2006, 3(1): 17-36. doi: 10.3934/mbe.2006.3.17

Nonlinear semelparous Leslie models


Department of Mathematics & Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ 85721, United States

Received  February 2005 Revised  April 2005 Published  November 2005

In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at $n=1$ despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at $n=1$. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.
Citation: J. M. Cushing. Nonlinear semelparous Leslie models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 17-36. doi: 10.3934/mbe.2006.3.17

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure and Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75


Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3345-3374. doi: 10.3934/dcdsb.2021188


Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155


El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883


Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355


Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009


Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014


Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009


Carol C. Horvitz, Anthony L. Koop, Kelley D. Erickson. Time-invariant and stochastic disperser-structured matrix models: Invasion rates of fleshy-fruited exotic shrubs. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1639-1662. doi: 10.3934/dcdsb.2015.20.1639


Jian Zhao, Fang Deng, Jian Jia, Chunmeng Wu, Haibo Li, Yuan Shi, Shunli Zhang. A new face feature point matrix based on geometric features and illumination models for facial attraction analysis. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1065-1072. doi: 10.3934/dcdss.2019073


Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617


Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems and Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030


Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015


K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026


Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012


Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012


Henry Adams, Lara Kassab, Deanna Needell. An adaptation for iterative structured matrix completion. Foundations of Data Science, 2021, 3 (4) : 769-791. doi: 10.3934/fods.2021028


Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065


Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial and Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036


Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

2018 Impact Factor: 1.313


  • PDF downloads (38)
  • HTML views (0)
  • Cited by (28)

Other articles
by authors

[Back to Top]