×

Consequences of double Allee effect on the number of limit cycles in a predator-prey model. (English) Zbl 1236.34067

Summary: Our main goal is to show a comparative analysis of simple continuous time predator-prey models considering the Allee effect affecting the prey population, also known as depensation in fisheries sciences. This phenomenon may be expressed by different mathematical forms, yielding a distinct number of limit cycles surrounding a positive equilibrium point, when two of these different formalizations are considered in the same system.
It is known that the Volterra predation model, using the most usual form to express the Allee effect, has a unique limit cycle. In this work, considering a more complex mathematical expression, the existence of two limit cycles is proved, by means of the Lyapunov quantities.
We argue that the second equation explains the existence of two Allee effects affecting the same population, which could justify the difference observed between the models.
These results imply that the election of mathematical formulation can have consequences on the fit of the observed data, thus leading to mistakes for ecologists.
We conclude that the oscillatory behaviors and overall dynamics depend strongly on the algebraic expression of the Allee effect, making difficult the proposition of general results. Nevertheless, the techniques reviewed in this paper emerge as key tools to analyze the existence of limit cycles in the presence of multiple Allee effects.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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
[2] Berryman, A. A.; Gutierrez, A. P.; Arditi, R., Credible, parsimonious and useful predator-prey models—a reply to Abrams, Gleeson, and Sarnelle, Ecology, 76, 1980-1985 (1995)
[3] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations (1998), World Scientific · Zbl 0605.92015
[4] Conway, E. D.; Smoller, J. A., Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46, 630-642 (1986) · Zbl 0608.92016
[5] 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
[6] Zu, J.; Mimura, M., The impact of Allee effect on a predator-prey system with Holling type II functional response, Applied Mathematics and Computation, 217, 3542-3556 (2010) · Zbl 1202.92088
[7] May, R. M., Stability and Complexity in Model Ecosystems (2001), Princeton University Press
[8] Hasík, K., On a predator-prey system of Gause type, Journal of Mathematical Biology, 60, 59-74 (2010) · Zbl 1311.92159
[9] Xiao, D.; Zhang, Z., On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16, 1185-1201 (2003) · Zbl 1042.34060
[10] Cheng, K. S., Uniqueness of a limit cycle for a predator-prey system, SIAM Journal on Applied Mathematics, 12, 541-548 (1981) · Zbl 0471.92021
[11] 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
[12] 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
[13] Aguirre, P.; González-Olivares, E.; Sáez, E., Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69, 1244-1262 (2009) · Zbl 1184.92046
[14] González-Olivares, E.; Meneses-Alcay, H.; González-Yañez, B.; Mena-Lorca, J.; Rojas-Palma, A.; Ramos-Jiliberto, Rodrigo, Multiple stability and uniqueness of limit cycle in a Gause-type predator-prey model considering Allee effect on prey, Nonlinear Analysis. Real World Applications, 12, 2931-2942 (2011) · Zbl 1231.34053
[15] 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
[16] González-Olivares, E.; Mena-Lorca, J.; Rojas-Palma, A.; Flores, J. D., Dynamical complexities in the Leslie-Gower predator-prey model considering a simple form to the Allee effect on prey, Applied Mathematical Modelling, 35, 366-381 (2011) · Zbl 1202.34079
[17] González-Yañez, B.; González-Olivares, E., Consequences of Allee effect on a Gause type predator-prey model with nonmonotonic functional response, (Mondaini, R., Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, vol. 2 (2004), E-Papers Serviços Editoriais Ltda.: E-Papers Serviços Editoriais Ltda. Río de Janeiro), 358-373
[18] Chicone, C., (Ordinary Differential Equations with Applications. Ordinary Differential Equations with Applications, Texts in Applied Mathematics, vol. 34 (2006), Springer) · Zbl 1120.34001
[19] 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
[20] Stephens, P. A.; Sutherland, W. J., Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14, 401-405 (1999)
[21] Stephens, P. A.; Sutherland, W. J.; Freckleton, R. P., What is the Allee effect?, Oikos, 87, 185-190 (1999)
[22] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse dependence and the Allee effect, Trends in Ecology & Evolution, 14, 405-410 (1999)
[23] Berec, L.; Angulo, E.; Courchamp, F., Multiple Allee effects and population management, Trends in Ecology & Evolution, 22, 185-191 (2007)
[24] Courchamp, F.; Berec, L.; Gascoigne, J., Allee Effects in Ecology and Conservation (2008), Oxford University Press
[25] 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)
[26] 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)
[27] Kot, M., Elements of Mathematical Ecology (2001), Cambridge University Press
[28] 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 effect, Mathematical Biosciences, 209, 451-469 (2007) · Zbl 1126.92062
[29] 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
[30] Clark, C. W., Mathematical Bioeconomics: The Mathematics of Conservation (2010), John Wiley and Sons Inc.
[31] Liermann, M.; Hilborn, R., Depensation: evidence, models and implications, Fish and Fisheries, 2, 33-58 (2001)
[32] 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)
[33] González-Olivares, E.; González-Yañez, B.; Mena-Lorca, J.; Ramos-Jiliberto, R., Modelling the Allee effect: are the different mathematical forms proposed equivalents?, (Mondaini, R., Proceedings of the 2006 International Symposium on Mathematical and Computational Biology (2007), E-Papers Serviços Editoriais Ltda.: E-Papers Serviços Editoriais Ltda. Río de Janeiro), 53-71
[34] 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, 36-147 (2007) · Zbl 1123.92034
[35] Zhou, S.-R.; Liu, Y.-F.; Wang, G., The stability of predator-prey systems subject to the Allee effect, Theoretical Population Biology, 67, 23-31 (2005) · Zbl 1072.92060
[36] Knipling, E. F., Possibilities of insect control or eradication through the use of sexually sterile males, Journal of Economic Entomology, 48, 4, 459-462 (1955)
[37] Barclay, H.; Mackauer, M., The sterile insect release method for pest control: a density-dependent model, Environmental Entomology, 9, 810-817 (1980)
[38] Flores, J. D.; Mena-Lorca, J.; González-Yañez, B.; González-Olivares, E., Consequences of depensation in a Smith’s bioeconomic model for open-access fishery, (Mondaini, R., Proceedings of the 2006 International Symposium on Mathematical and Computational Biology (2007), E-Papers Serviços Editoriais Ltda.: E-Papers Serviços Editoriais Ltda. Río de Janeiro), 219-232
[39] Freedman, H. I., Deterministic Mathematical Model in Population Ecology (1980), Marcel Dekker · Zbl 0448.92023
[40] Goh, B.-S., Management and Analysis of Biological Populations (1980), Elsevier Scientific Publishing Company
[41] Dumortier, F.; Llibre, J.; Artés, J. C., Qualitative Theory of Planar Differential Systems (2006), Springer · Zbl 1110.34002
[42] Arrowsmith, D. K.; Place, C. M., Dynamical System: Differential Equations, Maps and Chaotic Behaviour (1992), Chapman and Hall · Zbl 0754.34001
[43] Wolfram, S., (Mathematica: A System for Doing Mathematics by Computer (1991), Wolfram Research, Addison Wesley)
[44] Perko, L., (Differential Equations and Dynamical Systems. Differential Equations and Dynamical Systems, Texts in Applied Mathematics, vol. 7 (2001), Springer-Verlag) · Zbl 0973.34001
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.