MKLau commented 10 years ago

burn in for transience = 100
8 to chaotic threshold at r = 2.571451 (t = 571)
a sequence of r values with roughly equal time steps before and after chaos threshold
error was drawn from a normal distribution (mean = 0) and added to the r sequence
initial N = 10
K = 100
one simulation has sd = 0 and another has sd = 0.01
EWS calculated per Dakos (CRAN package "earlywarnings")
Window Size = 50% of sample

r over time for SD = 0, 0.005 and 0.01 horizontal = Chaos threshold

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MKLau commented 10 years ago

SD = 0

screen shot 2014-06-06 at 6 25 37 pm

MKLau commented 10 years ago

SD = 0.005

screen shot 2014-06-06 at 6 41 53 pm

MKLau commented 10 years ago

SD = 0.01

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MKLau commented 10 years ago

Conduct repeated simulations and get the average trendline.

MKLau commented 10 years ago

Modeling Questions

Where is the burn in?
Where are the thresholds?
What does it mean?

Data Possibilities

Transcriptomics (Warm Ants)
Proteomics (Pitcher Plants)
Food web abundances (Pitcher Plants)

MKLau commented 10 years ago

Where is the burn in? = first 100, but remember the window is 50% observations
Where are the thresholds? = rbar = 487 (8 to 16) and 530 (16 to 8)
What does it mean? From Wikipedia's Logistic Map description (http://en.wikipedia.org/wiki/Logistic_map#Chaos_and_the_logistic_map):

It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically,[8] the invariant measure is \pi ^{-1}x^{-1/2}(1-x)^{-1/2}. Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states a long time into the future, and use this knowledge to inform decisions based on the state of the system.

MKLau commented 10 years ago

No correlation between EWS stats between 8 to 16 vs 16 to 8

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