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Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms
Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain
1. | Mathematics and Statistics Department, University of North Carolina Wilmington, Wilmington, NC 28403-5970, United States, United States |
References:
[1] |
M. Candaten and S. Rinaldi, Peak-to-peak dynamics in food chain models, Theoretical Population Biology, 63 (2003), 257-267. |
[2] |
L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, Inc., 1977.
doi: 10.1137/1.9780898719147. |
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W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.
doi: 10.1080/00036819408840277. |
[4] |
W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478.
doi: 10.3934/cpaa.2011.10.1463. |
[5] |
O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains, The American Naturalist, 150 (1997), 328-345. |
[6] |
M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452.
doi: 10.1007/s11538-008-9368-4. |
[7] |
S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181 (2003), 55-83.
doi: 10.1016/S0025-5564(02)00127-X. |
[8] |
Y. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487.
doi: 10.1137/S0036139900378542. |
[9] |
Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics, Mathematical Biosciences, 134 (1996), 1-33.
doi: 10.1016/0025-5564(95)00104-2. |
[10] |
Y. Kuang, Some mechanistically derived population models, Mathematical Biosciences and Engineering, 4 (2007), 1-11. |
[11] |
C. Lu, W. Feng and X. Lu, Long-term survival in a 3 -species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. |
[12] |
S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics, Theoretical Population Biology, 66 (2004), 277-286.
doi: 10.1016/j.tpb.2004.07.002. |
[13] |
M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[14] |
D. M. Wrzosek, Limit cycles in predator-prey models, Mathematical Biosciences, 98 (1990), 1-12.
doi: 10.1016/0025-5564(90)90009-N. |
show all references
References:
[1] |
M. Candaten and S. Rinaldi, Peak-to-peak dynamics in food chain models, Theoretical Population Biology, 63 (2003), 257-267. |
[2] |
L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, Inc., 1977.
doi: 10.1137/1.9780898719147. |
[3] |
W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.
doi: 10.1080/00036819408840277. |
[4] |
W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478.
doi: 10.3934/cpaa.2011.10.1463. |
[5] |
O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains, The American Naturalist, 150 (1997), 328-345. |
[6] |
M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452.
doi: 10.1007/s11538-008-9368-4. |
[7] |
S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181 (2003), 55-83.
doi: 10.1016/S0025-5564(02)00127-X. |
[8] |
Y. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487.
doi: 10.1137/S0036139900378542. |
[9] |
Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics, Mathematical Biosciences, 134 (1996), 1-33.
doi: 10.1016/0025-5564(95)00104-2. |
[10] |
Y. Kuang, Some mechanistically derived population models, Mathematical Biosciences and Engineering, 4 (2007), 1-11. |
[11] |
C. Lu, W. Feng and X. Lu, Long-term survival in a 3 -species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213. |
[12] |
S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics, Theoretical Population Biology, 66 (2004), 277-286.
doi: 10.1016/j.tpb.2004.07.002. |
[13] |
M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[14] |
D. M. Wrzosek, Limit cycles in predator-prey models, Mathematical Biosciences, 98 (1990), 1-12.
doi: 10.1016/0025-5564(90)90009-N. |
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