Memory-based Movement and PDE/DDE Analysis
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Memory-based Movement and PDE/DDE Analysis
2024 Summer Schedule:
June 8, 2024 - Yurij (Turing instability + matlab)
June 15, 2024 - Zhenfeng
June 22, 2024 - Tianxu
June 29, 2024 - NO MEETING
July 6, 2024 - TBD (Di or Guodong?)
July 13, 2024 - Yulei
July 20, 2024 - TBD (Di or Guodong?)
July 27, 2024 - Haihui
August 3, 2024 - Zhenkun
August 10, 2024 - Shohel
Subgroup Meeting Materials
June 1, 2024
Activities:
Introductions (name, year/position, research interests, hobbies etc.)
Expectations for the group meeting / things people want to discuss
Determined we will focus on techniques for analysis, modelling, and numerical simulation. Not much focus will be on writing this term.
Meeting schedule determined (see below)
Yurij gave a mini presentation about some philosophy behind approximating solutions to linear / nonlinear parabolic PDE, a key step in proving well-posedness of solutions
June 8, 2024
Activities: Linear stability, diffusion driven instability, PDEPE in matlab (Yurij)
Topics discussed:
linear stability analysis for scalar (logistic) equation
compare boundary conditions (homogeneous Neumann vs. Dirichlet); critical domain size
linear stability analysis for general 2-species Lotka-Volterra system
linear stability analysis for general 2-species reaction diffusion system; Turing instability (diffusion driven instability)
How to use PDEPE to solve these problems numerically in matlab (one for each system)
References followed:
1. [9] J.D. Murray. Mathematical Biology. Springer-Verlag, 1989. There are at least two volumes. Most of the PDE stuff is in the second volume, much of the stability analysis for systems / Turing instability material is taken from here. Specifically, I follow Volume II, Ch. 2.3-2.4.
2. [10] Wei-Ming Ni. Mathematics of diffusion. SIAM, 2011. Chapter 4 has some good examples for stability analysis and global asymptotic stability for 2-species competition systems.
3. [13] Z. Wu, J. Yin, and C. Wang. Elliptic and Parabolic Equations. World Scientific Publishing, 2006. Chapter 12 covers monotone equations and systems (in the monotone comparison principle sense) relevant to some discussion on linear stability of the logistic problem.
4. [5] Cantrell and Cosner. Spatial Ecology via Reaction-Diffusion Equations. John Wiley and Sons, 2003. I don’t use material from here directly, but they cover in detail linear stability analysis for scalar equations and systems.
5. https://www.mathworks.com/help/matlab/ref/pdepe.html; it may seem silly, but this help page is a main reference for PDEPE and some basic (mostly) working examples
June 15, 2024
Activities: Asymptotic behavior of stochastic biomathematical models perturbed by the Ornstein-Uhlenbeck process (Zhenfeng)
Topics discussed:
• Ornstein-Uhlenbeck (OU) process equation and its properties
• Viral infection model
• SIS epidemic model
• Random walk within home range
• Two-dimensional random walk model
References:
1. [8] This is a basic textbook on stochastic differential equations.
2. [1] This paper describes the OU process and the log-OU process.
3. [11] In this paper, we obtain the threshold for the SIS epidemic model under log OU process perturbation for the first time.
June 22, 2024
Activities: A Prey-Taxis Food Chain Model with Bold and Shy Traits (Tianxu)
June 29, 2024: NO MEETING
July 6, 2024
Activities: Pattern Formation in a Generalised Keller-Segel Model on a Layered Domain (Dan)
Topics discussed:
• Recap of Turing instabilities in a 2-species reaction-diffusion system
• Salt and Pepper patterning
• Introduction to a generalized Keller-Segel model for chemotaxis
• Conditions for patterning of such models on a stratified domain
References:
1. [9] The second volume derives conditions for patterning in a two-species reaction-diffusion system with chemotaxis on several domains (including square domain with zero-flux boundary conditions).
2. [6] This paper derives conditions for domain-length driven instability of a diffusion-advection system for several boundary conditions, in particular providing an example of ’salt and pepper’ patterning.
3. [7] This paper extends classical Turing patterning results to a more complex stratified domain
July 13, 2024
Activities: TBD (Yulei)
Topics discussed:
• FREE BOUNDARY PROBLEM OF A PREDATOR-PREY MODEL WITH FEAR EFFECT
• the use of Schauder theory, Lp theory, maximum principle, Hopf boundary lemma in Nonlinear Parabolic Equations
• contraction mapping principle to obtain the local existence uniqueness, boundedness and global existence of classical solutions
• the spread - vanishing dichotomy between predator and prey is studied by using the comparison principle
References:
1. Wang M. Spreading and vanishing in the diffusive prey–predator model with a free boundary. Communications in Nonlinear Science and Numerical Simulation, 2015, 23(1-3): 311-327.
2. Textbook Lecture of Nonlinear Parabolic Equations by Professor Mingxin Wang
Schedule for 2023 Summer:
June 28: Yurij presents
July 12: Xiuli presents
July 26: Guodong presents
August 9: Shu presents
August 23: Yulei presents
Date: August 23, 2023
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Yuyan Wang, Guodong Liu
Activities: Shu gave the proof of some theorems in Evan’s PDE. Yulei discussed the progress and some questions about her recent work. Then Guodong talked about the zero-flux boundary conditions. Yurij gave the comments and suggestions.
Date: August 9, 2023
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Yuyan Wang, Guodong Liu
Activities: Yurij talked about the loop-argument to verify the global existence of the solution to parabolic equations. Then we discussed a paper regarding resource-dependent dispersal rates. At last, Yulei gave the proof of a theorem in Evan’s PDE.
Date: July 26, 2023
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Yuyan Wang, Guodong Liu
Activities: Guodong talked about the proof of the first three theorems of the heat equation in Evan’s PDE. Then Guodong gave a question regarding the boundedness of solution to a reaction-diffusion system.
Date: July 12, 2023
Participants: Xiuli Sun, Yulei Cheng, Yuyan Wang, Guodong Liu
Activities: Xiuli gave a talk on her recent investigation.
Date: June 28
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Guodong Liu
Activities: We made the schedule of the summer subgroup meetings. Then Yurij introduced some ideas to establish the existence of solutions to different PDE problems.
Date: Oct 13
Participants: Yurij Salmaniw, Kai Wang, Hua Zhang, Chunxi Feng, Di Liu, Dongxu Geng, Pegah Tagheie Karaji
Activities: Kai discussed variational characterizations of eigenvalues in the context of PDEs. After some introduction, his main question was "can we do this for a system instead of a scalar equation", which is a very difficult question but would certainly be useful if possible.
Date: Oct 20
Participants
Activities: A two dimensional nonlocal free boundary problem, given by Chunxi Feng
Date: November 3: Yurij will discuss some memory movement work (joint with Di Liu)
Date:November 17: Hua will discuss some of her work (topic has not yet been decided)
The existence of positive equilibria is basic for further dynamics analysis of mathematical models, which strongly depends on the structure of the connectivity matrix for patchy models. When the connectivity matrix includes memory or other cognition, the existence and uniqueness of positive equilibria is complex. In the sub-group meeting, I will talk about this problem for a SIS patchy model, where the dispersal of the susceptible population depends on the transmission rate and the dispersal of the infected population depends on the recovery rate.
There will be no subgroup meeting over the reading week.
Date: November 24
Participants: Yurij Salmaniw, Kai Wang, Hua Zhang, Chunxi Feng, Di Liu, Guodong Liu
Activities: Di briefly discussed her work on a density-suppressed motility model. She then proposed a question about the boundness of solutions of this model subject to the Dirichlet boundary condition.
Date: Jan 11
Participants: Everyone
Activities: Some new members gave introductions and some rough outlines of their research interests. Then we determined the leader for the next subgroup meetings.
Date: Jan 18
Participants: Everyone
Activities: We discussed chapter IV in "Elements of Style". We gave some words we usually misused. Then Yurij gave the presentation about how to describe the relationship between species mobility and food or other resource.
Date: Feb 1
Participants: Everyone
Activities: We discussed chapter V in "Elements of Style". We discussed 10 tips, whether they are useful or not etc. Then Valeria presented some of her work about population movement.
Date: Feb 8
Participants: Yurij Salmaniw, Valeria Giunta, Xiuli Sun, Di Liu, Yulei Cheng, Kai Wang, Guodong Liu
Activities: We discussed tips 11-21 in chapter V in "Elements of Style". We classified each tip whether it is necessary for academic writing. Then Guodong asked three concrete questions related to derivatives, elliptic regularity, and positive operators. Yurij gave his answers.
Date: Feb 15
Participants: Yurij Salmaniw, Xiuli Sun, Di Liu, Yulei Cheng, Guodong Liu
Activities: We discussed what might be most useful to focus on moving forward in terms of academic writing. Then Yulei asked some questions on reaction-diffusion equations and projection operators.
Mar 1
Participants: Yurij Salmaniw, Valeria Giunta, Xiuli Sun, Yulei Cheng, Kai Wang, Guodong Liu
Activities: Yurij and Guodong gave three abstracts of papers, and then analyzed the structure of these abstracts. Then Xiuli and Yulei asked some questions on the linearization of nonlinear equations and variational problems.
March 8
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Chunxi Feng, Guodong Liu
Activities: Yurij presented some key techniques dealing with PDEs, especially the existence of solutions for nonlinear PDEs.
March 15
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Di Liu, Chunxi Feng, Guodong Liu
Activities: Yurij shared his previous three abstracts, gave some feedback, and improved some areas. Guodong also shared two abstracts and rewrote some sentences of one abstract.
March 29
Participants: Yurij Salmaniw,Valeria Giunta, Yulei Cheng, Shu Li, Di Liu, Kai Wang, Guodong Liu
Activities: Guodong introduced two approaches to defining the basic reproduction number of reaction-diffusion epidemic models: the principal eigenvalue approach and the next generation operator approach. And he showed the connection between these two approaches.
April 5
Participants: Yurij Salmaniw, Xiuli Sun, Yulei Cheng, Shu Li, Di Liu, Kai Wang, Guodong Liu
Activities: Shu presented some work related to cognitive animal movement.
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