Dynamics of a predator-prey system with prey subject to Allee effects and disease
In this article, we propose a general predator-prey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (I-class) has no contribution; (ii) Consuming infected prey would contribu...
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AIMS Press
2014-02-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.877 |
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| author | Yun Kang Sourav Kumar Sasmal Amiya Ranjan Bhowmick Joydev Chattopadhyay |
| author_facet | Yun Kang Sourav Kumar Sasmal Amiya Ranjan Bhowmick Joydev Chattopadhyay |
| author_sort | Yun Kang |
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| description | In this article, we propose a general predator-prey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (I-class) has no contribution; (ii) Consuming infected prey would contribute less or negatively to the growth rate of predator (P-class) in comparison to the consumption of susceptible prey (S-class). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model (SIP-Allee Model) as well as its corresponding model in the absence of Allee effects (SIP-no-Allee Model); we obtain the complete dynamics of both models: (a) SIP-Allee Model may have only one attractor (extinction of all species), two attractors (bi-stability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors (tri-stability); (b) SIP-no-Allee Model may have either one attractor (only S-class survives or the persistence of S and I-class or the persistence of S and P-class) or two attractors (bi-stability with the persistence of S and I-class or the persistence of S and P-class). One of the most interesting findings is that neither models can support the coexistence of all three S, I, P-class. This is caused by the assumption (ii), whose biological implications are that I and P-class are at exploitative competition for S-class whereas I-class cannot be superior and P-class cannot gain significantly from its consumption of I-class. In addition, the comparison study between the dynamics of SIP-Allee Model and SIP-no-Allee Model lead to the following conclusions: 1) In the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; 2) In the presence of Allee effects, disease may be able to save prey from the predation-driven extinction and leads to the coexistence of S and I-class while predator can not save the disease-driven extinction. All these findings may have potential applications in conservation biology. |
| format | Article |
| id | doaj-art-219556b07ace4882b468bc3d8aa49a50 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-219556b07ace4882b468bc3d8aa49a502025-01-24T02:28:18ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-02-0111487791810.3934/mbe.2014.11.877Dynamics of a predator-prey system with prey subject to Allee effects and diseaseYun Kang0Sourav Kumar Sasmal1Amiya Ranjan Bhowmick2Joydev Chattopadhyay3Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108In this article, we propose a general predator-prey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (I-class) has no contribution; (ii) Consuming infected prey would contribute less or negatively to the growth rate of predator (P-class) in comparison to the consumption of susceptible prey (S-class). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model (SIP-Allee Model) as well as its corresponding model in the absence of Allee effects (SIP-no-Allee Model); we obtain the complete dynamics of both models: (a) SIP-Allee Model may have only one attractor (extinction of all species), two attractors (bi-stability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors (tri-stability); (b) SIP-no-Allee Model may have either one attractor (only S-class survives or the persistence of S and I-class or the persistence of S and P-class) or two attractors (bi-stability with the persistence of S and I-class or the persistence of S and P-class). One of the most interesting findings is that neither models can support the coexistence of all three S, I, P-class. This is caused by the assumption (ii), whose biological implications are that I and P-class are at exploitative competition for S-class whereas I-class cannot be superior and P-class cannot gain significantly from its consumption of I-class. In addition, the comparison study between the dynamics of SIP-Allee Model and SIP-no-Allee Model lead to the following conclusions: 1) In the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; 2) In the presence of Allee effects, disease may be able to save prey from the predation-driven extinction and leads to the coexistence of S and I-class while predator can not save the disease-driven extinction. All these findings may have potential applications in conservation biology.https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.877eco-epidemiological system.bi-stabilityfunctional responsesdisease/predation-driven extinctiontri-stabilityallee effect |
| spellingShingle | Yun Kang Sourav Kumar Sasmal Amiya Ranjan Bhowmick Joydev Chattopadhyay Dynamics of a predator-prey system with prey subject to Allee effects and disease Mathematical Biosciences and Engineering eco-epidemiological system. bi-stability functional responses disease/predation-driven extinction tri-stability allee effect |
| title | Dynamics of a predator-prey system with prey subject to Allee effects and disease |
| title_full | Dynamics of a predator-prey system with prey subject to Allee effects and disease |
| title_fullStr | Dynamics of a predator-prey system with prey subject to Allee effects and disease |
| title_full_unstemmed | Dynamics of a predator-prey system with prey subject to Allee effects and disease |
| title_short | Dynamics of a predator-prey system with prey subject to Allee effects and disease |
| title_sort | dynamics of a predator prey system with prey subject to allee effects and disease |
| topic | eco-epidemiological system. bi-stability functional responses disease/predation-driven extinction tri-stability allee effect |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.877 |
| work_keys_str_mv | AT yunkang dynamicsofapredatorpreysystemwithpreysubjecttoalleeeffectsanddisease AT souravkumarsasmal dynamicsofapredatorpreysystemwithpreysubjecttoalleeeffectsanddisease AT amiyaranjanbhowmick dynamicsofapredatorpreysystemwithpreysubjecttoalleeeffectsanddisease AT joydevchattopadhyay dynamicsofapredatorpreysystemwithpreysubjecttoalleeeffectsanddisease |