User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/02/24
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Derin & Alan Meeting12pm
Analytic calculations continuedExpected Variance across EnsemblesThe expected variance across ensembles can be calculated from the FokkerPlanck Equation derived from the linear noise approximation. This should agree with the variance computed over a time window when the system has reached stationarity. I do these calculations and compare: Equilibrium population size solves [math]\displaystyle{ e n  a = \frac{e K n^2}{n^2+h^2} }[/math] Equilibrium variance should be given by [math]\displaystyle{ \hat \sigma^2 = \frac{d(\hat n)}{b'(\hat n)d'(\hat n)} = \frac{e \hat n  a}{ 2 e K \hat n (1  \frac{\hat n}{\hat n^2 + h^2})  e} }[/math] Time averaging from simulation doesn't match analytic prediction!The analytic solution to [math]\displaystyle{ \hat n }[/math] is the general solution to a cubic so not very pretty, but easy to calculate numerically in order to give the variance estimate; using the values below I confirm that ODE solver and analytic solution above agree ([math]\displaystyle{ \hat \sigma^2 = 1352 }[/math]), which is only about half that computed over the time window (approx 2360)! Parameter values and solutions from analytics are:
Test of higher order correctionsI try the higher order correction to the mean dynamics (accounts for inflation effect of the variance): [math]\displaystyle{ \frac{d}{dt} E(n) = \alpha_1(n) + \frac{1}{2} \sigma^2 \alpha_1''(n) }[/math] Though not surprisingly, the effect is negligible. In this case the average is slightly higher (572.5) and variance slightly lower (1349), so the curvature at equilibrium must be positive (in contrast to the deflation example with logistic or Levins model). Perhaps this is a consequence of the ergodicity assumption failing; I'll check against simply computing ensemble variance rather than the (meanoverensembles) timeaveraged variance. Ensemble averaging simulation does match theory, though time averaging doesn't
Reflections on Warning Signals literature
Current Library(Also a chance to test Mendeley library embedding) <html> <div style="padding:0;margin:0;textalign:left"><iframe src="http://www.mendeley.com/collections/embed/1374711/A70805/" style="height:300px;width:450px;border:none;"></iframe><hr style="border:1px solid #E0E0E0;margin:0px 0px 5px 0px;padding:0;width:450px;"/><span style="color:#666666;fontsize:11px;"><a style="color:#A70805" href="http://www.mendeley.com">Research</a> collected using Mendeley</span></div> </html> Subversion Logr25  cboettig  20100225 00:12:09 0800 (Thu, 25 Feb 2010)  1 line made samplefreq into a double instead of size_t, still coopting the autocorrelations, seems to show successfully a different ensemble vs timeaveraged variance r24  cboettig  20100224 23:46:07 0800 (Wed, 24 Feb 2010)  2 lines coopted the autoregressive data for the moment to explore direct ensemble mean and variance (without any timeaveraging) r23  cboettig  20100224 21:53:12 0800 (Wed, 24 Feb 2010)  2 lines ode integrator seems to accurately compute mean and variance dynamics, though the timeaveraged variance doesn't seem to agree with the analytic ensemble variance; will have to compute that directly from simulation to compare. RungeKutta solver on adaptive mesh seems significantly slower than my simple Euler scheme; particularly with the (nearly irrelevant) higher order correction to the mean dynamics.
