Predator-prey system with strong Allee effect in prey

Journal of Mathematical Biology

Volumn 62, Issue 3, 2011, Pages 291-331

  Wang, Jinfeng ^a,b   Shi, Junping ^b,c   Wei, Junjie ^a  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

^b HARBIN NORMAL UNIVERSITY   (China)

^c The College of William and Mary   (United States)

Author keywords

Allee effect; Global bifurcation; Heteroclinic loop; Limit cycles; Predator prey model

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; ECOSYSTEM; FOOD CHAIN; MATHEMATICAL PHENOMENA; POPULATION DYNAMICS; PREDATION;

ANIMALS; ECOSYSTEM; FOOD CHAIN; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; POPULATION DYNAMICS; PREDATORY BEHAVIOR;

EID: 79951508642     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-010-0332-1     Document Type: Article

References (69)

1. Aguirre P, González-Olivares E, Sáez E (2009) Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM J Appl Math 69(5): 1244-1262.
2. Albrecht F, Gatzke H, Wax N (1973) Stable limit cycles in prey-predator populations. Science 181: 1074-1075.
3. Albrecht F, Gatzke H, Haddad A, Wax N (1974) The dynamics of two interacting populations. J Math Anal Appl 46: 658-670.
4. Allee WC (1931) Animal aggregations: a study in general sociology. University of Chicago Press.
5. Alexander JC, Yorke JA (1978) Global bifurcations of periodic orbits. Am J Math 100: 263-292.
6. Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol 139: 311-326.
7. Bazykin AD (1998) Nonlinear dynamics of interacting populations. World Scientific Series on nonlinear science. Series A: monographs and treatises, 11. World Scientific Publishing Co., Inc., River Edge.
8. Berec L, Angulo E, Courchamp F (2007) Multiple Allee effects and population management. Trends Ecol Evol 22: 185-191.
9. Boukal SD, Berec L (2002) Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J Theor Biol 218: 375-394.
10. Boukal SD, Sabelis WM, Berec L (2007) How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor Popul Biol 72: 136-147.
11. Burgman MA, Ferson S, Akcakaya HR (1993) Risk assessment in conservation biology. Chapman and Hall, London.
12. Cheng KS (1981) Uniqueness of a limit cycle for a predator-prey system. SIAM J Math Anal 12: 541-548.
13. Chow SN, Mallet-Paret J (1978) The Fuller index and global Hopf bifurcation. J Differ Equ 29: 66-85.
14. Conway ED, Smoller JA (1986) Global analysis of a system of predator-prey equations. SIAM J Appl Math 46: 630-642.
15. Courchamp F, Clutton-Brock T, Grenfell B (1999) Inverse density dependence and the Allee effect. Trends Ecol Evol 14: 405-410.
16. Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford.
17. Dennis B (1989) Allee effect: population growth, critical density, and chance of extinction. Nat Resour Model 3: 481-538.
18. Dhooge A, Govaerts W, Kuznetsov YA (2003) MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans Math Softw 29: 141-164.
19. Edelstein KL (1998) Mathematical models in biology. Random House, New York.
20. Gascoigne JC, Lipcius RN (2004) Allee effects driven by predation. J Appl Ecol 41: 801-810.
21. González-Olivares E, González-Yaez B, Sáez E, Szántó I (2006) On the number of limit cycles in a predator prey model with non-monotonic functional response. Discrete Contin Dyn Syst B 6(3): 525-534.
22. Gruntfest Y, Arditi R, Dombrovsky Y (1997) A fragmented population in a varying environment. J Theor Biol 185: 539-547.
23. Hassard BD, Kazarinoff ND, Wan Y (1981) Theory and applications of Hopf bifurcation. London Math Soc Lecture Note Ser, vol 41. Cambridge University Press, Cambridge.
24. Hilker FM, Langlais M, Malchow H (2009) The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations. Am Nat 173: 72-88.
25. Holling CS (1959) The components of predation as revealed by a study of small mammal predation of the European Pine Sawfly. Can Entomol 91: 293-320.
26. Hsu SB (1978) On global stability of a predator-prey system. Math Biosci 39: 1-10.
27. Hsu SB (2006) Ordinary differential equations with applications. Series on Applied Mathematics, vol 16. World Scientific Publishing Co., Hackensack.
28. Hsu SB, Shi J (2009) Relaxation oscillator profile of limit cycle in predator-prey system. Discrete Contin Dyn Syst B 11(4): 893-911.
29. Hsu SB, Hwang TW, Kuang Y (2001) Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system. J Math Biol 42: 489-506.
30. Ivlev VS (1955) Experimental ecology of the feeding of fishes. Yale University Press.
31. Jacobs J (1984) Cooperation, optimal density and low density thresholds: yet another modification of the logistic model. Oecologia 64: 389-395.
32. Jiang J, Shi J (2009) Bistability dynamics in some structured ecological models. In: Spatial ecology. CRC Press, Boca Raton.
33. Kazarinov N, van den Driessche P (1978) A model predator-prey systems with functional response. Math Biosci 39: 125-134.
34. Kuang Y, Beretta E (1998) Global qualitative analysis of a ratio-dependent predator-prey system. J Math Biol 36: 389-406.
35. Kuang Y, Freedman HI (1988) Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math Biosci 88: 67-84.
36. Kuznetsov YA (2004) Elements of applied bifurcation theory. Appl Math Sci, vol 112. Springer-Verlag, New York.
37. Lande R (1987) Extinction thresholds in demographic models of territorial populations. Am Nat 130: 624-635.
38. Lewis MA, Kareiva P (1993) Allee dynamics and the spread of invading organisms. Theor Popul Biol 43: 141-158.
39. Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, Baltimore.
40. Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in ecology and epidemiology. Theory, models, and simulation. Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall, Boca Raton.
41. May RM (1972) Limit cycles in predator-prey communities. Science 177: 900-902.
42. Morozov A, Petrovskii S, Li B-L (2004) Bifurcations and chaos in a predator-prey system with the Allee effect. Proc R Soc Lond B Biol Sci 271: 1407-1414.
43. Morozov A, Petrovskii S, Li B-L (2006) Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. J Theor Biol 238(1): 18-35.
44. Owen MR, Lewis MA (2001) How predation can slow, stop or reverse a prey invasion. Bull Math Biol 63: 655-684.
45. Petrovskii SV, Morozov A, Venturino E (2002) Allee effect makes possible patchy invasion in a predator-prey system. Ecol Lett 5: 345-352.
46. Petrovskii S, Morozov A, Li B-L (2005) Regimes of biological invasion in a predator-prey system with the Allee effect. Bull Math Biol 67(3): 637-661.
47. Polking JA, Arnold D (2003) Ordinary differential equations using MATLAB, 3rd edn. Prentice-Hall, Englewood Cliffs.
48. Rosenzweig ML (1969) Why the prey curve has a hump? Am Nat 103: 81-87.
49. Rosenzweig LM (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171(3969): 385-387.
50. Rosenzweig ML, MacArthur R (1963) Graphical representation and stability conditions of predator-prey interactions. Am Nat 97: 209-223.
51. Ruan S, Xiao D (2000) Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J Appl Math 61: 1445-1472.
52. Shi J, Shivaji R (2006) Persistence in reaction diffusion models with weak Allee effect. J Math Biol 52: 807-829.
53. Stephens PA, Sutherland WJ (1999) Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol Evol 14: 401-405.
54. Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect? Oikos 87: 185-190.
55. Takeuchi Y (1996) Global dynamical properties of Lotka-Volterra systems. World Scientific, Singapore.
56. Thieme HR, Dhirasakdanon T, Han Z, Trevino R (2009) Species decline and extinction: synergy of infectious disease and Allee effect? J Biol Dyn 3: 305-323.
57. Turchin P (2003) Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press.
58. van Voorn GAK, Hemerik L, Boer MP, Kooi BW (2007) Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect. Math Biosci 209: 451-469.
59. Volterra V (1926) Fluctuations in the abundance of species, considered mathmatically. Nature 118: 558.
60. Wang ME, Kot M (2001) Speeds of invasion in a model with strong or weak Allee effects. Math Biosci 171: 83-97.
61. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Texts Appl Math, vol 2. Springer, New York.
62. Wilson EO, Bossert WH (1971) A primer of population biology. Sinauer Assoxiates, Sunderland.
63. Wu J (1998) Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 350: 4799-4838.
64. Xiao D, Ruan S (2001) Global dynamics of a ratio-dependent predator-prey system. J Math Biol 43: 268-290.
65. Xiao D, Zhang Z (2003) On the uniqueness and nonexistence of limit cycles for predator-prey system. Nonlinearity 16: 1-17.
66. Xiao D, Zhang Z (2008) On the existence and uniqueness of limit cycles for generalized Liénard systems. J Math Anal Appl 343: 299-309.
67. Zeng X, Zhang Z, Gao S (1994) On the uniqueness of the limit cycle of the generalized Liénard equation. Bull Lond Math Soc 26: 213-247.
68. Zhang ZF (1986) Proof of the uniqueness theorem of limit cycles of generalized Liénard equations. Appl Anal 23(1-2): 63-76.
69. Zhou SR, Liu YF, Wang G (2005) The stability of predator-prey systems subject to the Allee effects. Theor Popul Biol 67(1): 23-31.