Publication Cover
Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 8
198
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Stationary pattern and Hopf bifurcation of a diffusive predator–prey model

, &
Pages 2141-2159 | Received 07 Apr 2021, Accepted 13 Dec 2021, Published online: 30 Dec 2021

References

  • Peng GJ, Jiang YL, Li CP. Bifurcations of a Holling-type II predator-prey system with constant rate harvesting. Int J Bifurc Chaos. 2009;19(8):2499–2514.
  • Zhang FR, Zhang XH, Li Y, et al. Hopf bifurcation of a delayed predator-prey model with nonconstant death rate and constant-rate prey harvesting. Int J Bifurc Chaos. 2018;28(14):1850179.
  • Luo JF, Zhao Y. Stability and bifurcation analysis in a predator-prey system with constant harvesting and prey group defense. Int J Bifurc Chaos. 2017;27(11):1750179.
  • Etoua RM, Rousseau C. Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III. J Differ Equ. 2010;249(9):2316–2356.
  • Lenzini P, Rebaza J. Non-constant predator harvesting on ratio-dependent predator-prey models. Appl Math Sci. 2010;4(16):791–803.
  • Leard B, Lewis C, Rebaza J. Dynamics of ratio-dependent predator-prey models with non-constant harvesting. Amer Instit Math Sci. 2008;1(2):303–315.
  • Das T, Mukherjee RN, Chaudhuri KS. Bioeconomic harvesting of a prey-predator fishery. J Biol Dyn. 2009;3(5):447–462.
  • Li Y, Wang MX. Dynamics of a diffusive predator-prey model with modified Leslie-Gower term and Michaelis-Menten type prey harvesting. Acta Appl Math. 2015;140(1):147–172.
  • Tchinda PM, Djidjou RD, Tewa JJ, et al. Bifurcation analysis and optimal harvesting of a delayed predator-prey model. Int J Bifurc Chaos. 2015;25(1):1550012.
  • Budick SA, Dickinson MH. Free-flight responses of Drosophila melanogaster to attractive odors. J Exper Biol. 2006;209(15):3001–3017.
  • Murray JD. Mathematical biology. 2nd ed. Berlin: Springer-Verlag; 1993. (Biomathematics Texts; 19).
  • Patlak CS. Random walk with persistence and external bias. Bullet Math Biophys. 1953;15:311–338.
  • Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. J Theor Biol. 1970;26:399–415.
  • Keller EF, Segel LA. Model for chemotaxis. J Theor Biol. 1971;30:225–234.
  • Stevens A. The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J Appl Math. 2000;61:183–212.
  • Ainseba B, Bendahmane M, Noussair A. A reaction-diffusion system modeling predator-prey with prey-taxis. Nonlinear Anal: Real World Appl. 2008;9:2086–2105.
  • Grünbaum D. Advection-diffusion equations for generalized tactic searching behaviours. J Math Biol. 1999;38:169–194.
  • Inkyung A, Changwook Y. Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis. J Differ Equ. 2020;268(8):4222–4255.
  • Jin HY, Wang ZA. Global stability of prey-taxis systems. J Differ Equ. 2017;262(3):1257–1290.
  • Kareiva P, Odell G. Swarms of predators exhibit prey-taxis if individual predators use Area-Restricted search. Am Nat. 1987;130(2):233–270.
  • Grunbaum D. Using spatially explicit models to characterize foraging performance in heterogeneous landscapes. Am Nat. 1998;15(1):97–113.
  • Braza PA. Predator-prey dynamics with square root functional responses. Nonlinear Anal: Real World Appl. 2012;13(4):1837–1843.
  • Song YL, Xu Z. Periodic travelling wave solution in a diffusive predator-prey system. J Hangzhou Normal University(Natural Science Edition). 2017;16(4):368–377.
  • Yuan SL, Xu CQ, Zhang TH. Spatial dynamics in a predator-prey model with herd behavior. Chaos(Woodbury, N.Y.). 2013;23(3):033102.
  • Xu Z, Song YL. Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality. Math Meth Appl Sci. 2015;38(14):2994–3006.
  • Xiao D, Jennings L. Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J Appl Math. 2005;65:737–753.
  • Xiao D, Li W, Han M. Dynamics in a ratio-dependent predator-prey model with predator harvesting. J Math Anal Appl. 2006;324:14–29.
  • Yodzis P. Predator-prey theory and management of multispecies fisheries. Ecol Appl. 1994;4:51–58.
  • Wei J, Huang QC. Bifurcation theory of functional differential equations development survey. Sci Bullet. 1997;24:2581–2586.
  • Yuan R, Jiang WH, Wang Y. Nonresonant double Hopf bifurcation in toxic phytoplankton-zooplankton model with delay. Int J Bifurc Chaos. 2017;27(2):1750028.
  • Li Y, Li SY, Zhang FR. Dynamics of a diffusive predator-prey model with herd behavior. Nonlinear Anal: Model Control. 2020;25(1):19–35.
  • Li L, Shi JP, Wang JF. Dynamic analysis of a predator-prey model with Holling-type reaction function. J Natural Sci Harbin Normal University. 2009;25(5):10–12.
  • Zuo WJ, Hu WM. Stability and branch analysis of a predator-prey model with diffusion and time delay. J Yili Normal University(Natural Science Edition). 2013;7(3):6–10.
  • Zhou J, Shi JP. Pattern formation in a general glycolysis reaction-diffusion system. IMA J Appl Math. 2015;80(6):1703–1738.
  • Painter KJ, Hillen T. Volume-filling and quorum-sensing in models for chemosensitive movement. Canadian Appl Math Q. 2002;10(10):501–543.
  • Hillen T, Painter KJ. A user's guide to PDE models for chemotaxis. J Math Biol. 2009;58(1-2):183–217.
  • Meng XY, Meng FL. Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting. AIMS Math. 2021;6(6):5695–5719.
  • Song YL, Tang X. Stability, steady-state bifurcations, and turing patterns in a predator-prey model with herd behavior and prey-taxis. Stud Appl Math. 2017;139(3):371–404.
  • Jiang HP, Tang XS. Hopf bifurcation in a diffusive predator-prey model with herd behavior and prey harvesting. J Appl Anal Comput. 2019;9(2):671–690.
  • Yi FQ, Wei J, Shi JP. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J Differ Equ. 2009;246(5):1944–1977.
  • Hassard BD, Kazarinoff ND, Wan YH. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.
  • Crandall MG, Rabinowitz PH. The Hopf bifurcation theorem in infinite dimensions. Arch Ration Mech Anal. 1977;67(1):53–72.
  • Ko W, Ryu K. A qualitative study on general Gause-type predator-prey models with non-monotonic functional response. Nonlinear Anal Real World Appl. 2009;10(4):2558–2573.
  • Li L. Coexistence theorems of steady states for predator-prey interacting systems. Trans Am Math Soc. 1988;305(1):143–166.
  • Dancer EN. On the indices of fixed points of mapping in cones and applications. J Math Anal Appl. 1983;91(1):131–151.
  • Julin LG. Positive periodic solutions of Lotka-Volterra reaction-diffusion systems. Differ Integral Equ. 1992;5(1):55–72.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. J Appl Math Mech. 1983;65(11):568–568.
  • Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. Siam Rev. 1976;18:620–709.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.