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03.19.2013: A summary so far of our recent modeling study

I summarize here the results of our recent modeling study.

Case without zooplankton

Without zooplankton, the final values of Ps, Pi and V are independent of the initial conditions. The value of β, that characterizes the efficiency of the infection, needs to be larger than a threshold value (about 0.12 in our case) otherwise the virus population dies off and phytoplankton blooms fully (final Ps is 1) (see 02.01.2013: Second analysis of run 001). For β larger than this threshold, the system displays damped oscillations (Fig. 1 in 11.02.2012: First analysis of run 001). The larger β, the faster and less damped the oscillations (see Figs. 6 and 7 in 02.01.2013: Second analysis of run 001). Final value of Ps decays with β while final values of Pi and V peaks for β equals about 0.35 (see Figs. 1 to 3 in 02.01.2013: Second analysis of run 001). The behavior is qualitatively the same even when initial Pi is non-zero (see 02.05.2013: Third analysis of run 001).

Case with zooplankton

The overall conclusion is that, for regime where the zooplankton is maintained without viruses, the dynamics will be ultimately Z-P dynamics and viruses can have a significant impact in these regimes only if a bloom is allowed without viruses. In these cases, the lower Z0 (Figs. 9 to 15 in 02.14.2013: Fourth analysis of run 001) or the lower Rm (Figs. 7 to 10 in 03.15.2013: Sixth analysis of run 001 - Recap), the longer the period over which the viruses can have an impact. In the case where there is a limit cycle between Z and P, the viruses can have an impact over the first few cycles but their influence slowly diminishes with time (see Figs. 2 to 5 in 03.15.2013: Sixth analysis of run 001 - Recap for different value of Rm). If no bloom is allowed by the Z-P dynamics but the zooplankton population is nonetheless maintained, the viruses have little impact (Figs. 5 to 8 in 03.05.2013: Fourth analysis of run 001. Part 3 for Ps0=0.1 and Z0=0.05).

For regime where the zooplankton population is not maintained (low Rm and/or very low Z0; see lower panel of Fig. 4 in 03.08.2013: Sixth analysis of run 001), the dynamics are ultimately V-P (Fig. 1 in 03.15.2013: Sixth analysis of run 001 - Recap) and the V and Pi population are maintained as in the case where there was no zooplankton (Figs. 2 and 4 in 03.08.2013: Sixth analysis of run 001).

Although the lower panel of Figs. 3 and 4 in 03.08.2013: Sixth analysis of run 001 may suggest that there is a regime in between where both the zooplankton and virus population can maintained together, I think this is an artifact due to the fact that I have only run the simulations until a finite time (I think 25000 or 50000 time steps). If I would have run the simulations longer, I predict that this transition would have shrunk.

To do

See if we can get the equilibrium values from the equations. If we can, start with initial values that are slightly off the equilibrium values. Find also the range of β for which there may be co-existence between viruses, phytoplankton and zooplankton. Read Rhodes and Martin that indeed claims that such co-existence is possible.