Lecture Room B, 4th Floor, The 3rd General Building, NTHU
Speaker(s):
Lei Niu (Donghua University)
Organizer(s):
Sze-Bi Hsu (National Tsing Hua University)
Jia-Yuan Dai (National Tsing Hua University)
1. Introduction & Purposes
Seasonal succession is a prevalent environmental feature in nature. Due to the seasonal alternate, populations experience a periodic dynamical environment driven by both internal dynamics of species interactions and external forcing. Ecological models with seasonal succession have attracted considerable attention. Exploring and predicting the long-term influences of such periodic forcing on the dynamics and structure of ecosystems has been a fascinating subject. Mathematically, the vector fields of such models are discontinuous and periodic in time. However, there are few analytical results on the possible effects of seasonal succession due to the complexity of such systems, e.g. the lack of explicit expressions and no information on the number of positive fixed points of their Poincaré maps. Therefore, studying their global dynamics, even in three-dimensional situations, is a very challenging problem.
We will introduce how to use the carrying simplex theory of competitive dynamical systems to study the global dynamics of three-dimensional Lotka-Volterra competitive systems with seasonal succession. The main idea is to use the geometric properties of the carrying simplex and the index theory of fixed points to study the global dynamics. The approach may also be inspiring for studying the global dynamics of other systems with seasonal succession.
Reference
[1]. L. Niu, Y. Wang and X. Xie, Global dynamics of three-dimensional Lotka-Volterra competition models with seasonal succession: I. Classification of dynamics, SIAM J. Appl. Math. 85 (2025) 499–523.
[2]. J. Jiang, L. Niu and Y.Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol. 72 (2016) 939–972.
[3]. J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol. 74 (2017) 1223–1261.
[4]. L. Niu and A. Ruiz-Herrera, Trivial dynamics in discrete-time systems: carrying simplex and translation arcs, Nonlinearity, 31 (2018), pp. 2633–2650.
2. Outline & Descriptions
The objective is to introduce methods to study the global dynamics of three-dimensional Lotka- Volterra systems with seasonal succession by using the geometric properties of the carrying simplex for competitive dynamical systems.
The outline is as follows:
Day 1 (July 21)
Morning Session (10am to 12am): Poincaré map and carrying simplex
We will study the properties of the Poincaré map for the Lotka-Volterra competition model with seasonal succession. The existence of the carrying simplex for the Poincaré map will be proved
Afternoon Session (2pm to 4pm): Stability and 2D dynamics
The existence and stability of fixed points for the Poincaré map will be studied. We will provide a detailed analysis of the global dynamics of the 2D models.
Day 2 (July 22)
Morning Session (10am to 12am): 3D dynamics: equivalence and the index formula
We will introduce the definition of the equivalence via the boundary of the carrying simplex. An index formula on the carrying simplex will be provided.
Afternoon Session (2pm to 4pm): 3D dynamics: equivalence classification
We will provide the complete equivalence classification for the 3D models.
Day 3 (July 23)
Morning Session (10am to 12am): 3D dynamics: global analysis
We will study the global dynamics on the carrying simplex for each equivalence class.
Afternoon Session (2pm to 4pm): 3D dynamics: heteroclinic cycle
We will study the existence of the heteroclinic cycle and its stability.
Day 4 (July 24)
Morning Session (10am to 12am): 3D dynamics: uniqueness of the positive fixed point
We will study the uniqueness of the positive fixed point for the case with identical growth rate and death rate. We then provide the global dynamics of the models with identical growth rate and death rate.
Afternoon Session (2pm to 4pm): Averaged systems and discussions
We will study the global dynamics of the associated averaged systems. We then give some further discussions.
3. Prerequisites
It is good to have some basic knowledge of ordinary differential equations and dynamical systems.
Reference
[5]. S.B. Hsu, Ordinary Differential Equations with Applications, 2nd Edition, World Scientific, 2013.
4. Registration
Contact:
Vickey Sun (vickeysun@ncts.ntu.edu.tw), Murphy Yu (murphyyu@ncts.tw)
Poster: events_3_3502506104918123892.pdf
