On a boom and bust year class cycle

Abstract

Dedicated to Jim Cushing for all the inspiration, ever since Cortona 1979.We motivate and describe a class of nonlinear Leslie matrix models for semelparous populations, like cicadas and Pacific salmon. We then focus on a Cushing-inspired special case for which one can show rigorously that a heteroclinic boundary cycle exists and attracts nearby orbits for a certain range of parameters. Along the way we formulate some open problems concerning a carrying simplex and global behaviour.Keywords: Heteroclinic cycle; Leslie matrix model; Semelparous species What is a year class?A species is called semelparous if reproduction also amounts to signing one's own death sentence. When there is exactly one reproduction opportunity per year (usually in the spring), we take one year as a natural discrete time unit. For several species, in particular many cicada species, the period in between being born and going to reproduce is strictly fixed at, say, k years. The population then subdivides into subpopulations according to the year of birth modulo k (or, equivalently, the year of reproduction modulo k). Such a subpopulation is called a year class (or, also, a brood). Note that whereas the age class to which an individual belongs increases by one when a year has passed, the year class to which it belongs is fixed once and for all.Year classes only mate among themselves and hence are reproductively isolated. In particular, if a year class goes extinct, it remains extinct. We then say that the year class is "missing". The periodical insects [1] are those for which all but one year classes are missing. The prime example is the Magicicadas (the k ¼ 17 species had its most recent emergence in the North-East United States in 2004).

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

“…Thus one year class can drive other year classes to extinction by inducing, for instance, periodic environmental conditions. The work of Davydova and co-workers (Davydova 2004;Davydova et al 2003Davydova et al , 2005Diekmann et al 2005) focused on this phenomenon.…”

“…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.…”

Carrying simplices in discrete competitive systems and age-structured semelparous populations

“…The main obstacle is that, although it can be easily checked for the Poincaré map associated with time-periodic differential equations (see [30]), the hypothesis (H6) is actually very difficult, sometimes more or less hopeless, to check in discrete-time models. This is indeed the situation, for example, when we investigate a class of nonlinear Leslie models, describing the population dynamics of an age-structured semelparous species (see [5,6,7]).…”

“…Working with Davydova and van Gils, the first author [6,7] investigated the dynamics of semelparous populations and found various phenomena, such as competitive exclusion (also called single year class (SYC) behaviour in [3,31], or synchronization in [20]), coexistence, vertical bifurcation and the possibility of an attracting heteroclinic boundary cycle. One of the main techniques of the analysis in [6,7] is to consider the "full-life-cycle map" T , which is defined as the kth-iterate of the map featuring in (1.1), i.e., define the map T :…”

Stable bifurcations in semelparous Leslie models

“…There is a considerable literature on semelparous Leslie models [1,2,7,8,11,[13][14][15][17][18][19]22,23,25,[27][28][29]. A substantial amount of this literature was stimulated by investigations into the long cyclic outbreaks of periodical insects (the highlight example being the famous periodical cicadas), which is exactly the dynamic of the synchronous cycles on the boundary of the positive cone of the semelparous Leslie model.…”