Abstract
In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.
Highlights
Phytoplankton is the main producer of aquatic ecosystems and forms the basis of food webs [1]. e growth of phytoplankton can determine the behavioral space of plankton directly and even a ects the development of the whole aquatic ecosystem
We mainly study the effects of time delay and, diffusion coefficients on the spatiotemporal dynamics of phytoplankton growth. e conclusions of theoretical studies and the results of numerical simulations can show that: (1) time delay τ does not affect the stability of the stable fixed point E0 (I/q, 0), but the time delay may affect the whole process; (2) when time delay exists and is greater than a certain critical value τ0, the time delay can lead to the instability of the stable fixed point E∗, and form a cloud Neimark–Sacker pattern through Neimark–Sacker instability; (3) when d1 ≠ d2, by changing the value of d1 or d2 to make all the eigenvalues of the diffusion term are greater than 1, Turing bifurcation occurs
A band-shaped Turing pattern can be formed. Both the continuous model and the discrete model in this research are reaction-diffusion models, and they have the same functional responses. e differences between the two models are as follows: in the continuous model, reaction process and diffusion process occurs at the same time, while in the discrete model, the occurring order of reaction process and diffusion process can be varied, to be exactly in this research, diffusion process
Summary
Phytoplankton is the main producer of aquatic ecosystems and forms the basis of food webs [1]. e growth of phytoplankton can determine the behavioral space of plankton directly and even a ects the development of the whole aquatic ecosystem. E growth of phytoplankton can determine the behavioral space of plankton directly and even a ects the development of the whole aquatic ecosystem. After the Monod model was proposed, it was widely used in the study of phytoplankton growth dynamics [13,14,15,16,17,18,19] and found that the Complexity. Studies have shown that it is feasible to describe the quantitative relationship between phytoplankton growth rate and nutrient salt concentration with Michaelis–Menten model under steady-state conditions [24]. Erefore, it is of great significance to further reveal the formation mechanism of space-time self-organization by discretizing the reaction-diffusion equation in space and time. We illustrate the results that can be obtained by the discrete model in the research are far more than those by the continuous model in [46]
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