# A Numerical Method for Advection-Reaction-Diffusion Equations

## with applications to plankton modeling

A mathematical model of plankton dynamics involves three ingredients:

**Population dynamics**: a set of ODEs that describe the
food web of the many species forming a spatially homogeneous
planktonic community; this is in general awfully complicated.
Usually only predator-prey dynamics between zooplankton and
phytoplankton is modeled, often with the addition of a nutrient
variable.
**Large-scale transport**: plankton is carried along by
mesoscale and basin-scale ocean currents. This produces highly
complicated lagrangian trajectories.
**Small-scale mixing**: at small scales (less than 100m)
plankton is mixed by turbulence of the ocean's mixed layer and by
the swimming of single individuals. In general, this leads to
non-linear diffusion, but here, for simplicity, we just use
Laplacian diffusion.

Putting all three together one gets an advection-reaction-diffusion
equation for each species considered in the population dynamics model.

If there were no diffusion, the solution of the advection-reaction
problem is straightforward using the method of characteristics.

In the presence of diffusion, we propose a numerical method that
handles the diffusion operator on an auxiliary eulerian grid, while
using a particle-based approach for the advection-reaction part. In the
limit of no diffusion, one recovers the method of characteristics.

For a description of the method, please refer to this paper.

An open issue is the lack of cycles in the satellite observations of
chlorophyll (phytoplankton) concentration. Limit cycles (or chaotic
non-linear oscillations with a well-defined time scale) are ubiquitous
in the solution of the population dynamics ODEs.

Stirring by mesoscale currents may disguise the oscillations: water
patches oscillating with largely different phases may be brought
together by mesoscale stirring: a coarse graining observation, such
one from a satellite, would average over the phases, canceling any
fluctuations.

Without diffusion this explanation appears to be robust, even without
invoking other factors such as patchy nutrient sources: here is a
simulation with no diffusion, notice the "out-of-tuning TV" effect at
the end. Stand back a few steps from the screen: you won't see any
oscillations.

With a relatively large diffusion coefficient (D=10^{-5} for
both species) we have a rather quick spatial synchronization. You will
see the screen flashing as the zooplankton and the phytoplankton
follow their own limit cycle at unison in the entire domain.

With a small diffusion coefficient (D=5*10^{-7}, in the long
run, we still observe spatial synchronization. However, the
out-of-tune TV effect is still there for quite a while...

Last update:
04/12/2007