A free boundary problem of competition-diffusion system with Dirichlet boundary condition
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Bibliographic record
Abstract
Abstract In this article, we investigate a free boundary problem for the Lotka–Volterra model consisting of an invasive species with density u and a native species with density v in one dimension. We assume that v undergoes diffusion and growth in $[0,+\infty )$ , and u invades into the environment with spreading front $x=h(t)$ satisfying free boundary condition $h'(t)=-u_x(t,h(t))-\alpha $ for some decay rate $\alpha>0$ , this is caused by the bad environment at the boundary. When u is an inferior competitor, $u(t,x)$ and $h(t)$ tend to 0 within a finite time, while another specie $v(t,x)$ tends to a stationary $\Lambda (x)$ defined on the half-line. When u is a superior competitor, we have a trichotomy result: spreading of u and vanishing of v (i.e., as $t \to +\infty $ , $h(t)$ goes to $+\infty $ and $(u,v)\to (\Lambda ,0)$ ); the transition case (i.e., as $t \to +\infty $ , $(u,v)\to (w_\alpha , \eta _\alpha )$ , $h(t)$ tends to a finite point); vanishing of u and spreading of v (i.e., $u(t,x)$ and $h(t)$ tends to 0 within a finite time, $v(t,x)$ converges to $\Lambda (x)$ ). Additionally, we show that this trichotomy result depends on the initial data $u(0,x)$ .
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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 it