Math 6104
Infinite Dimensional Dynamical Systems

Fall, 2024

Text:

X. Zhao, "Dynamical Systems in Population Biology", second edition, Springer-Verlag, New York, 2017.

Evaluation:

Course Contents

• Introduction: The dynamical systems approach to evolution equations
• Dissipative Dynamical Systems
  • Limit sets and global attractors
  • Chain transitive sets and limiting systems
  • Uniform persistence and coexistence states
• Monotone Dynamics
  • Attracting order intervals and connecting orbits
  • Global attractivity and convergence
  • Monotone and subhomogeneous systems
• Basic Reproduction Numbers
  • Ordinary differential systems
  • Stability and principal eigenvalue
  • Reaction-diffusion systems
• Traveling Waves and Spreading Speeds
  • Monostable case
  • The method of upper and lower solutions
  • Bistable case
• Applications

References

• X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Communications on Pure and Applied Math., 60(2007), 1-40.
• J. Fang and X. Zhao, Bistable traveling waves for monotone semiflows with applications, Journal of European Mathematical Society, 17(2015), 2243-2288.
• Z. Bai, Y. Lou and X. Zhao, Spatial dynamics of species with annually synchronized emergence of adults, Journal of Nonlinear Science 32, 78 (2022).

Assignments:

• Assignment 1, Deadline: Oct. 2
• Assignment 2, Deadline: Oct. 23
• Assignment 3, Deadline: Nov. 6
• Assignment 4, Deadline: Nov. 27

Reading Materials (for assignments):

• Krein-Rutman Theorem
• Perron-Frobenius Theorem
• Frechet Derivative

Project Topics (for final exam):

• Coexistence states (Section 1.3.4 in Zhao's 2017 book)
• Competitive systems on ordered Banach space (Section 2.4 in Zhao's 2017 book)
• Asymptotically periodic semiflows (Section 3.2 in Zhao's 2017 book)
• A discrete-time Chemostat model (Sections 4.1 and 4.2 in Zhao's 2017 book)
• Spatial dynamics of species with annually synchronized emergence of adults (bounded domain) in [Bai, Lou and Zhao, Journal of Nonlinear Science 32, 78 (2022)]
• Spatial dynamics of species with annually synchronized emergence of adults (unbounded domain) in [Bai, Lou and Zhao, Journal of Nonlinear Science 32, 78 (2022)]
• Global Dynamics of a reaction-diffusion model of Zika virus transmission with seasonality in [Li and Zhao, Bulletin of Mathematical Biology 83, 43(2021)]

Presentation Schedule:

• Nov. 27, Meng (topic 2, 2:00-2:35); Yue (topic 5, 2:35-3:10)
• Dec. 2, Hu (topic 1, 1:00-1:35, HH-3030); Jiang (topic 6, 2:00-2:35); Liang (topic 7,2:35-3:10)