Nonlinear semelparous Leslie models
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...
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AIMS Press
2005-10-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.17 |
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| author | J. M. Cushing |
| author_facet | J. M. Cushing |
| author_sort | J. M. Cushing |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a900a211979b4d8180913155171fc473 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2005-10-01 |
| publisher | AIMS Press |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-a900a211979b4d8180913155171fc4732025-01-24T01:51:11ZengAIMS PressMathematical Biosciences and Engineering1551-00182005-10-0131173610.3934/mbe.2006.3.17Nonlinear semelparous Leslie modelsJ. M. Cushing0Department of Mathematics & Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ 85721In 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.https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.17nonlinear matrix modelscyclesbifurcationstabilityleslie matrix.semelparity |
| spellingShingle | J. M. Cushing Nonlinear semelparous Leslie models Mathematical Biosciences and Engineering nonlinear matrix models cycles bifurcation stability leslie matrix. semelparity |
| title | Nonlinear semelparous Leslie models |
| title_full | Nonlinear semelparous Leslie models |
| title_fullStr | Nonlinear semelparous Leslie models |
| title_full_unstemmed | Nonlinear semelparous Leslie models |
| title_short | Nonlinear semelparous Leslie models |
| title_sort | nonlinear semelparous leslie models |
| topic | nonlinear matrix models cycles bifurcation stability leslie matrix. semelparity |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.17 |
| work_keys_str_mv | AT jmcushing nonlinearsemelparouslesliemodels |