Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response
We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditi...
Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2018-01-01
|
| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/2386954 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849308655470510080 |
|---|---|
| author | Huayong Zhang Shengnan Ma Tousheng Huang Xuebing Cong Zichun Gao Feifan Zhang |
| author_facet | Huayong Zhang Shengnan Ma Tousheng Huang Xuebing Cong Zichun Gao Feifan Zhang |
| author_sort | Huayong Zhang |
| collection | DOAJ |
| description | We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations. |
| format | Article |
| id | doaj-art-0eb6c38d2db64fb8ac0791ffe8d5eb82 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-0eb6c38d2db64fb8ac0791ffe8d5eb822025-08-20T03:54:24ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/23869542386954Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional ResponseHuayong Zhang0Shengnan Ma1Tousheng Huang2Xuebing Cong3Zichun Gao4Feifan Zhang5Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaResearch Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaResearch Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaResearch Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaResearch Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaResearch Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, ChinaWe present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.http://dx.doi.org/10.1155/2018/2386954 |
| spellingShingle | Huayong Zhang Shengnan Ma Tousheng Huang Xuebing Cong Zichun Gao Feifan Zhang Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response Discrete Dynamics in Nature and Society |
| title | Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response |
| title_full | Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response |
| title_fullStr | Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response |
| title_full_unstemmed | Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response |
| title_short | Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response |
| title_sort | complex dynamics on the routes to chaos in a discrete predator prey system with crowley martin type functional response |
| url | http://dx.doi.org/10.1155/2018/2386954 |
| work_keys_str_mv | AT huayongzhang complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse AT shengnanma complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse AT toushenghuang complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse AT xuebingcong complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse AT zichungao complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse AT feifanzhang complexdynamicsontheroutestochaosinadiscretepredatorpreysystemwithcrowleymartintypefunctionalresponse |