×

Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey. (English) Zbl 1231.34053

Nonlinear Anal., Real World Appl. 12, No. 6, 2931-2942 (2011); corrigendum ibid. 14, No. 1, 888-891 (2013).
Summary: A bidimensional differential equation system obtained by modifying the well-known predator-prey Rosenzweig-MacArthur model is analyzed by considering prey growth influenced by the Allee effect.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Murdoch, W. W.; Briggs, C. J.; Nisbet, R. M., Consumer-Resources Dynamics (2003), Princeton University Press
[2] Turchin, P., (Complex Population Dynamics. A Theoretical/Empirical Synthesis. Complex Population Dynamics. A Theoretical/Empirical Synthesis, Monographs in Population Biology, vol. 35 (2003), Princeton University Press) · Zbl 1062.92077
[3] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14, 10, 405-410 (1999)
[4] Dennis, B., Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3, 4, 481-538 (1989) · Zbl 0850.92062
[5] Stephens, P. A.; Sutherland, W. J., Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14, 10, 401-405 (1999)
[6] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker · Zbl 0448.92023
[7] Kuang, Y.; Freedman, H. I., Uniqueness of limit cycles in Gause type models of predator-prey systems, Mathematical Biosciences, 88, 67-84 (1988) · Zbl 0642.92016
[8] Xiao, D.; Zhang, Z., On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16, 1185-1201 (2003) · Zbl 1042.34060
[9] Hasík, K., On a predator-prey system of Gause type, Journal of Mathematical Biology, 60, 59-74 (2010) · Zbl 1311.92159
[10] Liu, Y., Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems, Proceedings of the American Mathematical Society, 133, 3619-3626 (2005) · Zbl 1077.34056
[11] Cheng, K. S., Uniqueness of a limit cycle for a predator-prey system, SIAM Journal on Mathematical Analysis, 12, 541-548 (1981) · Zbl 0471.92021
[12] May, R. M., Limit cycles in predator-prey communities, Science, 177, 900-902 (1972)
[13] Albrecht, F.; Gatzke, H.; Wax, N., Stable limit cycles in prey-predator populations, Science, 181, 1073-1074 (1973)
[14] Chicone, C., (Ordinary Differential Equations with Applications. Ordinary Differential Equations with Applications, Texts in Applied Mathematics, vol. 34 (2006), Springer) · Zbl 1120.34001
[15] Gaiko, V. A., (Global Bifurcation Theory and Hilbert’s Sixteenth Problem. Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Mathematics and its Applications, vol. 559 (2003), Kluwer Academic Publishers) · Zbl 1156.34316
[16] Coleman, C. S., Hilbert’s 16th. Problem: How Many Cycles?, (Braun, M.; Coleman, C. S.; Drew, D., Differential Equations Model (1983), Springer Verlag), 279-297
[17] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 185-190 (1999)
[18] Berec, L.; Angulo, E.; Courchamp, F., Multiple allee effects and population management, Trends in Ecology and Evolution, 22, 185-191 (2007)
[19] Courchamp, F.; Berec, L.; Gascoigne, J., Allee effects in Ecology and Conservation (2008), Oxford University Press
[20] Angulo, E.; Roemer, G. W.; Berec, L.; Gascoigne, J.; Courchamp, F., Double allee effects and extinction in the island fox, Conservation Biology, 21, 1082-1091 (2007)
[21] Bazykin, A. D.; Berezovskaya, F. S.; Isaev, A. S.; Khlebopros, R. G., Dynamics of forest insect density: bifurcation approach, Journal of Theoretical Biology, 186, 267-278 (1997)
[22] Conway, E. D.; Smoller, J. A., Global Analysis of a system of predator-prey Equations, SIAM Journal on Applied Mathematics, 46, 4, 630-642 (1986) · Zbl 0608.92016
[23] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations (1998), World Scientific
[24] J.D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares, Consequences of depensation in a Smith’s Bioeconomic model for open-access fishery, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., 2007, pp. 219-232.; J.D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares, Consequences of depensation in a Smith’s Bioeconomic model for open-access fishery, in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., 2007, pp. 219-232.
[25] González-Olivares, E.; Rojas-Palma, A., Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey, Bulletin of Mathematical Biology, 73, 1378-1397 (2011) · Zbl 1215.92061
[26] van Voorn, G. A.K.; Hemerik, L.; Boer, M. P.; Kooi, B. W., Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee, Mathematical Biosciences, 209, 451-469 (2007) · Zbl 1126.92062
[27] Wang, M.-H.; Kot, M., Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171, 83-97 (2001) · Zbl 0978.92033
[28] Wang, J.; Shi, J.; Wei, J., Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62, 291-331 (2011) · Zbl 1232.92076
[29] Clark, C. W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources (1990), John Wiley and Sons · Zbl 0712.90018
[30] Clark, C. W., The Worldwide Crisis in Fisheries: Economic Model and Human Behavior (2007), Cambridge University Press
[31] Liermann, M.; Hilborn, R., Depensation: evidence, models and implications, Fish and Fisheries, 2, 33-58 (2001)
[32] Boukal, D. S.; Sabelis, M. W.; Berec, L., How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72, 136-147 (2007) · Zbl 1123.92034
[33] Boukal, D. S.; Berec, L., Single-species models and the Allee effect: extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218, 375-394 (2002)
[34] E. González-Olivares, B. González-Yañez, J. Mena-Lorca, R. Ramos-Jiliberto, Modelling the Allee effect: are the different mathematical forms proposed equivalents? in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, 2007, pp. 53-71.; E. González-Olivares, B. González-Yañez, J. Mena-Lorca, R. Ramos-Jiliberto, Modelling the Allee effect: are the different mathematical forms proposed equivalents? in: R. Mondaini (Ed.), Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Serviços Editoriais Ltda., Rio de Janeiro, 2007, pp. 53-71.
[35] Goh, B.-S., Management and Analysis of Biological Populations (1980), Elsevier Scientific Publishing Company
[36] González-Olivares, E.; Ramos-Jiliberto, R., Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 166, 135 (2003)
[37] H. Meneses-Alcay, E. González-Olivares, Consequences of the Allee effect on Rosenzweig-MacArthur predator-prey model, in: R.Mondaini (Ed.), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003, E-papers Serviços Editoriais Ltda., vol. 2 2004, pp. 264-277.; H. Meneses-Alcay, E. González-Olivares, Consequences of the Allee effect on Rosenzweig-MacArthur predator-prey model, in: R.Mondaini (Ed.), Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003, E-papers Serviços Editoriais Ltda., vol. 2 2004, pp. 264-277.
[38] Dumortier, F.; Llibre, J.; Artés, J. C., Qualitative Theory of Planar Differential Systems (2006), Springer · Zbl 1110.34002
[39] Arrowsmith, D. K.; Place, C. M., (Dynamical Systems. Differential Equations, Maps and Chaotic Behaviour (1992), Chapman and Hall) · Zbl 0754.34001
[40] Coulson, T.; Rohani, P.; Pascual, M., Skeletons, noise and population growth: the end of an old debate?, Trends in Ecology and Evolution, 19, 359-364 (2004)
[41] Wolfram Research, Mathematica: a system for doing mathematics by computer, Champaing, IL, 1988.; Wolfram Research, Mathematica: a system for doing mathematics by computer, Champaing, IL, 1988. · Zbl 0671.65002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.