A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory
In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the no...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/9/1815 |
| _version_ | 1797482011300986880 |
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| author | Florian Munteanu |
| author_facet | Florian Munteanu |
| author_sort | Florian Munteanu |
| collection | DOAJ |
| description | In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view using the Kosambi–Cartan–Chern (KCC) geometric theory. We then determine the nonlinear connection, the Berwald connection, and the five KCC invariants which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Furthermore, we obtain the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points, and we point these out on a few illustrative examples. |
| first_indexed | 2024-03-09T22:22:11Z |
| format | Article |
| id | doaj.art-7108a99c56f44e7c974f66a01a1886d3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T22:22:11Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7108a99c56f44e7c974f66a01a1886d32023-11-23T19:11:22ZengMDPI AGSymmetry2073-89942022-09-01149181510.3390/sym14091815A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric TheoryFlorian Munteanu0Department of Applied Mathematics, University of Craiova, Al. I. Cuza, 13, 200585 Craiova, RomaniaIn this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view using the Kosambi–Cartan–Chern (KCC) geometric theory. We then determine the nonlinear connection, the Berwald connection, and the five KCC invariants which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Furthermore, we obtain the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points, and we point these out on a few illustrative examples.https://www.mdpi.com/2073-8994/14/9/1815predator–prey systemsKolmogorov systemsKCC theorythe deviation curvature tensorJacobi stability |
| spellingShingle | Florian Munteanu A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory Symmetry predator–prey systems Kolmogorov systems KCC theory the deviation curvature tensor Jacobi stability |
| title | A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory |
| title_full | A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory |
| title_fullStr | A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory |
| title_full_unstemmed | A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory |
| title_short | A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory |
| title_sort | study of the jacobi stability of the rosenzweig macarthur predator prey system through the kcc geometric theory |
| topic | predator–prey systems Kolmogorov systems KCC theory the deviation curvature tensor Jacobi stability |
| url | https://www.mdpi.com/2073-8994/14/9/1815 |
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