92D30.

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Nonlinear semelparous Leslie models

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

## Abstract    Related pages

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.
Figure/Table
Supplementary
Article Metrics

Citation: J. M. Cushing. Nonlinear semelparous Leslie models. Mathematical Biosciences and Engineering, 2006, 3(1): 17-36. doi: 10.3934/mbe.2006.3.17

• 1. Amy Veprauskas, J. M. Cushing, A juvenile–adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects, Journal of Biological Dynamics, 2017, 11, sup1, 1, 10.1080/17513758.2015.1131853
• 2. Luigi Monte, Characterisation of a nonlinear Leslie matrix model for predicting the dynamics of biological populations in polluted environments: Applications to radioecology, Ecological Modelling, 2013, 248, 174, 10.1016/j.ecolmodel.2012.10.005
• 3. Arild Wikan, Ørjan Kristensen, Nonstationary and Chaotic Dynamics in Age-Structured Population Models, Discrete Dynamics in Nature and Society, 2017, 2017, 1, 10.1155/2017/1964286
• 4. J. M. Cushing, Shandelle M. Henson, Stable bifurcations in semelparous Leslie models, Journal of Biological Dynamics, 2012, 6, sup2, 80, 10.1080/17513758.2012.716085
• 5. J. M. Cushing, A dynamic dichotomy for a system of hierarchical difference equations, Journal of Difference Equations and Applications, 2012, 18, 1, 1, 10.1080/10236198.2011.628319
• 6. Jim M. Cushing, , Applied Analysis in Biological and Physical Sciences, 2016, Chapter 3, 41, 10.1007/978-81-322-3640-5_3
• 7. Ryusuke Kon, Age-Structured Lotka–Volterra Equations for Multiple Semelparous Populations, SIAM Journal on Applied Mathematics, 2011, 71, 3, 694, 10.1137/100794262
• 8. Ryusuke Kon, Non-synchronous oscillations in four-dimensional nonlinear semelparous Leslie matrix models, Journal of Difference Equations and Applications, 2017, 1, 10.1080/10236198.2017.1365144
• 9. J. M. Cushing, Shandelle M. Henson, Lih-Ing Roeger, Coexistence of competing juvenile–adult structured populations, Journal of Biological Dynamics, 2007, 1, 2, 201, 10.1080/17513750701201372
• 10. Yuanshi Wang, Hong Wu, Shigui Ruan, Periodic orbits near heteroclinic cycles in a cyclic replicator system, Journal of Mathematical Biology, 2012, 64, 5, 855, 10.1007/s00285-011-0435-3
• 11. O. Diekmann, S. A. van Gils, On the Cyclic Replicator Equation and the Dynamics of Semelparous Populations, SIAM Journal on Applied Dynamical Systems, 2009, 8, 3, 1160, 10.1137/080722734
• 12. J. M. Cushing, Three stage semelparous Leslie models, Journal of Mathematical Biology, 2009, 59, 1, 75, 10.1007/s00285-008-0208-9
• 13. Amy Veprauskas, J. M. Cushing, Evolutionary dynamics of a multi-trait semelparous model, Discrete and Continuous Dynamical Systems - Series B, 2015, 21, 2, 655, 10.3934/dcdsb.2016.21.655
• 14. Ryusuke Kon, , Advances in Difference Equations and Discrete Dynamical Systems, 2017, Chapter 1, 3, 10.1007/978-981-10-6409-8_1
• 15. Ryusuke Kon, Yoh Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations, Journal of Mathematical Biology, 2007, 55, 5-6, 781, 10.1007/s00285-007-0111-9
• 16. R. Bravo de la Parra, M. Marvá, F. Sansegundo, A. Morozov, S. Petrovskii, Fast Dispersal in Semelparous Populations, Mathematical Modelling of Natural Phenomena, 2016, 11, 4, 120, 10.1051/mmnp/201611409
• 17. Yuanshi Wang, Hong Wu, Shigui Ruan, Global dynamics and bifurcations in a four-dimensional replicator system, Discrete and Continuous Dynamical Systems - Series B, 2012, 18, 1, 259, 10.3934/dcdsb.2013.18.259
• 18. Arild Wikan, An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities, Journal of Applied Mathematics, 2017, 2017, 1, 10.1155/2017/8934295
• 19. J. M. Cushing, Simon Maccracken Stump, Darwinian dynamics of a juvenile-adult model, Mathematical Biosciences and Engineering, 2013, 10, 4, 1017, 10.3934/mbe.2013.10.1017
• 20. A. Veprauskas, A nonlinear continuous-time model for a semelparous species, Mathematical Biosciences, 2018, 297, 1, 10.1016/j.mbs.2018.01.003
• 21. A. Veprauskas, Synchrony and the Dynamic Dichotomy in a Class of Matrix Population Models, SIAM Journal on Applied Mathematics, 2018, 78, 5, 2491, 10.1137/17M1136444
• 22. Ryusuke Kon, , Mathematical Modeling of Biological Systems, Volume II, 2008, Chapter 7, 75, 10.1007/978-0-8176-4556-4_7
• 23. Sophia Jang, Allee effects in an iteroparous host population and in host-parasitoid interactions, Discrete and Continuous Dynamical Systems - Series B, 2010, 15, 1, 113, 10.3934/dcdsb.2011.15.113
• 24. J. M. Cushing, , Dynamics, Games and Science, 2015, Chapter 12, 215, 10.1007/978-3-319-16118-1_12
• 25. Ryusuke Kon, Bifurcations of cycles in nonlinear semelparous Leslie matrix models, Journal of Mathematical Biology, 2020, 10.1007/s00285-019-01459-9
• 26. Yunshyong Chow, Ryusuke Kon, Global dynamics of a special class of nonlinear semelparous Leslie matrix models, Journal of Difference Equations and Applications, 2020, 1, 10.1080/10236198.2020.1777288