By Uffe Høgsbro Thygesen
This postdoc project concerns construction and analysis of stochastic individual-based models of marine ecosystems.
The motivation behind the project is that current ecosystem models do not benefit from what we know about individual behaviour. It is plausible that we could obtain better ecosystem models by including more detailed information about behaviour. This is not an easy task, and much current research - within the SLIP network and beyond - aims to move in this direction.
The objective of this projects is, generally, to investigate if and how recent advances in applied mathematics and statistics can contribute to the improvement of individual-based ecosystem models, by tying together the microscale of the individual and the macroscale of the ecosystem. Somewhat more specifically, I am interested in applying techniques from system theory, control theory, and stochastic geometry.
The individual-based models I intend to examine will be explicit in time and space. They will be based on stochastic differential equations, i.e. diffusion processes, with jumps. I am interested in modelling, simulation, model reduction, and estimation. A specific thread which I would like to investigate is the possible connection between the stochastic geometry of an ecosystem and the statistical mechanics approach to model reduction.
A proto-type ecosystem: Marine snow 
A proto-type of a problem which I have been looking at, is the case of marine snow. Marine snow is small aggregates which sink down the water column. As they do so, they leak nutrients (aminoacids) which form a long slender plume.
In the picture on the right you can see a picture of the plume. This figure was drawn by George Jackson based on the output from a piece of software I wrote for analysis of the problem.
These nutrients attract bacteria and zooplankters: The bacteria flourish on the nutrients; some in the plume while others colonize the aggregates. The zooplankters, in turn, graze the bacteria. The aggregates thus become the scene of abiotic as well as biotic processes and can therefore be termed ecosystems.
A proto-type analysis: From individual to population
The individual behaviour of the bacteria is fairly well known: They use a run-tumble strategy to perform chemotaxis, i.e. move towards the regions with high concentrations of nutrients.
This run-tumble strategy can be approximated by a diffusion process. The essence in these approximation schemes is classic; nevertheless there is still room for improvements in this field.
Once we have the diffusion approximation, it is possible to compute population-level fluxes etc. We can also examinate how well the bacteria can exploit plumes of different magnitudes, by means of a simplified analysis of scales.
An interesting sub-problem is that of information: What information is available to the bacteria, and how does their lack of information constrain them? In general, constraints on information are being neglected in much work on optimal behaviour of individuals, despite the obvious importance of these constraints. (This point was brought to my attention by Danny Grünbaum)
A nice introduction to these problems is the book by Berg. I became interested in the case of marine snow during collaboration with Thomas Kiørboe.
Schooling behaviour
The picture on the right picture was taken in Nordsøcentret, the oceanarium in Hirtshals. On the left you can see a school of mackerel circling; in the center you can (barely) see a school of herring which is exploding in order to avoid a penetrating predator.
For many species schooling is a very characteristic part of the behaviour. One of the principal motivations for schooling is, according to dominant beliefs, that it reduces the risk of predation. Hence, one should believe that schooling is one aspect of individual behaviour which has significance at the population level. More specifically, in many multipsecies models the amount of prey eaten by a predator is proportional to the product of their abundance. Considering the schooling behaviour of the prey, one can imagine that one should replace this quadratic rule with something else which reflects the number and size of the schools.
I intend to investigate schooling behaviour by means of simulation and optimization.
Referencer
- H.C. Berg. Random Walks in Biology. Second edition. Princeton University Press, 1993.
- T. Kiørboe, H. Ploug and U.H. Thygesen. Fluid motion and solute distribution around sinking aggregates. I. Small scale fluxes and heterogeneity of nutrients in the pelagic environment. Marine Ecology Progress Series, vol 211, p. 1-13, 2001
- T. Kiørboe and G. Jackson. Marine Snow, Organic Solute Plumes, and Optimal Chemosensory Behavior of Bacteria. Limnol. Ocean. vol 46, number 6, p. 1309-1318, 2001