Population dynamics under demographic and environmental stochasticity
Bibliographic record
Abstract
The present paper is devoted to the study of the long term dynamics of diffusion processes modelling a single species that experiences both demographic and environmental stochasticity. In our setting, the long term dynamics of the diffusion process in the absence of demographic stochasticity is determined by the sign of Λ0, the external Lyapunov exponent, as follows: Λ0<0 implies (asymptotic) extinction and Λ0>0 implies convergence to a unique positive stationary distribution μ0. If the system is of size 1ϵ2 for small ϵ>0 (the intensity of demographic stochasticity), demographic effects will make the extinction time finite almost surely. This suggests that to understand the dynamics one should analyze the quasi-stationary distribution (QSD) μϵ of the system. The existence and uniqueness of the QSD is well known under mild assumptions. We look at what happens when the population size is sent to infinity, that is, when ϵ→0. We show that the external Lyapunov exponent still plays a key role: (1) If Λ0<0, then μϵ→δ0, the mean extinction time is of order |lnϵ| and the extinction rate associated with the QSD μϵ has a lower bound of order 1|lnϵ|; (2) If Λ0>0, then μϵ→μ0, the mean extinction time is polynomial in 1ϵ2 and the extinction rate is polynomial in ϵ2. Furthermore, when Λ0>0 we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD μϵ and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in ϵ for the time spent close to μϵ. In contrast to models that only take into account demographic stochasticity, our results demonstrate the significant effect of environmental stochasticity—it turns an exponentially long mean extinction time to a sub-exponential one.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.022 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
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".