Afternoon
meeting of the
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.