Age-Structured Lotka-Volterra Equations for Multiple Semelparous Populations

This paper derives a Lotka–Volterra equation with a certain symmetry from a coupled nonlinear Leslie matrix model for interacting semelparous species. The global analysis focuses on the special case where the system is composed of two species, one species having two age-classes and the other species having a single age-class. This analysis almost completely describes its global dynamics and provides examples that the age-structure changes the destiny of the system.

[1]  Norihiko Adachi,et al.  The existence of globally stable equilibria of ecosystems of the generalized Volterra type , 1980 .

[2]  Single-class orbits in nonlinear Leslie matrix models for semelparous populations , 2007, Journal of mathematical biology.

[3]  J. Cushing.  Three stage semelparous Leslie models , 2009, Journal of mathematical biology.

[4]  Competitive Exclusion Between Year-Classes in a SemelparousBiennial Population , 2008 .

[5]  Willem H. Haemers.  Matrices and Graphs , 2005 .

[6]  W. M. Post,et al.  Analysis of compensatory Leslie matrix models for competing species. , 1980, Theoretical population biology.

[7]  O Diekmann,et al.  Year class coexistence or competitive exclusion for strict biennials? , 2003, Journal of mathematical biology.

[8]  J. Cushing.  An introduction to structured population dynamics , 1987 .

[9]  G. Webb.  The prime number periodical cicada problem , 2001 .

[10]  J. Cushing.  Nonlinear semelparous leslie models. , 2005, Mathematical biosciences and engineering : MBE.

[11]  E. Werner,et al.  THE ONTOGENETIC NICHE AND SPECIES INTERACTIONS IN SIZE-STRUCTURED POPULATIONS , 1984 .

[12]  Dmitriĭ Olegovich Logofet,et al.  Matrices and Graphs Stability Problems in Mathematical Ecology , 1993 .

[13]  O. Diekmann,et al.  On the Cyclic Replicator Equation and the Dynamics of Semelparous Populations , 2009, SIAM J. Appl. Dyn. Syst..

[14]  Xiao-Qiang Zhao,et al.  Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems , 2001 .

[15]  Jim M Cushing,et al.  THE NET REPRODUCTIVE VALUE AND STABILITY IN MATRIX POPULATION MODELS , 1994 .

[16]  Ivor Brown,et al.  Old and Young: , 1971 .

[17]  Y. Takeuchi.  Global Dynamical Properties of Lotka-Volterra Systems , 1996 .

[18]  H. Wilbur.  Complex Life Cycles , 1980 .

[19]  J M Cushing,et al.  A predator prey model with age structure , 1982, Journal of mathematical biology.

[20]  N. Davydova.  Old and Young. Can they coexist , 2004 .

[21]  J. Beddington,et al.  Age structure effects in predator-prey interactions. , 1976, Theoretical population biology.

[22]  M. G. Bulmer,et al.  Periodical Insects , 1977, The American Naturalist.

[23]  J. Hofbauer,et al.  Qualitative permanence of Lotka–Volterra equations , 2008, Journal of mathematical biology.

[24]  Mary Lou Zeeman,et al.  Hopf bifurcations in competitive three-dimensional Lotka-Volterra Systems , 1993 .

[25]  Josef Hofbauer,et al.  On the occurrence of limit cycles in the Volterra-Lotka equation , 1981 .

[26]  A. Wikan,et al.  On synchronization in semelparous populations , 2005, Journal of mathematical biology.

[27]  R. Macarthur.  Species packing and competitive equilibrium for many species. , 1970, Theoretical population biology.

[28]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .