Existence and Stability of a Spike in the Central Component for a Consumer Chain Model
Bibliographic record
Abstract
We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in Gierer and Meinhardt (Kybernetik 12:30–39, 1972 ) and Schnakenberg (J Theoret Biol 81:389–400, 1979 ) for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect : if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction–diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| 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".