Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems
There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka–Volterra model does not possess any limit cycles....
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| Format: | Article |
| Language: | English |
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Academy of Science of South Africa
2013-05-01
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| Series: | South African Journal of Science |
| Subjects: | |
| Online Access: | https://www.sajs.co.za/article/view/3888 |
| _version_ | 1818037593209569280 |
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| author | Quay van der Hoff Johanna C. Greeff P. Hendrik Kloppers |
| author_facet | Quay van der Hoff Johanna C. Greeff P. Hendrik Kloppers |
| author_sort | Quay van der Hoff |
| collection | DOAJ |
| description | There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka–Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka–Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator–prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where it is difficult, if not impossible, to prove analytically the existence of limit cycles. |
| first_indexed | 2024-12-10T07:29:18Z |
| format | Article |
| id | doaj.art-27524246841b4749a609999cdcf6a11f |
| institution | Directory Open Access Journal |
| issn | 1996-7489 |
| language | English |
| last_indexed | 2024-12-10T07:29:18Z |
| publishDate | 2013-05-01 |
| publisher | Academy of Science of South Africa |
| record_format | Article |
| series | South African Journal of Science |
| spelling | doaj.art-27524246841b4749a609999cdcf6a11f2022-12-22T01:57:36ZengAcademy of Science of South AfricaSouth African Journal of Science1996-74892013-05-011095/66610.1590/sajs.2013/11433888Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systemsQuay van der Hoff0Johanna C. Greeff1P. Hendrik Kloppers21 Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, South Africa 2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South AfricaDepartment of Mathematics and Statistics, Tshwane University of Technology, Pretoria, South AfricaDepartment of Mathematics and Statistics, Tshwane University of Technology, Pretoria, South AfricaThere has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka–Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka–Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator–prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where it is difficult, if not impossible, to prove analytically the existence of limit cycles.https://www.sajs.co.za/article/view/3888Lotka–Volterra modelspredator–prey systemsstable limit cyclePoincaré mappingnumerical method |
| spellingShingle | Quay van der Hoff Johanna C. Greeff P. Hendrik Kloppers Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems South African Journal of Science Lotka–Volterra models predator–prey systems stable limit cycle Poincaré mapping numerical method |
| title | Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems |
| title_full | Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems |
| title_fullStr | Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems |
| title_full_unstemmed | Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems |
| title_short | Numerical investigation into the existence of limit cycles in two-dimensional predator–prey systems |
| title_sort | numerical investigation into the existence of limit cycles in two dimensional predator prey systems |
| topic | Lotka–Volterra models predator–prey systems stable limit cycle Poincaré mapping numerical method |
| url | https://www.sajs.co.za/article/view/3888 |
| work_keys_str_mv | AT quayvanderhoff numericalinvestigationintotheexistenceoflimitcyclesintwodimensionalpredatorpreysystems AT johannacgreeff numericalinvestigationintotheexistenceoflimitcyclesintwodimensionalpredatorpreysystems AT phendrikkloppers numericalinvestigationintotheexistenceoflimitcyclesintwodimensionalpredatorpreysystems |