Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response
In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady stat...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Frontiers Media S.A.
2022-12-01
|
| Series: | Frontiers in Applied Mathematics and Statistics |
| Subjects: | |
| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2022.1083815/full |
| _version_ | 1811311373046513664 |
|---|---|
| author | Konstantin B. Blyuss Yuliya N. Kyrychko Oleg B. Blyuss |
| author_facet | Konstantin B. Blyuss Yuliya N. Kyrychko Oleg B. Blyuss |
| author_sort | Konstantin B. Blyuss |
| collection | DOAJ |
| description | In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state. |
| first_indexed | 2024-04-13T10:17:48Z |
| format | Article |
| id | doaj.art-c4b6693c366d4e97be1f77060d049026 |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-04-13T10:17:48Z |
| publishDate | 2022-12-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-c4b6693c366d4e97be1f77060d0490262022-12-22T02:50:40ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872022-12-01810.3389/fams.2022.10838151083815Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional responseKonstantin B. Blyuss0Yuliya N. Kyrychko1Oleg B. Blyuss2Department of Mathematics, University of Sussex, Brighton, United KingdomDepartment of Mathematics, University of Sussex, Brighton, United KingdomWolfson Institute of Population Health, Queen Mary University of London, London, United KingdomIn this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state.https://www.frontiersin.org/articles/10.3389/fams.2022.1083815/fullpredator-prey modeHolling type III functional responseratio dependenceextinctioncoexistence complex dynamics near extinction |
| spellingShingle | Konstantin B. Blyuss Yuliya N. Kyrychko Oleg B. Blyuss Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response Frontiers in Applied Mathematics and Statistics predator-prey mode Holling type III functional response ratio dependence extinction coexistence complex dynamics near extinction |
| title | Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response |
| title_full | Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response |
| title_fullStr | Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response |
| title_full_unstemmed | Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response |
| title_short | Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response |
| title_sort | complex dynamics near extinction in a predator prey model with ratio dependence and holling type iii functional response |
| topic | predator-prey mode Holling type III functional response ratio dependence extinction coexistence complex dynamics near extinction |
| url | https://www.frontiersin.org/articles/10.3389/fams.2022.1083815/full |
| work_keys_str_mv | AT konstantinbblyuss complexdynamicsnearextinctioninapredatorpreymodelwithratiodependenceandhollingtypeiiifunctionalresponse AT yuliyankyrychko complexdynamicsnearextinctioninapredatorpreymodelwithratiodependenceandhollingtypeiiifunctionalresponse AT olegbblyuss complexdynamicsnearextinctioninapredatorpreymodelwithratiodependenceandhollingtypeiiifunctionalresponse |