Symplectic methods applied to the Lotka-Volterra system
Bibliographic record
Abstract
We analyse the preservation of physical properties of numerical approximations tosolutions of the Lotka-Volterra system: its positivity and the conservation of theHamiltonian. We focus on two numerical methods : the symplectic Euler method andan explicit variant of it. We first state under which conditions they are symplectic andwe prove they are both Poisson integrators for the Lotka-Volterra system. Then, westudy under which conditions they stay positive. For the symplectic Euler method,we derive a simple condition under which the numerical approximation always stayspositive. For the explicit variant, there is no such simple condition. Using propertiesof Poisson integrators and backward error analysis, we prove that for initial conditionsin a given set in the positive quadrant, there exists a bound on the step size, such thatnumerical approximations with step sizes smaller than the bound stay positive overexponentially long time intervals. We also show how this bound can be estimated.We illustrate all our results by numerical experiments.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".