Condensation in large random Lotka-Volterra systems
Final Report Abstract
Condensation processes in large complex systems may occur under very different circumstances. Their trademark is the occupation of one particular state by an extensive part of the elements in the system. In ecology the extinction of a certain fraction of potentially viable species is a dramatic realization of such a condensation phenomenon. As a rule large systems depend on very many parameters a notable fraction only few of which are easily accessible. To model such a situation it is useful to draw these parameters at random from sensible distributions and to analyze typical properties of the system that depend on the distributions rather than on the individual realizations. In our work we have studied a well-known mathematical model of an ecological system in which different species compete for a limited selection of resources. For a large number of species and resources – a case hardly studied so far – the model exhibits a sharp transition from a collective phase in which the available resources are used in an optimal way to a vulnerable and sub-optimal phase. The transition occurs when the degree of biodiversity, in the model the ratio of the number of species and the number of resources, falls below a certain threshold. We have substantiated the existence of this transition by providing an alternative and more streamlined derivation. Moreover, we characterized the two phases quantitatively not only in the vicinity of the transition but also away from criticality. Our central tool is to determine under which conditions a large system of random linear equations typically has a solution with all components nonnegative. Our methods can therefore also be applied for the analysis of related problems as arising, e.g., in the optimization of numerical codes, game theory and models of financial markets. Our results are abstract and part of basic research in statistical mechanics. Accordingly, we are not aware of any direct application and believe, that a decent popular presentation would be rather difficult.
Publications
- EPL 124, 18004 (2018) “Systems of random linear equations and the phase transition in MacArthur’s resource-competition model”
S. Landmann and A. Engel
(See online at https://doi.org/10.1209/0295-5075/124/18004) - Phys. Rev. E 97, 032133 (2018) “Random matrices and condensation into multiple states”
S. Sadeghi and A. Engel
(See online at https://doi.org/10.1103/PhysRevE.97.032133) - Phys. Rev. E 101, 062119 (2020) “Large systems of random linear equations with nonnegative solutions: Characterizing the solvable and the unsolvable phase”
S. Landmann and A. Engel
(See online at https://doi.org/10.1103/PhysRevE.101.062119) - Physica A 552, 122544 (2020) “On nonnegative solutions to large systems of random linear equations”
S. Landmann and A. Engel
(See online at https://doi.org/10.1016/j.physa.2019.122544) - eLife, (2021) “A simple regulatory architecture allows learning the statistical structure of a changing environment”
S. Landmann, C. M. Holmes and M. Tikhonov
(See online at https://doi.org/10.7554/elife.67455)
