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

Odo Diekmann,Robert Planqué

doi:10.1007/s00285-019-01362-3

Odo Diekmann, Robert Planqué

Open Access

https://doi.org/10.1007/s00285-019-01362-3

Copy DOI

Abstract

The aim of this short note is to give a simple explanation for the remarkable periodicity of Magicicada species, which appear as adults only every 13 or 17 years, depending on the region. We show that a combination of two types of density dependence may drive, for large classes of initial conditions, all but 1 year class to extinction. Competition for food leads to negative density dependence in the form of a uniform (i.e., affecting all age classes in the same way) reduction of the survival probability. Satiation of predators leads to positive density dependence within the reproducing age class. The analysis focuses on the full life cycle map derived by iteration of a semelparous Leslie matrix.

Highlights

The first part of the title is inspired by the ABBA song, but the inspiration was catalyzed by M
As far as we are aware, we provide below the first analytical demonstration of 1 year class driving all other year classes to extinction without severe restrictions on the initial conditions and without any condition on the resulting single year class (SYC) dynamics, except for boundedness
It is tempting to conjecture that a fixed point of the full life cycle map, with more than 1 year class having positive density, is necessarily unstable, as was found to be the case for the model considered by Machta et al (2019)

Summary

Periodical insects: a conundrum

According to Bulmer (1977), “An insect population is said to be periodical if the life cycle has a fixed length of k years (k > 1) and if the adults do not appear every year but only every kth year.” He provides several examples, one by quoting Lloyd and Dybas (1966). Note that this holds independently of the resulting single year class (SYC) dynamics in the sense that the winners may exhibit steady state, periodic or even chaotic dynamics Note that it depends on the initial condition which one of the k year classes will win the competition. The upshot is that the combination of uniform negative density dependence and concentrated (in one point of the life cycle) positive density dependence, as assumed in (Machta et al 2019), leads, as a rule, to exclusion of all but at most 1 year class This does not prove, that this is the mechanism underlying the observed phenomenon of SYC dynamics in Magicicada. As far as we are aware, we provide below the first analytical demonstration of 1 year class driving all other year classes to extinction without severe restrictions on the initial conditions and without any condition on the resulting SYC dynamics, except for boundedness

Model formulation

The main results

Model ingredients

Discussion