Set up a few long runs of simulated data to better illustrate the effects of sampling time and density, and better resolve the distributions.
Longer baseline run, nice 19, indicator_vs_likelihood.R 1600 reps, m=-0.015, N=500, T=100
Longer sampling, same interval: nice 18, stronger_signal.R 160 reps m=-0.015, N = 2500, T=500
Fails to compute, looks like poor estimates of model fits.
Denser sampling, nice 17: indicator_vs_likelihood.R 160 reps, m=-0.015, N=2500, T=100
Longer less dense sampling, nice 16: indicator_vs_likelihood.R 160 reps, m=-0.015, N=500, T=500. Fails to compute, looks like poor estimates of model fits.
Less dense sampling, nice 16: indicator_vs_likelihood.R 160 reps, m=-0.015, N=50, T=100.
Results pulled in below:
[flickr-gallery mode=“search” tags=“warningsignals” min_upload_date=“2011-03-14 00:00:00” max_upload_date=“2011-03-15 23:59:59”]
Bifurcations
Transcritical bifurcation
- Canonical form:
\[ \frac{dx}{dt} = -r_t x^2 \]
driven by changing sign on $ r_t $
- Example biological model (with stochasticity):
\[ dX_t = r_t X_t (1-X_t/K)dt + \sigma dB_t \]
- Linearized form:
\[ dX_t = r_t (K-X_t)dt + \sigma dB_t \]
Saddle-node
- Canonical form:
\[ \frac{dx}{dt} = r - x^2 \]
Driven by changing sign on $ r_t $
- ** Example biological model:**
\[ dX_t = \left( \frac{e K x^2}{X^2 + h_t^2} - e X_t - a_t\right) dt + \sigma \sqrt{ \frac{e K x^2}{X^2 + h_t^2} + e X_t + a_t} dB_t \]
Either $ a_t$ or $ e$ decreasing through zero can force this through a saddle-node bifurcation, see this entry. Either of these can be approximated as a quadratic in the neighborhood of the stable point, which gives the bifurcation the canonical form,
\[ dX = r_t - (\theta - X_t)^2 dt + \sqrt{\sigma (r_t+ (\theta-X_t)^2) }dB_t \]
- Linearized:
\[ \phi_t = \sqrt{r_t} + \theta \] \[ dX = 2 \sqrt{ r_t } \left(\phi_t - X_t\right) + \sigma \sqrt{ \phi_t } dB_t \]
Previous entries:
Visualizing Saddle-node types – should add bifurcation diagram, and transcritical bifurcations to this set
Writing/figures
Need a panel and figure explaining the models
Need a panel and figure explaining the algorithm
Parametric bootstrap monte carlo
Notebook notes
Pasting in from word (or the visual editor?) turns LaTeX minus signs into longer dashes, causing the latex interpreter to fail. Rather annoying syntax error to find.