Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity

  • Xuebing Zhang
  • Hongyong Zhao
Original Paper


In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, we derive explicit conditions for the existence of nonconstant steady states that emerge through steady-state bifurcation from related constant steady states. Furthermore, the existence and nonexistence of nonconstant positive steady states of this model are studied by considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. These explicit conditions are numerically verified in detail and further compared to those conditions ensuring Turing instability. It is shown that the numerically observed behaviors are in good agreement with the theoretically proposed results. All theoretical analyses and numerical simulations show that toxic substances have a perceptible effect on the system.


Turing instability Pattern formation Bifurcation Nonconstant steady state 



This study was funded by National Science Foundation of China (Grant Number: 11571170). The study is partially supported by Natural Science Foundation of Jiangsu Province (Grant Number: BK20150420), and also supported by the Startup Foundation for Introducing Talent of NUIST.

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina
  2. 2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingChina

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