The Centre of Excellence in Analysis and Dynamics arranges a one day symposium "**Dynamics Day**” in Exactum, Kumpula on Friday 10 November 2017 in lecture room C124.

### Program:

**10.15 - 11.00** Jifa Jiang, Shanghai: The Decomposition Formula and Stationary Measures for Stochastic Lotka-Volterra Systems with Applications to Turbulent Convection

**11.15 - 12.00** Steve Baigent, London: Carrying simplices of competition ecology

**12 - 13** LUNCH

**13.15 - 14.00** Yi Wang, Hefei: Monotone systems with respect to high-rank cones

**14.15 - 14.45** Lei Niu, Helsinki: Permanence and impermanence for discrete-time competitive systems

COFFEE

**15.15 - 16.00** Yan Ping, Helsinki: On Zeeman's classification of the continuous model in a game of rock-paper-scissors

### ABSTRACTS:

**Jifa Jiang**: Motivated by the work of Busse et al. on turbulent convection in a rotating layer, we exploit the long-run behavior for stochastic Lotka-Volterra (LV) systems both in pullback trajectory and in stationary measure. It is proved stochastic decomposition formula describing the relation between solutions of stochastic and deterministic LV systems and stochastic Logistic equation. By virtue of this formula, it is verified that every pull-back omega limit set is an omega limit set for deterministic LV systems multiplied by the random equilibrium of the stochastic Logistic equation. This formula is used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness for the set of stationary measures and the invariance for their weak limits as the noise intensity vanishes, whose supports are contained in the Birkhoff center. Full abstract PDF

**Steve Baigent**: The carrying simplex is an attracting invariant hypersurface that is commonly found in competitive models from ecology. I will introduce the carrying simplex for both continuous- and discrete-time models, and outline conditions under which it is known to exist. I will also discuss techniques for determining the sign of the Gaussian curvature of the carrying simplex, illustrating with some well-known models from ecology. Finally, I will discuss how the curvature of the carrying simplex relates to stability of equilibria.

**Yi Wang:** In this talk, we will consider the semiflows which admit strong monotonicity properties with respect to “cones” of high ranks in a general Banach space. These semiflows are motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov Functionals. We will show that for a pseudo-ordered precompact semi-orbit the omega-limit set either is ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. We also establish a Poincare-Bendixson type theorem in the case where k=2. This is a joint work with Lirui Feng and Jianhong Wu.

**Lei Niu**: We study the permanence and impermanence for discrete-time competitive systems which admit a carrying simplex, that is, a globally attracting hypersurface of codimension one which captures the long-term dynamics. Sufficient conditions to guarantee permanence and impermanence are provided via the carrying simplex. We then focus on a class of systems with linear competition structure, which always admit a carrying simplex. We provide a universal classification via the boundary dynamics for 3D systems, which have a total of 33 stable equivalence classes, and we present the phase portrait on the carrying simplex for each class. Applying the permanence criteria to each class, we obtain that only systems in classes 29,31,33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Ricker models, Leslie/Gower models and Atkinson/Allen models are given.

**Ping Yan:** In this talk we give complete Zeeman's classification of the continuous model in a game of rock-paper-scissors (RPS). We construct two limit cycles with a heteroclinic polycycle for the continuous model of the RPS game. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist in the RPS game.