The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator
We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quanti...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2018/6146027 |
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| author | Jevgeņijs Carkovs Jolanta Goldšteine Kārlis Šadurskis |
| author_facet | Jevgeņijs Carkovs Jolanta Goldšteine Kārlis Šadurskis |
| author_sort | Jevgeņijs Carkovs |
| collection | DOAJ |
| description | We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances. |
| format | Article |
| id | doaj-art-dc57084dde524e1d8eeae2a1adfbfcb7 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-dc57084dde524e1d8eeae2a1adfbfcb72025-02-03T06:12:36ZengWileyJournal of Applied Mathematics1110-757X1687-00422018-01-01201810.1155/2018/61460276146027The Holling Type II Population Model Subjected to Rapid Random Attacks of PredatorJevgeņijs Carkovs0Jolanta Goldšteine1Kārlis Šadurskis2Department of Probability Theory and Mathematical Statistics, Riga Technical University, Kaļķu iela 1, Riga LV-1658, LatviaDepartment of Probability Theory and Mathematical Statistics, Riga Technical University, Kaļķu iela 1, Riga LV-1658, LatviaDepartment of Probability Theory and Mathematical Statistics, Riga Technical University, Kaļķu iela 1, Riga LV-1658, LatviaWe present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.http://dx.doi.org/10.1155/2018/6146027 |
| spellingShingle | Jevgeņijs Carkovs Jolanta Goldšteine Kārlis Šadurskis The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator Journal of Applied Mathematics |
| title | The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator |
| title_full | The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator |
| title_fullStr | The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator |
| title_full_unstemmed | The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator |
| title_short | The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator |
| title_sort | holling type ii population model subjected to rapid random attacks of predator |
| url | http://dx.doi.org/10.1155/2018/6146027 |
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