Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion
Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analy...
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/130851 |
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| author | Lei Zhang |
| author_facet | Lei Zhang |
| author_sort | Lei Zhang |
| collection | DOAJ |
| description | Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point (u2*,v2*,w2*), and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter c1, the sequence “holes → holes-stripe mixtures → stripes → spots-stripe mixtures → spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point (u1*,v1*,0), and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems. |
| format | Article |
| id | doaj-art-16acc8371ff5431ba96726a7e43344a7 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-16acc8371ff5431ba96726a7e43344a72025-02-03T01:30:12ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/130851130851Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-DiffusionLei Zhang0Computer Science and Technology Department, East China Normal University, Shanghai 200241, ChinaPredator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point (u2*,v2*,w2*), and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter c1, the sequence “holes → holes-stripe mixtures → stripes → spots-stripe mixtures → spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point (u1*,v1*,0), and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.http://dx.doi.org/10.1155/2014/130851 |
| spellingShingle | Lei Zhang Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion Abstract and Applied Analysis |
| title | Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion |
| title_full | Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion |
| title_fullStr | Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion |
| title_full_unstemmed | Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion |
| title_short | Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion |
| title_sort | spatiotemporal patterns in a ratio dependent food chain model with reaction diffusion |
| url | http://dx.doi.org/10.1155/2014/130851 |
| work_keys_str_mv | AT leizhang spatiotemporalpatternsinaratiodependentfoodchainmodelwithreactiondiffusion |