scholarly journals Nonlinear Dynamics of a Toxin-Phytoplankton-Zooplankton System with Self- and Cross-Diffusion

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Pengfei Wang ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Nan Wang ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Spatial Distribution ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Cross Diffusion ◽  
The Rich

A nonlinear system describing the interaction between toxin-producing phytoplankton and zooplankton was investigated analytically and numerically, where the system was represented by a couple of reaction-diffusion equations. We analyzed the effect of self- and cross-diffusion on the system. Some conditions for the local and global stability of the equilibrium were obtained based on the theoretical analysis. Furthermore, we found that the equilibrium lost its stability via Turing instability and patterns formation then occurred. In particular, the analysis indicated that cross-diffusion can play an important role in pattern formation. Subsequently, we performed a series of numerical simulations to further study the dynamics of the system, which demonstrated the rich dynamics induced by diffusion in the system. In addition, the numerical simulations indicated that the direction of cross-diffusion can influence the spatial distribution of the population and the population density. The numerical results agreed with the theoretical analysis. We hope that these results will prove useful in the study of toxic plankton systems.

scholarly journals Spatiotemporal Complexity of the Nutrient-Phytoplankton Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Debing Mei ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai
Keyword(s):  
Mathematical Formulation ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Critical Conditions ◽  
Horizontal Movement ◽  
Advection Diffusion ◽  
Advection Term ◽  
Formulation Analysis

We consider the mathematical formulation, analysis, and numerical solution of a nonlinear system of nutrient-phytoplankton, which consists of a series of reaction-advection-diffusion equations. We derive the critical conditions for Turing instability without an advection term and define the range of Turing instability with the change of nutrient concentration. We show that horizontal movement of phytoplankton could influence the system and that it is unstable when the horizontal velocity exceeds a critical value. We also compare reaction-diffusion equations with reaction-advection-diffusion equations through simulations, with spotted, banded, and crenulate patterns produced from our model. We found that different spatial constructions could occur, impacted by the diffusion and sinking of nutrients and phytoplankton. The new model may help us better understand the dynamics of an aquatic community.

scholarly journals Wavelet Solution for Nonlinear Reaction-Diffusion Equations

2019 ◽  
Vol 8 (6) ◽  
pp. 226-229
Keyword(s):  
Approximate Solution ◽  
Theoretical Analysis ◽  
Error Bound ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Nonlinear Pde ◽  
Diffusion Equations ◽  
Legendre Wavelets ◽  
Connection Coefficients ◽  
Nonlinear Reaction

In this paper, we consider Fisher’s equation to find the approximate solution to overcome the difficulty to handle its nonlinearity. For solving this nonlinear PDE, we propose a method based on Legendre wavelets with lesser number of connection coefficients. We also study the theoretical analysis and error bound for the proposed technique. Two examples are tested with the proposed method to show the applicability and efficiency. The outcomes show that this approach fulfils the error bound conditions.

Steady State Bifurcation and Patterns of Reaction–Diffusion Equations

2020 ◽  
Vol 30 (11) ◽  
pp. 2050215
Author(s):  
Chunrui Zhang ◽  
Baodong Zheng
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatial Region ◽  
Bifurcation Points ◽  
Steady State Bifurcation ◽  
Theoretical Results

In this paper, steady state bifurcations arising from the reaction–diffusion equations are investigated. Using the Lyapunov–Schmidt reduction on a square domain, a simple, and a double steady state bifurcation caused by the symmetry of spatial region is obtained. By examining the reduced bifurcation equations, complete bifurcation scenario and patterns at simple and double steady state bifurcation points are obtained. Numerical simulations support the theoretical results.

Numerical simulations of reaction–diffusion equations modeling prey–predator interaction with delay

2018 ◽  
Vol 11 (04) ◽  
pp. 1850054 ◽  
Author(s):  
Ishtiaq Ali ◽  
Ghulam Rasool ◽  
Saleh Alrashed
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Natural Systems ◽  
Nonlinear Reaction ◽  
The Rich ◽  
Delay Differential ◽  
Type Ii Functional Response

To model biological systems one often uses ordinary and partial differential equations. These equations can be quite good at approximating observed behavior, but they suffer from the downfall of containing many parameters, often signifying quantities which cannot be determined experimentally. For the better understanding of complicated phenomena, the delay differential equation approach to model such phenomena is becoming more and more essential to capture the rich variety of dynamics observed in natural systems. In this study, we investigated numerically the influence of delay on the dynamics of nonlinear reaction–diffusion equations modeling prey–predator interaction using finite difference scheme subject to appropriate initial and boundary conditions. We first consider the prey–predator with Holling type II functional response where the growth of prey is assumed to be logistic in the sense of predator in one-dimensional space. The effect of delay was investigated with the help of simulations and is compared with the model equation without delay. The proposed method is then extended to two-dimensional space.

scholarly journals Synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo type with nonlinear coupling

2021 ◽  
Vol 13 (2) ◽  
pp. 43-51
Author(s):  
Van Long Em Phan
Keyword(s):  
Theoretical Result ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sufficient Condition ◽  
Nonlinear Coupling ◽  
Complete Network ◽  
Complete Networks

The synchronization in complete network consisting of  nodes is studied in this paper. Each node is connected to all other ones by nonlinear coupling and is represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the sufficient condition on the coupling strength to achieve the synchronization is found. The result shows that the networks with bigger in-degrees of nodes synchronize more easily. The paper also presents the numerical simulations for theoretical result and shows a compromise between the theoretical and numerical results.

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