Bifurcation Analysis of a Predator-Prey System With Nonmonotonic Function Response

dc.contributor.authorZhu, Huaiping
dc.contributor.authorCampbell, Sue Ann
dc.contributor.authorWolkowicz, Gail S. K.
dc.date.accessioned2007-02-12T23:45:44Z
dc.date.available2007-02-12T23:45:44Z
dc.date.issued2002
dc.description.abstractWe consider a predator-prey system with nonmonotonic functional response: p(x) = mx /(ax2 +bx+1) . By allowing b to be negative (b > −2√a), p(x) is concave up for small values of x > 0 as it is for the sigmoidal functional response. We show that in this case there exists a Bogdanov–Takens bifurcation point of codimension 3, which acts as an organizing center for the system. We study the Hopf and homoclinic bifurcations and saddle-node bifurcation of limit cycles of the system. We also describe the bifurcation sequences in each subregion of parameter space as the death rate of the predator is varied. In contrast with the case b ≥ 0, we prove that when −2√a < b < 0, a limit cycle can coexist with a homoclinic loop. The bifurcation sequences involving Hopf bifurcations, homoclinic bifurcations, as well as the saddle-node bifurcations of limit cycles are determined using information from the complete study of the Bogdanov–Takens bifurcation point of codimension 3 and the geometry of the system. Examples of the predicted bifurcation curves are also generated numerically using XPPAUT. Our work extends the results in [F. Rothe and D. S. Shafer, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 313–347] and [S. Ruan and D. Xiao, SIAM J. Appl. Math., 61 (2001), pp. 1445–1472].
dc.format.extent517881 bytes
dc.format.mimetypeapplication/pdf
dc.identifier.citationHuaiping Zhu, Sue Ann Campbell and Gail S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic function response. SIAM J. Appl. Math. 63 (2002), no. 2, 636--682
dc.identifier.issn0036-1399
dc.identifier.urihttp://hdl.handle.net/10315/912
dc.identifier.urihttps://doi.org/10.1137/S0036139901397285
dc.language.isoen
dc.publisherSIAM Journal on Applied Mathematics
dc.rightsCopyright © 2002 Society for Industrial and Applied Mathematics
dc.subjectpredator-prey system
dc.subjectHopf bifurcation
dc.subjecthomoclinic bifurcation
dc.subjectBogdanov–Takens bifurcation
dc.subjectsaddle-node bifurcation of limit cycles
dc.subjectlimit cycle
dc.titleBifurcation Analysis of a Predator-Prey System With Nonmonotonic Function Response
dc.typeArticle

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