A mathematical model of plankton dynamics involves three ingredients:
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