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October  2016, 21(8): 2867-2881. doi: 10.3934/dcdsb.2016077

Global dynamics of three species omnivory models with Lotka-Volterra interaction

Ting-Hui Yang 1, , Weinian Zhang 2, and  Kaijen Cheng 3,

1. 

Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137
2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
3. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan

Received  June 2016 Revised  July 2016 Published  September 2016

In this work, we consider the community of three species food web model with Lotka-Volterra type predator-prey interaction. In the absence of other species, each species follows the traditional logistical growth model and the top predator is an omnivore which is defined as feeding on the other two species. It can be seen as a model with one basal resource and two generalist predators, and pairwise interactions of all species are predator-prey type. It is well known that the omnivory module blends the attributes of several well-studied community modules, such as food chains (food chain models), exploitative competition (two predators-one prey models), and apparent competition (one predator-two preys models). With a mild biological restriction, we completely classify all parameters. All local dynamics and most parts of global dynamics are established corresponding to the classification. Moreover, the whole system is uniformly persistent when the unique coexistence appears. Finally, we conclude by discussing the strategy of inferior species to survive and the mechanism of uniform persistence for the three species ecosystem.
Keywords: ecosystem., food web model, three species, omnivory models, Lotka-Volterra interaction.
Mathematics Subject Classification: Primary: 37N25, 92D25, 92D4.
Citation: Ting-Hui Yang, Weinian Zhang, Kaijen Cheng. Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2867-2881. doi: 10.3934/dcdsb.2016077
References:
[1]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proceedings of the American Mathematical Society, 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[2]

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H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Mathematical Biosciences, 68 (1984), 213-231. doi: 10.1016/0025-5564(84)90032-4.  Google Scholar

[4]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology. An International Journal, 12 (1977), 197-229. doi: 10.1016/0040-5809(77)90042-9.  Google Scholar

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R. D. Holt and J. H. Lawton, The Ecological Consequences of Shared Natural Enemies, Annual Review of Ecology and Systematics, 25 (1994), 495-520. doi: 10.1146/annurev.es.25.110194.002431.  Google Scholar

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R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.  Google Scholar

[7]

S.-B. Hsu, S. Ruan and T.-H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, Journal of Mathematical Analysis and Applications, 426 (2015), 659-687. doi: 10.1016/j.jmaa.2015.01.035.  Google Scholar

[8]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, Journal of Mathematical Biology, 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z.  Google Scholar

[9]

N. Krikorian, The Volterra model for three species predator-prey systems: Boundedness and stability, Journal of Mathematical Biology, 7 (1979), 117-132. doi: 10.1007/BF00276925.  Google Scholar

[10]

T. Namba and K. Tanabe, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85. doi: 10.1016/j.ecocom.2008.02.001.  Google Scholar

[11]

E. R. Pianka, On $r$- and $K$-selection, American Naturalist, 104 (1970), 592-597. doi: 10.1086/282697.  Google Scholar

[12]

J. Vandermeer, Omnivory and the stability of food webs, Journal of Theoretical Biology, 238 (2006), 497-504. doi: 10.1016/j.jtbi.2005.06.006.  Google Scholar

[13]

S.-R. Zhou, W.-T. Li and G. Wang, Persistence and global stability of positive periodic solutions of three species food chains with omnivory, Journal of Mathematical Analysis and Applications, 324 (2006), 397-408. doi: 10.1016/j.jmaa.2005.12.021.  Google Scholar

show all references

References:
[1]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proceedings of the American Mathematical Society, 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[2]

H. I. Freedman, S. G. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set, Journal of Dynamics and Differential Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.  Google Scholar

[3]

H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Mathematical Biosciences, 68 (1984), 213-231. doi: 10.1016/0025-5564(84)90032-4.  Google Scholar

[4]

R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology. An International Journal, 12 (1977), 197-229. doi: 10.1016/0040-5809(77)90042-9.  Google Scholar

[5]

R. D. Holt and J. H. Lawton, The Ecological Consequences of Shared Natural Enemies, Annual Review of Ecology and Systematics, 25 (1994), 495-520. doi: 10.1146/annurev.es.25.110194.002431.  Google Scholar

[6]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.  Google Scholar

[7]

S.-B. Hsu, S. Ruan and T.-H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, Journal of Mathematical Analysis and Applications, 426 (2015), 659-687. doi: 10.1016/j.jmaa.2015.01.035.  Google Scholar

[8]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, Journal of Mathematical Biology, 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z.  Google Scholar

[9]

N. Krikorian, The Volterra model for three species predator-prey systems: Boundedness and stability, Journal of Mathematical Biology, 7 (1979), 117-132. doi: 10.1007/BF00276925.  Google Scholar

[10]

T. Namba and K. Tanabe, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85. doi: 10.1016/j.ecocom.2008.02.001.  Google Scholar

[11]

E. R. Pianka, On $r$- and $K$-selection, American Naturalist, 104 (1970), 592-597. doi: 10.1086/282697.  Google Scholar

[12]

J. Vandermeer, Omnivory and the stability of food webs, Journal of Theoretical Biology, 238 (2006), 497-504. doi: 10.1016/j.jtbi.2005.06.006.  Google Scholar

[13]

S.-R. Zhou, W.-T. Li and G. Wang, Persistence and global stability of positive periodic solutions of three species food chains with omnivory, Journal of Mathematical Analysis and Applications, 324 (2006), 397-408. doi: 10.1016/j.jmaa.2005.12.021.  Google Scholar

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