Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T17:36:48.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Allee Effect in Gause Type Predator-Prey Models: Existence of Multiple Attractors, Limit cycles and Separatrix Curves. A Brief Review

Published online by Cambridge University Press:  28 November 2013

E. González-Olivares  and
A. Rojas-Palma
Get access

Abstract

This work deals with the consequences on structural stability of Gause type predator-prey models, when are considered three standard functional responses and the prey growth rate is subject to an Allee effect.

An important consequence of this ecological phenomenon is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane. The origin is an attractor for any set of parameters and the existence of heteroclinic curves can be also shown.

Conditions on the parameter values are established to ensure the existence of a unique positive equilibrium, which can be either an attractor or a repellor surrounded by one or more limit cycles.

The influence of the Allee effect on the number of limit cycles is analyzed and the results are compared with analogous models without this phenomenon, and which main features have been given in various above works. Ecological interpretations of these results are also given.

Keywords

Allee effectfunctional responsepredator-prey modelslimit cyclebifurcationsseparatrix curve.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 143 - 164
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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) (2009), 1244-1269. CrossRefGoogle Scholar
Angulo, E., Roemer, G. W., Berec, L., Gascoigne, J., Courchamp, F.. Double Allee effects and extinction in the island fox. Conser. Biol., 21 (2007), 1082-1091. CrossRefGoogle Scholar
D. K. Arrowsmith, C. M. Place. Dynamical System. Differential equations, maps and chaotic behaviour, Chapman and Hall, London, 1992.
Bazykin, A. D., Berezovskaya, F. S., Isaev, A. S., Khlebopros, R. G., Dynamics of forest insect density: Bifurcation approach, J. Theor. Biol., 186 (1997), 267-278. CrossRefGoogle ScholarPubMed
A.D. Bazykin. Nonlinear dynamics of interacting populations. Nonlinear Sciences Series A Vol. 11, World Scientific, Singapore, 1998.
Berec, L., Angulo, E., Courchamp, F.. Multiple Allee effects and population management. Trends Ecol. Evol., 22 (2007), 185-191.CrossRefGoogle ScholarPubMed
Berezovskaya, F., Karev, G., Arditi, R.. Parametric analysis of the ratio-dependent predator-prey model. J. Math. Biol., 43 (2001), 221-246. CrossRefGoogle ScholarPubMed
Boukal, D. S., Berec, L.. Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters. J. Theor. Biol., 218 (2002), 375-394. CrossRefGoogle ScholarPubMed
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 (2007), 136-147. CrossRefGoogle ScholarPubMed
Cheng, K. S.. Uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. Anal., 12 (1981), 541-548. CrossRefGoogle Scholar
C. Chicone, Ordinary differential equations with applications (2nd edition). Texts in Applied Mathematics 34, Springer, New York, 2006.
Clark, C. W.. Mathematical models in the economics of renewable resources. SIAM Rev., 21(1) (1979), 81-99. CrossRefGoogle Scholar
C. W. Clark, Mathematical Bioeconomics: The optimal management of renewable resources (2nd edition), John Wiley and Sons, New York, 1990.
C. W. Clark. The Worldwide Crisis in Fisheries: Economic Model and Human Behavior. Cambridge University Press, Cambridge, 2007.
C. S. Coleman. Hilbert’s 16th problem: How many cycles?. In: M. Braun, C. S. Coleman and D. Drew (Eds.) Differential Equations Models, Springer-Verlag, New York, (1983), 279-297.
Collings, J. B., Wollkind, D. J.. A global analysis of a temperature-dependent model system for a mite predator-prey interaction. SIAM J. Appl. Math., 50 (1990), 1348-1372. CrossRefGoogle Scholar
Conway, E. D., Smoller, J. A.. Global Analysis of a system of predator-prey equations. SIAM J. Appl. Math., 46 (1986), 630-642. CrossRefGoogle Scholar
Courchamp, F., Clutton-Brock, T., Grenfell, B.. Inverse density dependence and the Allee effect. Trends Ecol. Evol., 14(10) (1999), 405-410. CrossRefGoogle ScholarPubMed
F. Courchamp, L. Berec, J. Gascoigne. Allee effects in Ecology and Conservation. Oxford University Press, Oxford, 2007.
F. Dumortier, J. Llibre, J. C. Artés.Qualitative theory of planar differential systems. Springer, Berlin, 2006.
J. D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares. Consequences of depensation in a Smithś bioeconomic model for open-access fishery. In (R. Mondaini and R. Dilao Eds.) Proceedings of International Symposium on Mathematical and Computational Biology. E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 219-232.
H. I. Freedman, Deterministic mathematical model in Population Ecology, Marcel Dekker, New York, 1980.
Fussmann, G. F., Blasius, B.. Community response to enrichment is highly sensitive to model structure. Biol. Lett. 1 (2004), 9-12. CrossRefGoogle ScholarPubMed
V. Gaiko. Global bifurcation Theory and Hilbertś sexteenth problem. Kluwer Academic Press, Boston, 2003.
Gascoigne, J. C., Lipcius, R.. Allee effects driven by predation. J. Appl. Ecol. 41 (2004), 801-810. CrossRefGoogle Scholar
G. F. Gause. The Struggle for Existence. Dover Publications Inc, London, 1971.
Getz, W. M.. A hypothesis regarding the abruptness of density dependence and the growth rate populations. Ecology 77 (1996), 2014-2026. CrossRefGoogle Scholar
L. Ginzburg, M. Colyvan. Ecological Orbits: How Planets Move and Populations Grow. Oxford University Press, Oxford, 2004.
B-S. Goh, Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, Amsterdam, 1980.
González-Olivares, E., Ramos-Jiliberto, R.. Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model. 166 (2003), 135-146. CrossRefGoogle Scholar
González-Olivares, E., González-Yañez, B., Sáez, E., Szantó N, I.. On the number of limit cycles in a predator prey model with non-monotonic functional response. Discrete Cont. Dyn-B., 6 (2006), 525-534. Google Scholar
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 International Symposium on Mathematical and Computational Biology. E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 53-71.
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. Appl. Math. Model., 3 (2011), 366-381. CrossRefGoogle Scholar
González-Olivares, E., Meneses-Alcay, H., González-Yañez, B., Mena-Lorca, J., Rojas-Palma, A., Ramos-Jiliberto, R.. Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey. Nonlinear Anal-Real., 12 (2011), 2931-2942. CrossRefGoogle Scholar
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. Bull. Math. Biol., 73 (2011), 1378-1397. CrossRefGoogle Scholar
González-Olivares, E., Rojas-Palma, A.. Limit cycles in a Gause type predator-prey model with sigmoid functional response and weak Allee effect on prey. Math. Method App. Sci. 35 (2012), 963-975. CrossRefGoogle Scholar
González-Olivares, E., González-Yañez, B., Mena-Lorca, J., Rojas-Palma, A., D. Flores, J., Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comp. Math. App., 62 (2011), 3449-3463. CrossRefGoogle Scholar
B. González-Yañez, E. González-Olivares, Consequences of Allee effect on a Gause type predator-prey model with nonmonotonic functional response, In R. Mondaini (Ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais Ltda, Río de Janeiro, Vol. 2 (2004), 358-373.
Hasík, K.. Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline. Math. Biosci., 164 (2000), 203-215. CrossRefGoogle ScholarPubMed
Hasík, K.. On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59-74. CrossRefGoogle ScholarPubMed
Hesaaraki, M., Moghadas, S. M.. Existence of limit cycles for predator-prey systems with a class of functional responses. Ecol. Model. 142 (2001), 1-9. CrossRefGoogle Scholar
Hilker, F. M., Langlais, M., Malchow, H.. The Allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations. Am. Nat., 173(1) (2009), 72-88. CrossRefGoogle ScholarPubMed
Huang, X-C., Wang, Y., Zhu, L.. One and three limit cycles in a cubic predator-prey system. Math. Method App. Sci., 30 (2007), 501-511. CrossRefGoogle Scholar
Hwang, T-W.. Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl., 290 (2004), 113-122. CrossRefGoogle Scholar
Hsu, S-B., Hwang, T-W., Kuang, Y.. Global dynamics of a predator-prey model with Hassell-Varley type functional response. Discrete Cont. Dyn. S., 10 (2008), 857-871. Google Scholar
Hui, C., Li, Z.. Distribution patterns of metapopulation determined by Allee effects. Popul. Ecol., 46 (2004), 55-63. CrossRefGoogle Scholar
M. Kot, Elements of Mathematical Ecology, Cambridge University Press 2001.
Kuang, Y., I. Freedman, H., Uniqueness of limit cycles in Gause type models of predator-prey systems, Math. Biosci. 88 (1988) 67-84. CrossRefGoogle Scholar
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (2nd Edition), Springer-Verlag, New York, 1998.
Lamontagne, Y., Coutu, C., Rousseau, C.. Bifurcation analysis of a predator-prey system with generalised Holling type III functional response. J. Dyn. Differ. Equ., 20(3) (2008), 535-571. CrossRefGoogle Scholar
Liermann, M., Hilborn, R.. Depensation: evidence, models and implications. Fish Fish 2 (2001), 33-58. CrossRefGoogle Scholar
Ludwig, D., Jones, D. D., Holling, C. S.. Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Ecol., 36 (1978), 204-221. Google Scholar
R. M. May, Stability and complexity in model ecosystems(2nd Edition), Princeton University Press, Princeton, NJ, 2001.
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., Río de Janeiro, Volumen 2 (2004), 264-277.
Middlemas, S. J., Barton, T. R., Armstrong, J. D., Thompson, P. M.. Functional and aggregative responses of harbour seals to changes in salmonid abundance. P. Roy. Soc. B., 273 (2006), 193-198. CrossRefGoogle ScholarPubMed
Moghadas, S. M., Corbett, B. D.. Limit cycles in a generalized Gause-type predator-prey model. Chaos Solitons Fract., 37 (2008), 1343-1355. CrossRefGoogle Scholar
Morozov, A., Arashkevich, E.. Patterns of zooplankton functional response in communities with vertical heterogeneity: a model study. Math. Mod. Nat. Phenom., 3(3) (2008) 131-148. CrossRefGoogle Scholar
W. W. Murdoch, C. J. Briggs, R. M. Nisbet. Consumer-Resources Dynamics. Monographs in Population Biology 36, Princeton University Press, Princeton, 2003.
Myerscough, M. R., Darwen, M. J., Hogarth, W. L.. Stability, persistence and structural stability in a classical predator-prey model. Ecol. Model., 89 (1996) 31-42. CrossRefGoogle Scholar
L. Perko. Differential Equations and Dynamical Systems (3rd. Edition). Springer-Verlag, New York, 2001.
A. Rojas-Palma, E. González-Olivares, B. González-Yañez, Metastability in a Gause type predator-prey models with sigmoid functional response and multiplicative Allee effect on prey, In R. Mondaini (Ed.) Proceedings of International Symposium on Mathematical and Computational Biology, E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 295-321.
Seo, G. and Kot, M.. A comparison of two predator-prey models with Holling’s type I functional response. Math. Biosci., 212 (2008) 161-179. CrossRefGoogle ScholarPubMed
Sen, M., Banerjee, M., Morozov, A.. Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect. Ecol. Complex., 11 (2013), 12-27. CrossRefGoogle Scholar
Spencer, P. D., Collie, J. S.. A simple predator-prey model of exploited marine fish populations incorporating alternative prey. ICES J. Mar. Sci., 53 (1995), 615-628. CrossRefGoogle Scholar
Stephens, P. A., Sutherland, W. J.. Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol. Evol., 14 (1999), 401-405. CrossRefGoogle ScholarPubMed
Stephens, P. A., Sutherland, W. J., Freckleton, R. P.. What is the Allee effect?. Oikos 87 (1999), 185-190. CrossRefGoogle Scholar
Sugie, J., Miyamoto, K., Morino, K.. Absence of limits cycle of a predator-prey system with a sigmoid functional response. Appl. Math. Lett., 9 (1996), 85-90. CrossRefGoogle Scholar
Sugie, J., Kohno, R., Miyazaki, R., On a predator-prey system of Holling type. P. Am. Math. Soc., 125 (1997), 2041-2050. CrossRefGoogle Scholar
R. J. Taylor. Predation. Chapman and Hall, New York,1984.
P. Turchin. Complex population dynamics. A theoretical/empirical synthesis. Mongraphs in Population Biology 35 Princeton University Press, Princeton, NJ, 2003.
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. Math. Biosci., 209 (2007), 451-469. CrossRefGoogle ScholarPubMed
K. Vilches-Ponce, E. González-Olivares. A Gause-type predator-prey model with a non-differentiable functional response. (2013) submitted.
Wang, M-H., Kot, M., Speeds of invasion in a model with strong or weak Allee effects, Math. Bioci., 171 (2001), 83-97. CrossRefGoogle ScholarPubMed
Wang, J., Shi, J., Wei, J.. Predator-prey system with strong Allee effect in prey. J. Math. Biol., 62 (2011), 291-331. CrossRefGoogle ScholarPubMed
S. Wolfram Research, Mathematica: A System for Doing Mathematics by Computer (2nd edition), Wolfram Research, Addison Wesley 1991.
Wollkind, D. J., Collings, J. B., Logan, J. A.. Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees. Bull. Math. Biol., 50(4) (1988), 379-109. CrossRefGoogle Scholar
Xiao, D., Zhang, Z.. On the uniqueness and nonexistence of limit cycles for predator-prey systems. Nonlinearity 16 (2003), 1185-1201. CrossRefGoogle Scholar
Zu, J., Mimura, M.. The impact of Allee effect on a predator-prey system with Holling type II functional response. Appl. Math. Comp., 217 (2010), 3542-3556. CrossRefGoogle Scholar