with the lowest possible toxic phytoplankton numbers, which could lower the public health risk.

Dynamics of a stochastic phytoplankton-toxic phytoplankton-zooplankton system under regime switching

In this paper, a stochastic phytoplankton-toxic phytoplankton-zooplankton system with Beddington-DeAngelis functional response, where both the white noise and regime switching are taken into account, is studied analytically and numerically. The aim of this research is to study the combined effects of the white noise, regime switching and toxin-producing phytoplankton (TPP) on the dynamics of the system. Firstly, the existence and uniqueness of global positive solution of the system is investigated. Then some sufficient conditions for the extinction, persistence in the mean and the existence of a unique ergidoc stationary distribution of the system are derived. Significantly, some numerical simulations are carried to verify our analytical results, and show that high intensity of white noise is harmful to the survival of plankton populations, but regime switching can balance the different survival states of plankton populations and decrease the risk of extinction. Additionally, it is found that an increase in the toxin liberation rate produced by TPP will increase the survival change of phytoplankton, while it will reduce the biomass of zooplankton. All these results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton system in randomly disturbed aquatic environments.

Spatiotemporal Complexity Analysis for a Space-Time Discrete Generalized Toxic-Phytoplankton-Zooplankton Model with Self-Diffusion and Cross-Diffusion

In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system.