Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (4): 1204-1217.

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Stability and Bifurcation of a Pathogen-Immune Model with Delay and Diffusion Effects

Jingnan Wang*(),Dezhong Yang   

  1. Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080
  • Received:2019-09-28 Online:2021-08-26 Published:2021-08-09
  • Contact: Jingnan Wang
  • Supported by:
    the NSFC(11801122);the NSF of Heilongjiang Province(A2018008)


In order to understand the effects of diffusion and time-delay factors on the dynamics between pathogens and immune cells, a delayed pathogen-immune reaction diffusion model with homogeneous Neumann boundary condition is established. By using the diffusion ratio of pathogen-immune cells and immune delay as two parameters, the characteristic root distribution of the linearized system at the positive steady state is analyzed and the necessary and sufficient conditions for the positive steady state to undergo Turing instability and Hopf bifurcation are obtained by using the bifurcation theory of functional differential equations. In addition, the dynamic behavior close to the critical value of Turing instability and Hopf bifurcation is intuitively shown by Matlab numerical simulation. The biological and medicinal significance of corresponding dynamic behaviors are discussed. Furthermore, the obtained results provide certain theoretical support for controlling the growth of pathogen.

Key words: Pathogen immunity, Reaction diffusion, Delay, Turing bifurcation, Hopf bifurcation

CLC Number: 

  • O175.29