Dynamics of a class of 3-dimensional Lotka-Volterra Systems
We provide the complete dynamics of the Lotka-Volterra differential system x˙ = x(ay - cz), y˙ = y(bz - ax), z˙ = z(cx - by), where a, b, c are positive parameters and x, y, z are in the positive octant of R3. In particular we show that this system is completely integrable, i.e. it has two independe...
| Authors: | ; |
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| Format: | Article |
| Publication date: | 2023 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:301150 |
| Online Access: | https://ddd.uab.cat/record/301150 |
| Access Level: | Open access |
| Keyword: | Lotka-Volterra system Invariant Global dynamics Phase portrait |
| Summary: | We provide the complete dynamics of the Lotka-Volterra differential system x˙ = x(ay - cz), y˙ = y(bz - ax), z˙ = z(cx - by), where a, b, c are positive parameters and x, y, z are in the positive octant of R3. In particular we show that this system is completely integrable, i.e. it has two independent first integrals. Fixing one of these first integrals we obtain invariant triangles in the positive octant of R3. The dynamics of the system on each one of these invariant triangles is given by an equilibrium point surrounded by periodic orbits, i.e. by a center. In short all the orbits of these system are either equilibrium points, or periodic orbits. This nonlinear differential system models, under the conservation of mass, a cycle ofirreversible autocatalytic reactions between the different states of three macromolecules and allows to describe stable chemical oscillations. |
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