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03.08.2013: Sixth analysis of run 001ΒΆ

I here fix the initial value of P0 and V0 but varies Z0 as well as, for the first time, Rm, the maximum grazing rate. The goal was to see if by changing Rm we can get a regime where P-V dynamics dominates at the end and/or a regime where P, V and Z co-exist.

The value of Rm=0.699 was used so far. We see that for low values of Rm, the final values start to be dependent on the initial values of Z0, which was not the case so far. As usual, it would be interesting to look at specific time series. We do that below.

[Nota: I have not reported that I have also studied the sensitivity of the solution to the value of beta (with the value Rm=0.699). I just want to say that I did not find anything interesting, with the long-term solution being dominated by Z-P dynamics.]

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Figure 1: Maximum value of Ps and its timing and the final value of Ps.

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Figure 2: Maximum value of Pi and its timing, and the final value of Pi.

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Figure 3: Maximum value of V and its timing, and the final value of V.

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Figure 4: Maximum value of Z and its timing, and the final value of Z.

Below are time series for different values of Rm and Z0. The first pair of values (Rm=0.35, Z0=0.12; Fig. 5) was chosen for the maximum in Z appearing in Fig. 4. We see that viruses and infected phytoplankton disappear and the dynamics is dominated by the Z-P dynamics. As in the case in other examples below, it seems we have affair here at a limit cycle.

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Figure 5: Time series for Rm=0.35 and Z0=0.12.

The second pair of values (Rm=0.1, Z0=0.12; Fig. 6) was chosen because of the relatively large value of V and Pi obtained in this region of phase space (Figs. 2 and 3). In this case, it seems that the dynamics are quickly dominated by V-Pi-Ps dynamics with a slow disappearance of Z with time.

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Figure 6: Time series for Rm=0.1 and Z0=0.12.

The third pair (Rm=0.25, Z0=0.15; Fig. 7) was chosen because of the anomalous large timing of the maximum in V and Pi (Figs. 2 and 3). This is an interesting case in which, for a long time (10000 time steps), the dynamics are dominated by Z-P but then, rather suddenly, the dynamics shifts and becomes dominated by V-Pi-Ps. It seems that the two types of dynamics never co-exist at the same time and that this case is at the edge where there may be long (chaotic?) shift between the two types of dynamics. It would be interesting to know what happens at a much longer time as viruses and infected phytoplankton seem to slowly decay. A shift to Z-P dynamics?

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Figure 7: Time series for Rm=0.25 and Z0=0.15.

The fourth pair (Rm=0.425, Z0=0.14; Fig. 8a) was chosen because of the anomalous large timing of the maximum in Ps (Fig. 1). The result is similar as for the first chosen pair (Fig. 5) with a rapid decay of viruses and an apparent limit cycle between Z and Ps. It would be interesting to see what is going on for even larger value of Rm for which, according to Fig. 1, Ps never blooms. This is shown in Fig. 8b below. The long term is dominated by small values of Z and P. This suggests that, as for different values of Z0 and P0, a bloom can suddenly disappear for a slightly larger predation pressure.

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Figure 8a: Time series for Rm=0.425 and Z0=0.14.

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Figure 8b: Time series for Rm=0.55 and Z0=0.14. [Note that the time series stopped at time step 21000 as a numerical stability arose after that].

The fifth pair (Rm=0.3, Z0=0.14; Fig. 9a) was used to explore why around these values, the final Ps is much larger than in any other region of the phase space (Fig. 1). Notice here that viruses and infected phytoplankton never totally disappear but temporary re-surface after time step 14000. Fig. 9b shows that time series when Rm is reduced to 0.2. We see that there is a sharp change in the dynamics between Rm=0.3 and Rm=0.2 that is due to the presence of viruses. We indeed see that transition in Figs. 2 and 3. For Rm larger than 0.3, there is no V and Pi but they appear for Rm smaller than 0.3.

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Figure 9a: Time series for Rm=0.3 and Z0=0.14.

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Figure 9b: Time series for Rm=0.2 and Z0=0.14.

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Figure 9c: Time series for Rm=0.25 and Z0=0.14.

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Figure 9d: Time series for Rm=0.275 and Z0=0.14.

In conclusion so far, for large values of Z0 (say, Z0=0.12) and with increasing value of Rm, we have first a region dominated by V-Pi-Ps dynamics. Then for values of Rm around 0.3, we have a co-existence of the two types of dynamics (like in Figs. 7, 9a and possibly 9b). For values of Rm larger than 0.3 but smaller than about 0.425, the dynamics are dominated by Z-P and there is a bloom; for values of Rm larger than about 0.5, dynamics are Z-P but there is no bloom. It may be interesting to fix Z0=0.12 but to chance now beta with Rm to see if we can move the two transitions around.

In Fig. 10, I look at the behavior when Z0 is initially low (Z0=0.02) but Rm is high. Because the value of Rm chosen (0.6) is close to the value we use in previous analyses (0.699), it is not surprising to see a familiar behavior where initially the dynamics are dominated by V-P but are finally dominated by Z-P over the long term.

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Figure 10: Time series for Rm=0.6 and Z0=0.02.

In Fig. 11, I look at the behavior still for low Z0 (Z0=0.02) but also a lower value of Rm (0.35) than in Fig. 11. Here, we first have the V-P dynamics that is then replaced by a limit cycle of Z-P. With Rm=0.25 (Fig. 12), we have the dynamics dominate by V-P although the zooplankton never totally disappears. The beginning is like in Fig. 11 except that instead of seeing V and Pi disappearing, they are maintained while Z never takes off. We might wonder what happens when Rm is in between these two different situations; this is shown in Fig. 13 for Rm=0.3: This figure simply tells us that the V-P and Z-P dynamics will probably never co-exist at low Z0 and what changes is how long the V-P dynamics can “survive”.

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Figure 11: Time series for Rm=0.35 and Z0=0.02.

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Figure 12: Time series for Rm=0.25 and Z0=0.02.

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Figure 13: Time series for Rm=0.3 and Z0=0.02.


To produce Figs. 1 to 4, manyruns_6.py in RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_001 has been used and were plotted with plot_analysis_many_runs_6.py in RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_001/analysis/manyruns_6/. Everything is on the main ipu1 disk.