On the Fundamental Bifurcation Theorem for Semelparous Leslie Models

Abstract

This brief survey of nonlinear Leslie models focuses on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as R ₀ increases through 1. Of particular interest is the bifurcation that occurs when only the oldest age class is reproductive, in which case the Leslie projection matrix is not primitive. This case is distinguished by the invariance of the boundary of the positive cone on which orbits contain temporally synchronized, missing age classes and by the bifurcation of oscillatory attractors, lying on the boundary of the positive cone, in addition to the bifurcation of positive equilibria. The lack of primitivity of the Leslie projection matrix, while seemingly only a mathematically technicality, corresponds to a fundamental life history strategy in population dynamics, namely, semelparity (when individuals have one reproductive event before dying). The study of semelparous Leslie models was historically motivated by the synchronized outbreak cycles of periodical insects, the most famous being the long-lived cicadas (C. magicicada spp).

Appendix

Theorem 1.20 in [56] implies the existence of two, globally distinct continua $\mathcal{C}_{+}^{e}$ and $\mathcal{C}_{-}^{e}$ of nonzero equilibrium pairs each of which satisfies the two alternatives, i.e. is unbounded in $R^{1} \times R^{m}$ or contains a point $\left (\lambda,0\right )$ where λ ≠ 1 is a characteristic value of $M\left (0\right )$.

² In a neighborhood of $\left (1,0\right )$, $\mathcal{C}_{+}^{e}$ and $\mathcal{C}_{-}^{e}$ consist of positive and negative equilibrium pairs respectively. Since second alternative is ruled out by the fact that $M\left (0\right )$ has no characteristic value other than 1, $\mathcal{C}_{+}^{e}$ and $\mathcal{C}_{-}^{e}$ are globally distinct distinct continua that are unbounded in $R \times R^{m}$. For purposes of contradiction we assume the unbounded continuum $C_{+}^{e}$, which in a neighborhood of $\left (1,0\right )$ lies in $R_{+}^{1} \times R_{+}^{m}$, does not remain in $R_{+}^{1} \times R_{+}^{m}$. In this case, it must contain a point $\left (R_{0}^{{\ast}},x^{{\ast}}\right ) \in \partial \left (R_{+} \times R_{+}^{m}\right )$ other than $\left (1,0\right )$ and we can find a sequence of points $\left (R_{0n},x_{n}\right ) \in \mathcal{C}_{+}^{e} \cap \left (R_{+} \times R_{+}^{m}\right )$ such that $\lim _{n\rightarrow \infty }\left (R_{0n},x_{n}\right ) = \left (R_{0}^{{\ast}},x^{{\ast}}\right )$ where $R_{0}^{{\ast}}\geq 0$ and $x^{{\ast}}\in \bar{ R}_{+}^{m}$. We want to arrive at a contradiction.

The points $\left (R_{0n},x_{n}\right )$ satisfy (12)

$$\displaystyle{ x_{n} = R_{0n}M\left (0\right )x_{n} + R_{0n}h\left (x_{n}\right ). }$$

(29)

First, suppose x ^∗ = 0. We can extract a subsequence from the sequence of unit vectors

$$\displaystyle{ u_{n} = \frac{x_{n}} {\left \vert x_{n}\right \vert } \in R_{+}^{m} }$$

that converges to a nonnegative unit vector u: 

$$\displaystyle{ \lim _{n\rightarrow \infty }u_{n} = u \in \bar{ R}_{+} }$$

Passing to the limit in

$$\displaystyle{ \frac{x_{n}} {\left \vert x_{n}\right \vert } = R_{0n}M\left (0\right )\frac{x_{n}} {\left \vert x_{n}\right \vert } + R_{0n}\frac{h\left (x_{n}\right )} {\left \vert x_{n}\right \vert }. }$$

we obtain $u = R_{0}^{{\ast}}M\left (0\right )u.$ This leads to an immediate contradiction if $R_{0}^{{\ast}} = 0$. If $R_{0}^{{\ast}}\neq 0$ then since the only characteristic value of $M\left (0\right )$ is 1 we obtain another contradiction, namely, $R_{0}^{{\ast}} = 1$. Having ruled out $x^{{\ast}} = 0,$ we conclude that $x^{{\ast}}\in \partial R_{+}^{m}\setminus \{0\}.$ Passing to the limit in Eq. (29) we conclude that x ^∗ is an equilibrium of the nonlinear Leslie model (with $R_{0} = R_{0}^{{\ast}}$). However, an inspection of components of the equilibrium equation (10) shows that if one component equals 0 then all components equal 0, i.e. x ^∗ = 0. This is a contradiction to $x^{{\ast}}\in \partial R_{+}^{m}\setminus \{0\}$.