__London Dynamical Systems Group__

Supported by the

__London Mathematical Society__

__Organised by Steve Baigent__

To be held on Tuesday, 16 February 2010, room 706,

__Department of Mathematics__, UCL

Organizational details:

All talks will be held in room 706 (seventh floor) of the Department of Mathematics UCL.

Directions of how to get there

__can be found here__.

Pre-registration is not necessary. For any further information please contact Steve Baigent

Programme:

14:00 - 14:50 David Angeli (Imperial) "Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates"

14:50 - 15:40 Zhanyuan Hou (London Met) "Global Asymptotic Behaviour of Autonomous Competitive Lotka-Volterra Systems"

15:40 - 16:00 Tea and Coffee break

16:00 - 16:50 Murad Banaji (UCL) "Convergence in strongly monotone systems with an increasing first integral"

16:50 - 17:40 Markus Kirkilionis (Warwick) "Monotone Dynamical Systems Seen As A Special Class Of Non-Linearities"

Abstracts

David Angeli: This talk illustrates new results for certain classes of chemical reaction networks, linking structure to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the form of the kinetics (e.g., mass-action) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, we show that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.

Zhanyuan Hou: For an autonomous system of differential equations modelling the population dynamics of a community of species competing for the same resources, the global asymptotic behaviour of the system is completely determined by the interacting parameters between the species. One of the important aspects of the dynamics is whether the system has a globally attracting equilibrium point. That is, in modelling terms, whether the species in this community can live in harmony and gradually settle down to a constant population size for each species. Mathematically, a condition on the parameters has been sought for the system to have a globally attracting equilibrium point. Although some known conditions are available, improvement for a weaker condition is needed. On the opposite direction is the existence of an equilibrium as a global repellor. In this talk, some recent results/methods will be presented on global attraction, global repulsion and permanence.

Murad Banaji: Monotone dynamical systems, i.e. systems which preserve some partial order on the state space, have been intensively studied. Monotonicity has been shown to constrain system behaviour in various ways, for example ruling out attracting nontrivial periodic orbits, under fairly general assumptions. When the system is strongly monotone, behaviour is constrained further: for almost all initial conditions bounded solutions converge to the set of equilibria. Sometimes such generic convergence claims can be strengthened: for instance, convergence of every bounded orbit can be obtained in a variety of special cases. Here the following situation is discussed: consider a local semiflow on a proper cone Y in Euclidean space, strongly monotone with respect to the order generated by a proper cone K containing Y. Assume further that the local semiflow preserves a K-increasing first integral. Then there can be no more than one equilibrium on any level set of the integral, and each equilibrium attracts its entire level set. The result is a considerable generalisation of results in the literature. (This is joint work with D. Angeli)

Markus Kirkilionis: In the talk I start looking at (time-continuous) Monotone Dynamical Systems by considering them as being defined by a special class of non-linearities, just like the ones given by Lotka-Volterra type, polynomial (including Lotka-Volterra), or rational (often derived by time-scaling arguments). For each class of non-linearities defining the system (where 'system' can mean system of ordinary differential equations, reaction-diffusion equations, other PDE, or more general integral equations) there are different mathematical constructions and techniques available that give (partial) answer to a deep mathematical question: How does the type of non-linearity restrict the possible qualitative behaviour of a given (class of) system? The talk will give an overview how these methods look like, and stress the fact that the methods available for monotone dynamical systems are quite special in many respects. We will carefully differentiate between competetive, cooperative and quasi-monotone systems, where the latter can be easily shown to be a much larger class of systems when compared to the first two ones. Therefore also much less powerful results are to be expected which is indeed the case. Nevertheless via so called comparison methods for solutions of such systems much can be said about their qualitative behaviour in special cases.