Allee effect in gause type predator-prey models: Existence of multiple attractors, limit cycles and separatrix curves. A brief review

Mathematical Modelling of Natural Phenomena

Volumn 8, Issue 6, 2013, Pages 143-164

  Morozov, A ^a   Tyutyunov, Yu V ^a   Gonzalez Olivares E ^a   Rojas Palma, A ^a  

^a PONTIFICIA UNIVERSIDAD CATÓLICA DE VALPARAÍSO   (Chile)

Author keywords

Allee effect; Bifurcations; Functional response; Limit cycle; Predator prey models; Separatrix curve.

Indexed keywords

ALLEE EFFECTS; FUNCTIONAL RESPONSE; LIMIT CYCLE; PREDATOR-PREY MODELS; SEPARATRIX;

BIFURCATION (MATHEMATICS); DELTA SIGMA MODULATION; STABILITY;

ECOLOGY;

EID: 84889841792     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20138610     Document Type: Article

References (79)

1. P. Aguirre, E. González-Olivares, E. Sáez. Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM J. Appl. Math., 69 (5) (2009), 1244-1269.
2. E. Angulo, G. W. Roemer, L. Berec, J. Gascoigne, F. Courchamp. Double Allee effects and extinction in the island fox. Conser. Biol., 21 (2007), 1082-1091. (Pubitemid 47076323)
3. D. K. Arrowsmith, C. M. Place. Dynamical System. Differential equations, maps and chaotic behaviour, Chapman and Hall, London, 1992.
4. A. D. Bazykin, F. S. Berezovskaya, A. S. Isaev, R. G. Khlebopros, Dynamics of forest insect density: Bifurcation approach, J. Theor. Biol., 186 (1997), 267-278. (Pubitemid 27281517)
5. A.D. Bazykin. Nonlinear dynamics of interacting populations. Nonlinear Sciences Series A Vol. 11, World Scientific, Singapore, 1998.
6. L. Berec, E. Angulo, F. Courchamp. Multiple Allee effects and population management. Trends Ecol. Evol., 22 (2007), 185-191. (Pubitemid 46437275)
7. F. Berezovskaya, G. Karev, R. Arditi. Parametric analysis of the ratio-dependent predator-prey model. J. Math. Biol., 43 (2001), 221-246. (Pubitemid 33773343)
8. D. S. Boukal, L. Berec. Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters. J. Theor. Biol., 218 (2002), 375-394. (Pubitemid 35191835)
9. D. S. Boukal, M. W. Sabelis, L. Berec. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol., 72 (2007), 136-147. (Pubitemid 46879817)
10. K. S. Cheng. Uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. Anal., 12 (1981), 541-548.
11. C. Chicone, Ordinary differential equations with applications (2nd edition). Texts in Applied Mathematics 34, Springer, New York, 2006.
12. C. W. Clark. Mathematical models in the economics of renewable resources. SIAM Rev., 21 (1) (1979), 81-99.
13. C. W. Clark, Mathematical Bioeconomics: The optimal management of renewable resources (2nd edition), John Wiley and Sons, New York, 1990.
14. C. W. Clark. The Worldwide Crisis in Fisheries: Economic Model and Human Behavior. Cambridge University Press, Cambridge, 2007.
15. 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.
16. J. B. Collings, D. J. Wollkind. A global analysis of a temperature-dependent model system for a mite predator-prey interaction. SIAM J. Appl. Math., 50 (1990), 1348-1372.
17. E. D. Conway, J. A. Smoller. Global Analysis of a system of predator-prey equations. SIAM J. Appl. Math., 46 (1986), 630-642. (Pubitemid 16607826)
18. F. Courchamp, T. Clutton-Brock, B. Grenfell. Inverse density dependence and the Allee effect. Trends Ecol. Evol., 14 (10) (1999), 405-410. (Pubitemid 29428466)
19. F. Courchamp, L. Berec, J. Gascoigne. Allee effects in Ecology and Conservation. Oxford University Press, Oxford, 2007.
20. F. Dumortier, J. Llibre, J. C. Artés.Qualitative theory of planar differential systems. Springer, Berlin, 2006.
21. J. D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares. Consequences of depensation in a Smith 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.
22. H. I. Freedman, Deterministic mathematical model in Population Ecology, Marcel Dekker, New York, 1980.
23. G. F. Fussmann, B. Blasius. Community response to enrichment is highly sensitive to model structure. Biol. Lett. 1 (2004), 9-12.
24. V. Gaiko. Global bifurcation Theory and Hilbert sexteenth problem. Kluwer Academic Press, Boston, 2003.
25. J. C. Gascoigne, R. Lipcius. Allee effects driven by predation. J. Appl. Ecol. 41 (2004), 801-810. (Pubitemid 39348275)
26. G. F. Gause. The Struggle for Existence. Dover Publications Inc, London, 1971.
27. W. M. Getz. A hypothesis regarding the abruptness of density dependence and the growth rate populations. Ecology 77 (1996), 2014-2026.
28. L. Ginzburg, M. Colyvan. Ecological Orbits: How Planets Move and Populations Grow. Oxford University Press, Oxford, 2004.
29. B-S. Goh, Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, Amsterdam, 1980.
30. E. González-Olivares, R. Ramos-Jiliberto. Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model. 166 (2003), 135-146. (Pubitemid 37537755)
31. E. González-Olivares, B. González-Yañez, E. Sáez, I. Szantó N. On the number of limit cycles in a predator prey model with non-monotonic functional response. Discrete Cont. Dyn-B., 6 (2006), 525-534.
32. 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.
33. E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores. 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.
34. E. González-Olivares, H. Meneses-Alcay, B. González- Yañez, J. Mena-Lorca, A. Rojas-Palma, R. Ramos-Jiliberto. 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.
35. E. González-Olivares, A. Rojas-Palma. 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.
36. E. González-Olivares, A. Rojas-Palma. 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.
37. E. González-Olivares, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comp. Math. App., 62 (2011), 3449-3463.
38. 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.
39. K. Hasík. Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline. Math. Biosci., 164 (2000), 203-215. (Pubitemid 30211017)
40. K. Hasík. On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59-74.
41. M. Hesaaraki, S. M. Moghadas. Existence of limit cycles for predator-prey systems with a class of functional responses. Ecol. Model. 142 (2001), 1-9. (Pubitemid 32818021)
42. F. M. Hilker, M. Langlais, H. Malchow. The Allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations. Am. Nat., 173 (1) (2009), 72-88.
43. X-C. Huang, Y. Wang, L. Zhu. One and three limit cycles in a cubic predator-prey system. Math. Method App. Sci., 30 (2007), 501-511. (Pubitemid 46382714)
44. T-W. Hwang. Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl., 290 (2004), 113-122. (Pubitemid 38236206)
45. S-B. Hsu, T-W. Hwang, Y. Kuang. Global dynamics of a predator-prey model with Hassell-Varley type functional response. Discrete Cont. Dyn. S., 10 (2008), 857-871.
46. C. Hui, Z. Li. Distribution patterns of metapopulation determined by Allee effects. Popul. Ecol., 46 (2004), 55-63. (Pubitemid 38991433)
47. M. Kot, Elements of Mathematical Ecology, Cambridge University Press 2001.
48. Y. Kuang, H. I. Freedman, Uniqueness of limit cycles in Gause type models of predator-prey systems, Math. Biosci. 88 (1988) 67-84.
49. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (2nd Edition), Springer-Verlag, New York, 1998.
50. Y. Lamontagne, C. Coutu, C. Rousseau. Bifurcation analysis of a predator-prey system with generalised Holling type III functional response. J. Dyn. Differ. Equ., 20 (3) (2008), 535-571.
51. M. Liermann, R. Hilborn. Depensation: evidence, models and implications. Fish Fish 2 (2001), 33-58. (Pubitemid 32437481)
52. D. Ludwig, D. D. Jones, C. S. Holling. Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Ecol., 36 (1978), 204-221.
53. R. M. May, Stability and complexity in model ecosystems(2nd Edition), Princeton University Press, Princeton, NJ, 2001.
54. In R. Mondaini (ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003. E-papers Serviços Editoriais Ltda.
55. 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.
56. S. J. Middlemas, T. R. Barton, J. D. Armstrong, P. M. Thompson. Functional and aggregative responses of harbour seals to changes in salmonid abundance. P. Roy. Soc. B., 273 (2006), 193-198. (Pubitemid 46768093)
57. S. M. Moghadas, B. D. Corbett. Limit cycles in a generalized Gause-type predator-prey model. Chaos Solitons Fract., 37 (2008), 1343-1355. (Pubitemid 351418150)
58. A. Morozov, E. Arashkevich. Patterns of zooplankton functional response in communities with vertical heterogeneity: a model study. Math. Mod. Nat. Phenom., 3 (3) (2008) 131-148.
59. W. W. Murdoch, C. J. Briggs, R. M. Nisbet. Consumer-Resources Dynamics. Monographs in Population Biology 36, Princeton University Press, Princeton, 2003.
60. M. R. Myerscough, M. J. Darwen, W. L. Hogarth. Stability, persistence and structural stability in a classical predator-prey model. Ecol. Model., 89 (1996) 31-42.
61. L. Perko. Differential Equations and Dynamical Systems (3rd. Edition). Springer-Verlag, New York, 2001.
62. 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.
63. G. Seo and M. Kot. A comparison of two predator-prey models with Holling's type I functional response. Math. Biosci., 212 (2008) 161-179.
64. M. Sen, M. Banerjee, A. Morozov. Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect. Ecol. Complex., 11 (2013), 12-27.
65. P. D. Spencer, J. S. Collie. A simple predator-prey model of exploited marine fish populations incorporating alternative prey. ICES J. Mar. Sci., 53 (1995), 615-628.
66. P. A. Stephens, W. J. Sutherland. Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol. Evol., 14 (1999), 401-405. (Pubitemid 29428465)
67. P. A. Stephens, W. J. Sutherland, R. P. Freckleton. What is the Allee effect. Oikos 87 (1999), 185-190.
68. J. Sugie, K. Miyamoto, K. Morino. Absence of limits cycle of a predator-prey system with a sigmoid functional response. Appl. Math. Lett., 9 (1996), 85-90. (Pubitemid 126365308)
69. J. Sugie, R. Kohno, R. Miyazaki, On a predator-prey system of Holling type. P. Am. Math. Soc., 125 (1997), 2041-2050.
70. R. J. Taylor. Predation. Chapman and Hall, New York,1984.
71. P. Turchin. Complex population dynamics. A theoretical/empirical synthesis. Mongraphs in Population Biology 35 Princeton University Press, Princeton, NJ, 2003.
72. G. A. K. van Voorn, L. Hemerik, M. P. Boer, B. W. Kooi. Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect. Math. Biosci., 209 (2007), 451-469. (Pubitemid 47390913)
73. K. Vilches-Ponce, E. González-Olivares. A Gause-type predator-prey model with a non-differentiable functional response. (2013) submitted.
74. M-H. Wang, M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Bioci., 171 (2001), 83-97. (Pubitemid 32429044)
75. J. Wang, J. Shi, J. Wei. Predator-prey system with strong Allee effect in prey. J. Math. Biol., 62 (2011), 291-331.
76. S. Wolfram Research, Mathematica: A System for Doing Mathematics by Computer (2nd edition), Wolfram Research, Addison Wesley 1991.
77. D. J. Wollkind, J. B. Collings, J. A. Logan. Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees. Bull. Math. Biol., 50 (4) (1988), 379-109.
78. D. Xiao, Z. Zhang. On the uniqueness and nonexistence of limit cycles for predator-prey systems. Nonlinearity 16 (2003), 1185-1201.
79. J. Zu, M. Mimura. The impact of Allee effect on a predator-prey system with Holling type II functional response. Appl. Math. Comp., 217 (2010), 3542-3556.