Kang, Yun; Armbruster, Dieter Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions. (English) Zbl 1247.37091 Int. J. Biomath. 4, No. 3, 255-274 (2011). This paper considers a class of discrete-time models for the growth dynamics of a plant species and for the dynamics resulting from the interaction of one plant and one herbivore species. The growth of a single plant species is modelled by a one-dimensional map, and assumptions on the monotonicity of the functional form of the map are shown to have various dynamical consequences. The model for the interaction of plants and herbivores has one free function, and the consequences of the monotonicity of this function are investigated. The presented examples involve the maximum value of the variables as \(t\rightarrow\infty\), the number of equilibria and their stability, and the bifurcations they undergo. The results are applied to two specific models, and periodic orbits and a heteroclinic bifurcation are found. The effects of noise near the heteroclinic bifurcation are also investigated. Reviewer: Carlo Laing (Auckland) Cited in 4 Documents MSC: 37N25 Dynamical systems in biology 39A28 Bifurcation theory for difference equations 39A30 Stability theory for difference equations 92B05 General biology and biomathematics Keywords:monotone growth models; uniformly persistent; Neimark-Sacker bifurcation; heteroclinic bifurcation; periodic infestations; bistability; noise bursting; crisis of chaos PDF BibTeX XML Cite \textit{Y. Kang} and \textit{D. Armbruster}, Int. J. Biomath. 4, No. 3, 255--274 (2011; Zbl 1247.37091) Full Text: DOI arXiv References: [1] DOI: 10.1111/j.1365-2656.2007.01263.x [2] Beddington J. R., Nature (London) 225 pp 58– [3] DOI: 10.1038/378610a0 [4] Crawley M. J., Studies in Ecology 10, in: Herbivory: The Dynamics of Animal–Plant Interactions (1983) [5] DOI: 10.1098/rstb.1990.0187 [6] DOI: 10.1080/10236190108808308 · Zbl 1002.39003 [7] DOI: 10.1007/BF01982351 · Zbl 0429.58012 [8] J. L. Harper, Population Biology of Plants (Academic Press, New York, 1977) pp. 1035–1039. [9] Henson S. M., J. Math. Biol. 34 pp 1416– [10] DOI: 10.1007/BF00276199 · Zbl 0638.92019 [11] DOI: 10.1090/S0002-9939-1989-0984816-4 [12] DOI: 10.1007/BF01540776 · Zbl 0542.34043 [13] DOI: 10.1080/10236190500539238 · Zbl 1088.92058 [14] DOI: 10.1080/17513750801956313 · Zbl 1140.92322 [15] DOI: 10.1016/j.tpb.2010.04.002 [16] DOI: 10.1890/0012-9658(1999)080[1789:WDPCAS]2.0.CO;2 [17] DOI: 10.1007/s00285-003-0224-8 · Zbl 1050.92045 [18] DOI: 10.1016/j.mbs.2005.12.010 · Zbl 1093.92056 [19] DOI: 10.2307/2260211 [20] DOI: 10.1007/BF02514974 [21] DOI: 10.1086/283161 [22] DOI: 10.1139/f54-039 [23] Rinaldi S., Ecol. Lett. 4 [24] DOI: 10.1007/978-1-4612-0873-0 [25] DOI: 10.1016/S0960-0779(01)00063-7 · Zbl 1022.92042 [26] DOI: 10.1023/A:1006550817371 [27] DOI: 10.1016/0022-5193(80)90297-0 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.