Abstract
We consider a multi-trait evolutionary (game theoretic) version of a two class (juvenile-adult) semelparous Leslie model. We prove the existence of both a continuum of positive equilibria and a continuum of synchronous 2-cycles as the result of a bifurcation that occurs from the extinction equilibrium when the net reproductive number $R_{0}$ increases through $1$. We give criteria for the direction of bifurcation and for the stability or instability of each bifurcating branch. Semelparous Leslie models have imprimitive projection matrices. As a result (unlike matrix models with primitive projection matrices) the direction of bifurcation does not solely determine the stability of a bifurcating continuum. Only forward bifurcating branches can be stable and which of the two is stable depends on the intensity of between-class competitive interactions. These results generalize earlier results for single trait models. We give an example that illustrates how the dynamic alternative can change when the number of evolving traits changes from one to two.
Highlights
Biological semelparity is a life history adaptation in which an individual organism reproduces once and or shortly thereafter, dies
We examine the evolutionary dynamics of a semelparous population by considering a juvenile-adult staged, discrete time model x1 = f φ(x1, x2)x2 (1a) x2 = sσ(x1, x2)x1 (1b) where x1 and x2 are the population densities of juveniles and adults respectively and where prime denotes that the density variable is evaluated at the value of the time variable, which is taken as the maturation period
Theorems 4.1 and 5.1 describe the dual bifurcation of a continuum of positive equilibria and a continuum of synchronous 2-cycles from an extinction state that occurs in the evolutionary semelparous model (3) as the parameter R00 increases through 1
Summary
Biological semelparity is a life history adaptation in which an individual organism reproduces once and or shortly thereafter, dies. Numerous theoretical studies utilize Leslie matrix models to describe the (discrete time) dynamics of semelparous populations with the goal of understanding the dynamic consequences of semelparity as a life history strategy [3], [4], [5], [9], [10], [11], [14], [16], [17], [19], [20], [21], [20], [21], [22], [23], [29], [30], [31], [32], [33], [37], [46] These studies have shown that there are two broad and contrasting categories of dynamics fundamentally implied by semelparous Leslie models, namely, equilibration in which generations overlap and synchronized periodic cycles in which they do not.
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