Ocean Plankton Populations as Excitable Media
by Mike Martin
On this page, you can vary the parameters in Truscott & Brindley's model for phytoplankton and zooplankton population dynamics. In a simpler, non-dimensionalized form, the model can be expressed as
The variable P (t) represents the proportion of the phytoplankton to a carrying capacity and Z (t) represents the proportion of the zooplankton to a carrying capacity. The other parameters are all positive. If the parameter ω is increased slightly from the default value, then periodic solutions emerge and a limit cycle is evidenced in the phase-plane. It is claimed by the original authors that these periodic solutions help to account for red tides or blooms (explosive growth of phytoplankton). Submit real, positive values for all of the parameters to obtain graphical solutions of the system as functions of time and in the phase plane. Nullclines (or isoclines) are plotted in the phase plane. A numeric solution on the interval [0, T] is produced using Mathematica's numerical ODE solver, NDSolve. Initial values for both the phyloplankton proportion, P, and zooplankton, Z, must be input. The first graph below is that of the phytoplankton proportion, P (in red), the second graph is that of the zooplankton proportion, Z (in green). The last graph is that of the phaseplane with the solution (in black) and streamlines and nullclines, too.
Enter in the value of the parameters and select the plotting bounds, then press the "Plot" button.
Truscott, JE, and J Brindley (1994). Ocean Plankton Populations as Excitable Media. Bull. of Math. Bio., 56:5: 981-998.