Dynamics in a diffusive predator-prey system with strong Allee effect and Ivlev-type functional response

Journal of Mathematical Analysis and Applications

Volumn 422, Issue 2, 2015, Pages 1447-1462

  Wang, Xuechen ^a   Wei, Junjie ^a  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

Author keywords

Allee effect; Hopf bifurcation; Ivlev type functional response; Periodic solutions; Prey predator

Indexed keywords

EID: 84908180764     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2014.09.051     Document Type: Article

References (42)

1. Allee W.C. Animal Aggregations: A Study in General Sociology 1931, University of Chicago Press.
2. Berec L., Angulo E., Courchamp F. Multiple Allee effects and population management. Trends Ecol. Evol. 2007, 22:185-191.
3. Boukal S.D., Sabelis W.M., Berec L. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theoret. Popul. Biol. 2007, 72:136-147.
4. Burgman M.A., Ferson S., Akcakaya H.R. Risk Assessment in Conservation Biology 1993, Chapman and Hall, London.
5. Chen X.-Y., Poláčik P. Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations. J. Dynam. Differential Equations 1995, 7(1):73-107.
6. Courchamp F., Berec L., Gascoigne J. Allee Effects in Ecology and Conservation 2008, Oxford University Press, Oxford.
7. Dennis B. Allee effect: population growth, critical density, and chance of extinction. Nat. Resour. Model. 1989, 3:481-538.
8. Garvie M.R. Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB. Bull. Math. Biol. 2007, 69:931-956.
9. Hale J.K. Asymptotic Behavior of Dissipative Systems. Math. Surveys Monogr. 1988, vol. 25. Amer. Math. Soc., Providence, RI.
10. Hassard B.D., Kazarinoff N.D., Wan Y.H. Theory and Applications of Hopf Bifurcation 1981, Cambridge University Press, Cambridge, MA.
11. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 1981, vol. 840. Springer-Verlag, Berlin/New York.
12. Ivlev V.S. Experimental Ecology of the Feeding of Fishes 1961, Yale University Press, New Haven.
13. Jiang J., Shi J. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete Contin. Dyn. Syst. 2008, 21(1):245-258.
14. Kay A.L., Sherratt J.A. Spatial noise stabilizes periodic wave patterns in oscillatory systems on finite domains. SIAM J. Appl. Math. 2000, 61:1013-1041.
15. Kooij R., Zegeling A. A predator-prey model with Ivlev's functional response. J. Math. Anal. Appl. 1996, 198:473-489.
16. Lande R. Extinction thresholds in demographic models of territorial populations. Am. Nat. 1987, 130:624-635.
17. Lewis M.A., Karevia P. Allee dynamics and the spread of invading organisms. Theoret. Popul. Biol. 1993, 43:141-158.
18. May R.M. Complexity and Stability in Model Ecosystems 1973, Princeton University Press, Princeton, NJ.
19. Mischaikow K., Smith H., Thieme H.R. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc. 1995, 347(5):1669-1685.
20. Murray J.D. Mathematical Biology. Biomathematics Texts 1993, vol. 19. Springer, Berlin.
21. Owen M.R., Lewis M.A. How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 2001, 63:655-684.
22. Pao C.-V. Nonlinear Parabolic and Elliptic Equations 1992, Plenum Press, New York.
23. Pearce I.G., Chaplain M.A.J., Schofield P.G., Anderson A.R.A., Hubbard S.F. Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications. J. Theoret. Biol. 2006, 241:876-886.
24. Peng R., Shi J., Wang M. On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law. Nonlinearity 2008, 21:1471-1488.
25. Petrovskii S.V., Morozov A., Venturino E. Allee effect makes possible patchy invasion in a predator-prey system. Ecol. Lett. 2002, 5:345-352.
26. Preedy K.F., Schofield P.G., Chaplain M.A.J., Hubbard S.F. Disease induced dynamics in host-parasitoid systems: chaos and coexistence. J. R. Soc. Interface 2007, 4:463-471.
27. Sherratt J.A. Periodic travelling waves in cyclic predator-prey systems. Ecol. Lett. 2001, 4:30-37.
28. Sherratt J.A., Eagan B.T., Lewis M.A. Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?. Philos. Trans. R. Soc. B Biol. Sci. 1997, 352:21-38.
29. Sherratt J.A., Lewis M.A., Fowler A.C. Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 1995, 92:2524-2528.
30. Shi J., Shivaji R. Persistence in reaction diffusion models with weak Allee effect. J. Math. Biol. 2006, 52(6):807-829.
31. Stephens P.A., Sutherland W.J., Freckleton R.P. What is the Allee effect?. Oikos 1999, 87:185-190.
32. Sugie J. Two-parameter bifurcation in a predator-prey system of Ivlev type. J. Math. Anal. Appl. 1998, 217:349-371.
33. Taylor C.M., Hastings A. Allee effects in biological invasions. Ecol. Lett. 2005, 8:895-908.
34. Tian R.C. Toward standard parameterizations in marine biological modeling. Ecol. Model. 2006, 193:363-386.
35. Uriu K., Iwasa Y. Turing pattern formation with two kinds of cells and a diffusive chemical. Bull. Math. Biol. 2007, 69:2515-2536.
36. Wang M.E., Kot M. Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 2001, 171:83-97.
37. Wang J., Shi J., Wei J. Predator-prey system with strong Allee effect in prey. J. Math. Biol. 2011, 3(3):291-331.
38. Wang J., Shi J., Wei J. Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. J. Differential Equations 15 August 2011, 251(4-5):1276-1304.
39. Wang H., Wang W. The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect. Chaos Solitons Fractals 2007, 38:1168-1176.
40. Wang X., Wei J. Diffusion-driven stability and bifurcation in a predator-prey system with Ivlev-type functional response. Appl. Anal. 2013, 92(4):752-775.
41. Ye Q., Li Z. Introduction to Reaction-Diffusion Equations 1994, Science Press, China, (in Chinese).
42. Yi F., Wei J., Shi J. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differential Equations 2009, 246:1944-1977.