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Weak Allee effects and species coexistence. (English) Zbl 1231.39008

Summary: We study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.

MSC:

39A22 Growth, boundedness, comparison of solutions to difference equations
39A60 Applications of difference equations
92D25 Population dynamics (general)
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[1] Begon, M.; Harper, J. L.; Townsend, C. R., Ecology. Individuals, Populations and Communities (1996), Blackwell Science: Blackwell Science Oxford
[2] Etiemme, R.; Werthei, B.; Schneider, Hemerik L.; Powell, J., The interaction between dispersal, the Allee effect and scramble competition affects population dynamics, Ecological Modeling, 148, 153-168 (2002)
[3] McCarthy, M. A., The Allee effect, finding mates and theoretical models, Ecological Modelling, 103, 99-102 (1997)
[4] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 185-190 (1999)
[5] Schreiber, S., Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64, 201-209 (2003) · Zbl 1104.92053
[6] Courchamp, F.; Berec, L.; Gascoigne, J., Allee Effects in Ecology and Conservation (2009), Oxford University Press
[7] Zhou, S. R.; Liu, C. Z.; Wang, G., The competitive dynamics of metapopulation subject to the Allee-like effect, Theoretical Population Biology, 65, 29-37 (2004) · Zbl 1105.92029
[8] Dennis, B., Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3, 481-538 (1989) · Zbl 0850.92062
[9] Dennis, B., Allee effects in stochastic populations, Oikos, 96, 389-401 (2002)
[10] Selgrade, J. F.; Namkoong, G., Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses, Journal of Mathematical Biology, 30, 815-826 (1992) · Zbl 0782.92010
[11] Shigesada, N.; Kawasaki, K., Introduction, (Shigesada, N.; Kawasaki, K., Biological Invasions: Theory and Practice (1997), Oxford University Press: Oxford University Press New York, USA), 1-5
[12] Greene, C.; Stamps, J. A., Habitat selection at low population densities, Ecology, 82, 2091-2100 (2001)
[13] Keitt, T. H.; Lewis, M. A.; Holt, R. D., Allee effects, invasion pinning, and species’ borders, The American Naturalist, 157, 203-216 (2001)
[14] Fagan, W. F.; Lewis, M. A.; Neubert, M. G.; van den, Driessche P., Invasion theory and biological control, Ecology Letters, 5, 148-157 (2002)
[15] Wang, M. H.; Kot, M.; Neubert, M. G., Integrodifference equations, Allee effects, and invasions, Journal of Mathematical Biology, 44, 150-168 (2002) · Zbl 0991.92032
[16] Liebhold, A.; Bascompte, J., The Allee effect, stochastic dynamics and the eradication of alien species, Ecology Letters, 6, 133-140 (2003)
[17] Drake, J. M., Allee effects and the risk of biological invasion, Risk Analysis, 24, 795-802 (2004)
[18] Petrovskii, S.; Morozov, A.; Li, B.-L., Regimes of biological invasion in a predator-prey system with the Allee effect, Bulletin of Mathematical Biology, 67, 637-661 (2005) · Zbl 1334.92363
[19] Taylor, C. M.; Hastings, A., Allee effects in biological invasions, Ecology Letters, 8, 895-908 (2005)
[20] Jang, S. R.J., Allee effects in a discrete-time host-parasitoid model, Journal of Difference Equations and Applications, 12, 165-181 (2006) · Zbl 1088.92058
[21] Aguirre, P.; González-Olivares, E.; Sáez, E., Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10, 1401-1416 (2009) · Zbl 1160.92038
[22] Egami, C., Positive periodic solutions of nonautonomous delay competitive systems with weak Allee effect, Nonlinear Analysis: Real World Applications, 10, 494-505 (2009) · Zbl 1154.34370
[23] Egami, C., Permanence of delay competitive systems with weak Allee effects, Nonlinear Analysis: Real World Applications, 11, 3936-3945 (2010) · Zbl 1205.34105
[24] Thieme, H.; Dhirasakdanon, T.; Han, Z.; Trevino, R., Species decline and extinction: synergy of infectious disease and Allee effect?, Journal of Biological Dynamics, 3, 305-323 (2009) · Zbl 1342.92206
[25] Elaydi, S. N.; Sacker, R. J., Population models with Allee effect: a new model, Journal of Biological Dynamics, 4, 397-408 (2010) · Zbl 1342.92166
[26] Nieto, J. J.; Otero-Espinar, M. V.; Rodríguez-López, R., Dynamics of the fuzzy logistic family, (Discrete and Continuous Dynamical Systems: Series B, vol. 14 (2010)), 699-717 · Zbl 1221.37083
[27] Padhi, S.; Srinivasu, P. D.N.; Kumar Kiran, G., Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects, Nonlinear Analysis: Real World Applications, 11, 2610-2618 (2010) · Zbl 1197.34078
[28] Wang, L. M.; Chen, L-S.; Nieto, J. J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Analysis: Real World Applications, 11, 1374-1386 (2010) · Zbl 1188.93038
[29] Liu, J.; Wang, D.; Teng, Z., The persistence and global attractivity in general nonautonomous discrete single-species Kolmogorov model with delays, Journal of Mathematical Analysis and Applications, 378, 403-417 (2011) · Zbl 1209.92049
[30] Amarasekare, P., Allee effects in metapopulation dynamics, The American Naturalist, 152, 298-302 (1998)
[31] Amarasekare, P., Interactions between local dynamics and dispersal: insights from single species models, Theoretical Population Biology, 53, 44-59 (1998) · Zbl 0894.92029
[32] Gyllenberg, M.; Hemminki, J.; Tammaru, T., Allee effects can both conserve and create spatial heterogeneity in population densities, Theoretical Population Biology, 56, 231-242 (1999) · Zbl 0962.92038
[33] Ackleh, A. S.; Allen, L. J.S; Carter, J., Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion, Theoretical Population Biology, 71, 290-300 (2007) · Zbl 1124.92046
[34] Kang, Y.; Lanchier, N., Expansion or extinction: deterministic and stochastic two-patch models with Allee effects, Journal of Mathematical Biology, 62, 6, 925-973 (2011) · Zbl 1230.92040
[35] Kang, Y.; Armbruster, D., Dispersal effects on a two-patch discrete model for plant-herbivore interactions, Journal of Theoretical Biology, 268, 84-97 (2011) · Zbl 1411.92249
[36] Scheuring, I., Allee effect increases dynamical stability in populations, Journal of Theoretical Biology, 199, 407-414 (1999)
[37] Thieme, H., Mathematics in Population Biology (2003), Princeton University Press: Princeton University Press Princeton · Zbl 1054.92042
[38] Hassell, M. P.; Lawton, J. H.; Beddington, J. R., The components of arthropod predation: I. The prey death-rate, Journal of Animal Ecology, 45, 135-164 (1976)
[39] Kang, Y.; Chesson, P., Relative nonlinearity and permanence, Theoretical Population Biology, 78, 26-35 (2010) · Zbl 1403.92249
[40] Y. Kang, 2011, Permanence of a general discrete two-species interaction model with non-monotonic per capita growth rates, 2011, Preprint.; Y. Kang, 2011, Permanence of a general discrete two-species interaction model with non-monotonic per capita growth rates, 2011, Preprint.
[41] Jiang, H.; Rogers, T. D., The discrete dynamics of symmetric competition in the plane, Journal of Mathematical Biology, 25, 573-596 (1987) · Zbl 0668.92011
[42] Cushing, J.; Levarge, S.; Chitnis, N.; Henson, S. M., Some discrete competition models and competitive exclusion principle, Journal of Difference Equations and Applications, 10, 1139-1151 (2004) · Zbl 1071.39005
[43] Cushing, J.; Henson, S. M.; Blackburn, C. C., Multiple mixed-type attractors in a competition model, Journal of Biological Dynamics, 1, 347-362 (2007) · Zbl 1284.92108
[44] R. Luis, S. Elaydi, H. Oliveira, 2011, Stability of a Ricker-type competition model and the competition exclusion principle. Journal of Biological Dynamics, in press (doi:10.1080/17513758.2011.581764; R. Luis, S. Elaydi, H. Oliveira, 2011, Stability of a Ricker-type competition model and the competition exclusion principle. Journal of Biological Dynamics, in press (doi:10.1080/17513758.2011.581764 · Zbl 1236.92070
[45] Edelstein-Keshet, L., Mathematical Models in Biology (2005), SIAM: SIAM Philadelphia · Zbl 1100.92001
[46] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Mathematical Biosciences, 111, 1-71 (1992) · Zbl 0783.92002
[47] Franke, J.; Yakubu, A.-A., Global attractors in competitive systems, Nonlinear Analysis, Theory, Methods & Application, 16, 111-129 (1991) · Zbl 0724.92024
[48] Franke, J.; Yakubu, A.-A., Mutual exclusion versus coexistence for discrete competitive systems, Journal of Mathematical Biology, 30, 161-168 (1991) · Zbl 0735.92023
[49] Luis, R.; Elaydi, S.; Oliveira, H., Nonautonomous periodic systems with Allee effects, Journal of Difference Equations and Applications, 1, 1-16 (2009)
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