2016 Fiscal Year Annual Research Report
Mathematical analysis of species coexistence and segregating pattern formation
| Project/Area Number |
16H07254
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| Research Institution | Meiji University |
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Principal Investigator |
コンテント ロレンゾ 明治大学, 研究・知財戦略機構, 研究推進員 (50782562)
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| Project Period (FY) |
2016-08-26 – 2018-03-31
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| Keywords | competition-diffusion / competitive exclusion / species coexistence / singular limit / traveling waves |
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| Outline of Annual Research Achievements |
With growing deterioration of the global environment, research on species diversity is gaining importance also from a theoretical point of view. We are interested in the study of a 3-species competition-diffusion (CD) system, in which coexistence of two otherwise incompatible species can occur thanks to the mediating influence of a third competitor (competitor-mediated coexistence). Such coexistence can be attained with complex spatio-temporal patterns in which two different traveling waves interact with each other.
Ideally, one would want to prove analytically under which conditions on the parameters coexistence occurs in the CD system. We started from discussing which situations cannot result in coexistence, thus limiting the parameter space inside which interesting phenomena may be observed. We showed that when the growth rate of a species tends to infinity, then its density will tend to its carrying capacity while the other species will become extinct. On the other hand, if the growth rate tends to zero, then such species will go extinct.
We also investigated simpler models which display the same qualitative behaviour of the 3-species CD system; in particular they should admit two different stable traveling waves. We believe there is no reason why such behaviour should be restricted to 3 component systems only. Since such a simpler system may be more amenable to singular limit analysis, we may hope to better understand the mechanism behind traveling wave interactions.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The project about proofs of competitive exclusion in competition-diffusion systems has been slightly delayed due to the postponement of my visit to one of my collaborators on said project.
Regarding the search for simpler mathematical models approximating the three-species competition-diffusion system, we have found a candidate better than our initial expectations. However, since this model may also have ecological applications we needed to dedicate more time to the modeling process, by also consulting with experts in theoretical ecology to find the most plausible formulation. We think this was necessary, but has delayed the start of the mathematical study of the model.
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| Strategy for Future Research Activity |
We aim to conclude to study of competitive exclusion in the competition-diffusion system by studying the large time behaviour, in addition to the singular limit problems we already considered. This will allow to prove the same results even in the case the growth rate is very large but still finite or very small but still positive, which is more useful from a modeling point of view than taking the limits to infinity or zero.
We intend to study the behaviour of the simplified model we derived, both in one and two spatial dimensions, seeing whether it can completely reproduce the three-species competition-diffusion model and whether new phenomena appear. We will start investigating the possibility of taking a singular limit and reducing the problem to a moving interface.
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