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Dynamics of predator-prey models with a strong Allee effect on the prey and predator-dependent trophic functions. (English) Zbl 1366.92100
Summary: The complex dynamics of a two-trophic chain are investigated. The chain is described by a general predator-prey system, in which the prey growth rate and the trophic interaction functions are defined only by some properties determining their shapes. To account for undercrowding phenomena, the prey growth function is assumed to model a strong Allee effect; to simulate the predator interference during the predation process, the trophic function is assumed predator-dependent. A stability analysis of the system is performed, using the predation efficiency and a measure of the predator interference as bifurcation parameters. The admissible scenarios are much richer than in the case of prey-dependent trophic functions, investigated in the authors’ paper [ibid. 12, No. 5, 2871–2887 (2011; Zbl 1227.34046)]. General conditions for the number of equilibria, for the existence and stability of extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated. Numerical results illustrate the qualitative behaviours of the system, in particular the presence of limit cycles, of global bifurcations and of bistability situations.

MSC:
92D25 Population dynamics (general)
34D20 Stability of solutions to ordinary differential equations
Software:
MATCONT
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[1] Logan, J. A., Derivation and analysis of composite models for insect populations, (Estimation and Analysis of Insect Populations, Lecture Notes in Statistic, vol. 55, (1989), Springer), 278-288
[2] Buffoni, G.; Gilioli, G., A lumped parameter model for acarine predator-prey population interactions, Ecol. Modell., 170, 2, 155-171, (2003)
[3] Aguirre, P.; González-Olivares, E.; Sáez, E., Two limit cycles in a Leslie-gower predator-prey model with additive allee effect, Nonlinear Anal. RWA, 10, 3, 1401-1416, (2009) · Zbl 1160.92038
[4] Boukal, D. S.; Berec, L., Modelling mate-finding allee effects and populations dynamics, with applications in pest control, Popul. Ecol., 51, 3, 445-458, (2009)
[5] Gilpin, M. E., A model of the predator-prey relationship, Theor. Popul. Biol., 5, 3, 333-344, (1974) · Zbl 0291.92032
[6] Kuno, E., Principles of predator-prey interaction in theoretical, experimental and natural population system, Adv. Ecol. Res., 16, 249-337, (1987)
[7] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the allee effect?, Oikos, 87, 185-190, (1999)
[8] Taylor, C. M.; Hastings, A., Allee effects in biological invasions, Ecol. Lett., 8, 8, 895-908, (2005)
[9] Gilpin, M. E.; Soulé, M. E., Minimum viable populations: processes of species extinction, (Soulé, M., Conservation Biology: The Science of Scarcity and Diversity, (1986), Sinauer Associates Sunderland, Massachusetts), 13-34
[10] Yablokov, A., Population biology: progress and problems of studies on natural populations, (1986), MIR Publishers Moscow
[11] Svirezhev, Y.; Logofet, D., Stability of biological communities, vol. 112, (1983), MIR Publishers Moscow
[12] Buffoni, G.; Groppi, M.; Soresina, C., Effects of prey over-undercrowding in predator-prey systems with prey-dependent trophic functions, Nonlinear Anal. RWA, 12, 5, 2871-2887, (2011) · Zbl 1227.34046
[13] Bazykin, A. D., Nonlinear dynamics of interacting populations, vol. 11, (1998), World Scientific Singapore
[14] Holling, C. S., The functional response of invertebrate predators to prey density, Mem. Entomol. Soc. Can., 98, 5-86, (1966)
[15] Ivlev, V., Experimental ecology of the feeding of fishes, vol. 42, (1961), Yale University Press New Haven, CT
[16] Royama, T., A comparative study of models for predation and parasitism, Res. Popul. Ecol., S1, 1-91, (1971)
[17] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139, 3, 311-326, (1989)
[18] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 5, 1530-1535, (1992)
[19] Ginzburg, L. R.; Akçakaya, H. R., Consequences of ratio-dependent predation for steady-state properties of ecosystems, Ecology, 73, 5, 1536-1543, (1992)
[20] 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, Theor. Popul. Biol., 72, 1, 136-147, (2007) · Zbl 1123.92034
[21] Gutierrez, A., Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm, Ecology, 73, 5, 1552-1563, (1992)
[22] Berezovskaya, F.; Karev, G.; Arditi, R., Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43, 3, 221-246, (2001) · Zbl 0995.92043
[23] Aguirre, P., A general class of predation models with multiplicative allee effect, Nonlinear Dynam., 78, 1, 629-648, (2014) · Zbl 1314.92121
[24] Van Coller, L., Automated techniques for the qualitative analysis of ecological models: continuous models, Conserv. Ecol., 1, 1, 5, (1997), Available from http://www.ecologyandsociety.org/vol1/iss1/art5/
[25] Buffoni, G.; Cassinari, M. P.; Groppi, M.; Serluca, M., Modelling of predator-prey trophic interactions. part I: two trophic levels, J. Math. Biol., 50, 6, 713-732, (2005) · Zbl 1066.92048
[26] Abrams, P. A.; Ginzburg, L. R., The nature of predation: prey dependent, ratio dependent or neither?, Trends Ecol. Evol., 15, 8, 337-341, (2000)
[27] Beddington, J., Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44, 331-340, (1975)
[28] DeAngelis, D. L.; Goldstein, R.; O’Neill, R., A model for trophic interaction, Ecology, 56, 881-892, (1975)
[29] Gutierrez, A.; Baumgärtner, J., Multitrophic level models of predator-prey energetics: II. A realistic model of plant-herbivore-parasitoid-predator interactions, Can. Entomol., 116, 7, 933-949, (1984)
[30] Seo, G.; DeAngelis, D. L., A predator-prey model with a Holling type I functional response including a predator mutual interference, J. Nonlinear Sci., 21, 6, 811-833, (2011) · Zbl 1238.92049
[31] Freedman, H.; So, J.-H., Global stability and persistence of simple food chains, Math. Biosci., 76, 1, 69-86, (1985) · Zbl 0572.92025
[32] Perko, L., Differential equations and dynamical systems, (1991), Springer-Verlag New York · Zbl 0717.34001
[33] Arnol’d, V. I., Catastrophe theory, (1992), Springer-Verlag Berlin · Zbl 0746.58001
[34] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol. 42, (1983), Springer Verlag New York
[35] Kuznetsov, Y. A., Elements of applied bifurcation theory, vol. 112, (1998), Springer Verlag New York
[36] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A., Matcont: A MATLAB package for numerical bifurcation analysis of odes, ACM Trans. Math. Software, 29, 2, 141-164, (2003), Software available from http://www.matcont.ugent.be/ · Zbl 1070.65574
[37] Aguirre, P.; González-Olivares, E.; Sáez, E., Three limit cycles in a Leslie-gower predator-prey model with additive allee effect, SIAM J. Appl. Math., 69, 5, 1244-1262, (2009) · Zbl 1184.92046
[38] Aguirre, P.; Flores, J. D.; González-Olivares, E., Bifurcations and global dynamics in a predator-prey model with a strong allee effect on the prey, and a ratio-dependent functional response, Nonlinear Anal. RWA, 16, 235-249, (2014) · Zbl 1298.34078
[39] Sen, M.; Banerjee, M.; Morozov, A., Bifurcation analysis of a ratio-dependent prey-predator model with the allee effect, Ecol. Complex., 11, 12-27, (2012)
[40] Flores, J. D.; González-Olivares, E., Dynamics of a predator-prey model with allee effect on prey and ratio-dependent functional response, Ecol. Complex., 18, 59-66, (2014)
[41] Gao, Y.; Li, B., Dynamics of a ratio-dependent predator-prey system with a strong allee effect, Discrete Contin. Dyn. Syst. Ser. B, 18, 9, 2283-2313, (2013) · Zbl 1281.34080
[42] Adamson, M.; Morozov, A. Y., Bifurcation analysis of models with uncertain function specification: how should we proceed?, Bull. Math. Biol., 76, 5, 1218-1240, (2014) · Zbl 1297.92059
[43] Van Voorn, G. A.; Hemerik, L.; Boer, M. P.; Kooi, B. W., Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong allee effect, Math. Biosci., 209, 2, 451-469, (2007) · Zbl 1126.92062
[44] Zu, J., Global qualitative analysis of a predator-prey system with allee effect on the prey species, Math. Comput. Simulation, 94, 33-54, (2013)
[45] Nundloll, S.; Mailleret, L.; Grognard, F., Influence of intrapredatory interferences on impulsive biological control efficiency, Bull. Math. Biol., 72, 8, 2113-2138, (2010) · Zbl 1201.92061
[46] Zu, J.; Mimura, M., The impact of allee effect on a predator-prey system with Holling type II functional response, Appl. Math. Comput., 217, 7, 3542-3556, (2010) · Zbl 1202.92088
[47] Buffoni, G.; Cassinari, M. P.; Groppi, M., Modelling of predator-prey trophic interactions. part II: three trophic levels, J. Math. Biol., 54, 5, 623-644, (2007) · Zbl 1114.92063
[48] Kuznetsov, Y. A.; Rinaldi, S., Remarks on food chain dynamics, Math. Biosci., 134, 1, 1-33, (1996) · Zbl 0844.92025
[49] Adamson, M.; Morozov, A. Y., When can we trust our model predictions? unearthing structural sensitivity in biological systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469, 20120500, (2012) · Zbl 1371.92057
[50] McCann, K.; Yodzis, P., Biological conditions for chaos in a three-species food chain, Ecology, 75, 561-564, (1994)
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