An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities

Abstract

A discrete age-structured semelparous Leslie matrix model where density dependence is included both in the fecundity and in the survival probabilities is analysed. Depending on strength of density dependence, we show in the precocious semelparous case that the nonstationary dynamics may indeed be rich, ranging from SYC (a dynamical state where the whole population is in one age class only) dynamics to cycles of low period where all age classes are populated. Quasiperiodic and chaotic dynamics have also been identified. Moreover, outside parameter regions where SYC dynamics dominates, we prove that the transfer from stability to instability goes through a supercritical Neimark−Sacker bifurcation, and it is further shown that when the population switches from possessing a precocious to a delayed semelparous life history both stability properties and the possibility of periodic dynamics become weaker.

Stage-Dependent Predation on Prey Species Who Possess Different Life Histories

“…Let us now in somewhat more detail scrutinize the dynamics in the region where the bifurcation is supercritical. Our frst observation, confer (10), is that the value of F (or x * � ln R 0 ) at instability threshold is smaller in the precocious cases compared to the delayed cases which signals that species who possess a delayed semelparous or iteroparous life history have better stability properties than species with precocious semelparous or iteroparous life histories.…”

“…Instead, single year class (SYC) dynamics dominates where the whole population is in one age class only at each time. Papers [6][7][8][9][10] focus on bifurcations, SYC dynamics as well as ergodic properties, and the dynamics and remarkable periodicity of Magicicada species has been scrutinized in [11,12] by use of semelparous models.…”

“…Theorem . In case of μ 1 sufciently large (not too close to zero), the fxed point (x * 1 , x * 2 ) of map (1) will undergo a supercritical fip bifurcation when (10) becomes an equality.…”

The winner takes it all: how semelparous insects can become periodical

“…Finally, the same three species-the same as nearly as anyone can tell by looking at them or listening to the songs of their males-exist both as 17 and as 13-year periodical cicadas! ; Note that the three species are now considered to be seven species, each with their own period of either 13 or 17 years (Marshall 2008)] In the wake of Bulmer's pioneering paper a rich literature on discrete time models for semelparous organisms developed, see (Behncke 2000;Webb 2001;Diekmann et al 2005;Cushing 2009;Cushing and Henson 2012;Kon 2012Kon , 2017Cushing 2015;Wikan 2017) and the references given there. The population splits into year classes according to the year of birth (equivalently: the year of reproduction) counted modulo k. As a year class is reproductively isolated from the other year classes, it forms a population by itself.…”

The winner takes it all: how semelparous insects can become periodical

“…In the wake of Bulmer's pioneering paper a rich literature on discrete time models for semelparous organisms developed, see (Behncke 2000, Cushing 20092015, Cushing and Henson 2012, Kon 2012, Webb 2001, Wikan 2017 and the references given there. The population splits into year classes according to the year of birth (equivalently: the year of reproduction) counted modulo k. As a year class is reproductively isolated from the other year classes, it forms a population by itself.…”