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

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. Keywords Age-structured population models • Nonlinear dynamics • Nonlinear Leslie matrix models • Single year class dynamics • Semelparous insects Mathematics Subject Classification 37N25 • 92B25 Dedicated to Hans Othmer, a winner who shares rather than deprives The first part of the title is inspired by the ABBA song, but the inspiration was catalyzed by M. Deijfen, R. van der Hofstad, The winner takes it all,

Global dynamics of a special class of nonlinear semelparous Leslie matrix models

“…It is clear that assumption (A1) implies both R n + and intR n + are forward invariant for system (14). The following lemma shows that any solution of system (14) converges to neither 0 nor infinity under assumption (A3). Lemma 3.1: Assume that (A1) and (A3) hold.…”

“…They conjectured that every positive solution converges. We are thus motivated to find some special systems like (14) and (18) such that this conjecture holds.…”

“…Equation ( 14) means that the effect of the other species on the growth rate of species i is determined by their total population size j =i x j (t). We study system (14) under the assumptions that for 1 ≤ ≤ m, (A1) h (x) are positive, continuous and strictly decreasing functions on [0, ∞), (A2) there are positive constants α with…”

“…We discuss in this section the local asymptotic stability of some equilibria in system (14). For this purpose, we have to assume that besides (A1)-(A3), all h in system ( 14) are differentiable.…”

“…Besides these bifurcation studies, Davydova et al [13] examined the asymptotic behaviour of (1) mathematically and numerically for the special case n = 2 and σ i (u 1 , u 2 ) = exp(−a i (c 1 u 1 + c 2 u 2 )), i = 1, 2. In addition, given that the coordinate axes include every cycle associated with the periodic behaviour in periodical insects, the attractivity of the coordinate axes was studied by Mjølhus et al [25], Kon [18], Kon and Iwasa [20] and Diekmann and Planqué [14]. All these studies only reveal the local behaviour around equilibria, cycles or the coordinate axes (but see [14] for an example of (1) where the coordinate axes attract a large set of initial conditions).…”