Mathematical aspects of fragmentation and coagulation models with
application to ecology
Ovide Arino
UR GEODES, IRD-Bondy,
32 avenue Henri Varagnat, 93143 Bondy, France
Ovide.Arino@bondy.ird.fr
Ryszard Rudnicki
Institute of Mathematics, Polish Academy of Sciences
and Institute of Mathematics, Silesian University,
Bankowa 14, 40-007 Katowice, Poland
The coagulation-fragmentation processes describe dynamics of populations of
particles distributed into groups of different sizes that coagulate and divide
at some given rates. They arise in a veriety of phenomena including polymer
degradation and growth, gelation, clusters performing coalescence, erosion,
crushing and griding of rocks, and blood coagulation. These processes have
recently received a considerable interest in ecology since they model the
process of grouping in plant and animal populations. They are intrinsic to such
phenomena as habitat destruction and fragmentation, invasion of exotic species
and extinction of native spacies, phytoplankton dynamics and sedimentation and
coagulation of algae.
For example phytoplankton cells have the ability of forming aggregates which are
dispersed in the water column as a result of currents and turbulence, leading to
a patchy distribution of phytoplankton. Phytoplankton is the first level of food
accessible to animals. It is in particular the main food available to the early
larval stages of many fish species, including the anchovy. At such stages,
larvae are passive and can only eat the prey passing in a very close vicinity.
The best situation is when the larva is near a phytoplankton aggregate, while on
the other hand larvae which stay far from aggregates are not likely to survive.
Thus, being able to describe the distribution in numbers of phytoplankton
aggregates of different sizes as well as locating them in the space turn out to
be of up-most importance in connection with the study of fish recrutment.
Recently, several authors have addressed the issue of modeling the dynamics of
phytoplankton in such a way as to exhibit such structure.
There are several techniques which can be applied to describe models of
coagulation-fragmentation processes including stochastic differential equations,
partial differential equations of transport type, advection-diffusion-reaction
models. But since these models are rather complicated their mathematical
treatment is still out of reach and their investigation mainly base on computer
simulations.
In the proposed session we expect papers and lectures which will present
ecological models based on fragmentation-coagulation processes. Especially,
insights and predictions from mathematical modelling and new theoretical results
concerning fragmentation-coagulation processes would be welcome.