# American Institute of Mathematical Sciences

July  2010, 14(1): 233-250. doi: 10.3934/dcdsb.2010.14.233

## The saddle-node-transcritical bifurcation in a population model with constant rate harvesting

 1 Faculty of Sciences and Mathematics, Universitas Pelita Harapan, Jl. M.H. Thamrin Boulevard, Tangerang, 15811, Indonesia 2 Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St. N., L1H 7K4 Oshawa, Ontario, Canada 3 Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

Received  April 2009 Revised  March 2010 Published  April 2010

We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system.
Citation: Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
 [1] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130 [2] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [3] Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115 [4] Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147 [5] Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056 [6] Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062 [7] Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130 [8] Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 [9] Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 [10] Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256 [11] Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027 [12] Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078 [13] Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014 [14] Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5633-5671. doi: 10.3934/dcdsb.2019076 [15] Wentao Meng, Yuanxi Yue, Manjun Ma. The minimal wave speed of the Lotka-Volterra competition model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021265 [16] Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75 [17] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [18] Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823 [19] Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999 [20] Fang Li, Liping Wang, Yang Wang. On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 669-686. doi: 10.3934/dcdsb.2011.15.669

2020 Impact Factor: 1.327