Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

Chao Xing; Jiaojiao Pan; Hong Luo

doi:10.3934/cpaa.2020275

Article Contents

2021, Volume 20, Issue 1: 427-448. Doi: 10.3934/cpaa.2020275

This issue Previous Article Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials Next Article The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

1.
College of Mathematics, Sichuan University, Sichuan 610065, China
2.
College of Applied Mathematics, Chengdu University of Information Technology, Sichuan 610225, China
3.
School of Mathematical Sciences, Sichuan Normal University, Sichuan 610068, China

^* Corresponding author
^* Corresponding author

Received: March 2020

Revised: September 2020

Early access: November 2020

Published: January 2021

The first author is supported by NSF grant 11901408 and 11711306

Abstract / Introduction Full Text(HTML) Figure(8) Related Papers Cited by

Abstract

The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.

Keywords:

Hopf bifurcation,
dynamic transition,
center manifold,
Phytoplankton-zooplankton model with additional food.

Mathematics Subject Classification: Primary: 35B35, 37K50, 37L10; Secondary: 35K40, 35Q92.

Citation:

Full Text(HTML)

Figure 1. The topological structure of phase portrait of continuous transition as control parameter $ \Lambda>\Lambda_{c} $

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Figure 2. The topological structure of phase portrait of continuous transition as control parameter $ \Lambda<\Lambda_{c} $

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Figure 3. The topological structure of phase portrait of jump transition as control parameter $ \Lambda<\Lambda_{c} $

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Figure 4. The topological structure of phase portrait of jump transition as control parameter $ \Lambda>\Lambda_{c} $

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Figure 5. The graph of critical parameter $ \Lambda_{c} $ and $ \lambda_{c} $ as $ n\in [0.1,1.5] $ and $ d\in [6,13] $

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Figure 6. The regions separating two types of transitions. Region A, continuous transitions from a real simple eigenvalue; Region B, jump transitions from a pair of simple complex eigenvalues

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Figure 7. The numerical solutions $ u_{1} $ and $ u_{2} $ at time T = 600. The parameter $ \lambda = 0.52 $

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Figure 8. The numerical solutions $ u_{1} $ and $ u_{2} $ at time T = 600. The parameter $ \lambda = 0.58 $

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