Bifurcations of cycles in nonlinear semelparous Leslie matrix models

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This paper develops a method for studying bifurcations that occur in a neighborhood of the extinction equilibrium in nonlinear semelparous Leslie matrix models. The method uses a Lotka–Volterra equation with cyclic symmetry to detect the existence and to evaluate the stability of bifurcating equilibria and cycles. An application of the method provides sharp stability conditions for both a single-class cycle and a positive equilibrium bifurcating from the extinction equilibrium. The stability condition for a bifurcating single-class cycle confirms that the periodicity observed in periodical insects occurs if competition is more severe between than within age-classes. The developed method is also used to investigate two examples of nonlinear semelparous Leslie matrix models incorporating predator satiation. The investigation shows that a single-class cycle, which is associated with the periodicity in periodical insects, is a unique stable cycle in a neighborhood of the extinction equilibrium if the density effects in survival probabilities are identical among age-classes.

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This work was supported by JSPS KAKENHI Grant Number 16K05279.

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Correspondence to Ryusuke Kon.

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Appendix A: The proof for the uniqueness of the function \(\hat{\varvec{p}}_0\) in Theorem 3

Appendix A: The proof for the uniqueness of the function \(\hat{\varvec{p}}_0\) in Theorem 3

For the sake of contradiction, suppose that, in addition to \(\hat{\varvec{p}}_0\), there exists a function \(\tilde{\varvec{q}}_0: [0,\epsilon _0) \rightarrow {\mathbb {R}}_+^n\) satisfying (a), (b), and (c). Let \({\tilde{m}}\) be the minimal period of the periodic orbit of \(\varvec{\xi }_\epsilon \) with the initial vector \(\tilde{\varvec{q}}_0(\epsilon )\). Then both \(\hat{\varvec{p}}_0(\epsilon )\) and \(\tilde{\varvec{q}}_0(\epsilon )\) are equilibria of the map \(\varvec{\xi }_\epsilon ^{{\tilde{m}}n}\) since \(\hat{\varvec{p}}_0(\epsilon )\) is an equilibrium of (4), i.e., \(\varvec{\xi }_\epsilon ^n\). The map \(\varvec{\xi }_\epsilon ^{{\tilde{m}}n}\) represents a Kolmogorov difference equation

$$\begin{aligned} x_{i,k+1}=x_{i,k} {\tilde{g}}_i(\epsilon ,\varvec{x}_k) \quad (i=1,2,\ldots ,n) \end{aligned}$$

and \({\tilde{g}}_i(0,\varvec{0})=1\) is satisfied. By the same argument as in Sect. 5, we can show that

$$\begin{aligned} \frac{\partial \tilde{\varvec{g}}}{\partial \epsilon }(\varvec{0})= & {} ({\tilde{m}},{\tilde{m}},\ldots ,{\tilde{m}})^\top \\ \frac{\partial \tilde{\varvec{g}}}{\partial \varvec{x}}(\varvec{0})= & {} BD+P^{-1}BDP+\cdots +P^{-{\tilde{m}}n+1} BD P^{{\tilde{m}}n-1}\\= & {} {\tilde{m}} (BD+P^{-1}BDP+\cdots +P^{-n+1} BD P^{n-1}). \end{aligned}$$

By rescaling time t, we can reduce the Lotka–Volterra equation (6) satisfying \(\varvec{r}=\frac{\partial \tilde{\varvec{g}}}{\partial \epsilon }(\varvec{0})\) and \(A=\frac{\partial \tilde{\varvec{g}}}{\partial \varvec{x}}(\varvec{0})\) to (6) satisfying (10). This implies that \(\varvec{x}^*\) is also an equilibrium of (6) satisfying \(\varvec{r}=\frac{\partial \tilde{\varvec{g}}}{\partial \epsilon }(\varvec{0})\) and \(A=\frac{\partial \tilde{\varvec{g}}}{\partial \varvec{x}}(\varvec{0})\). Since both the functions \(\hat{\varvec{p}}_0\) and \(\tilde{\varvec{q}}_0\) satisfy the properties that

  1. (i)


  2. (ii)

    \(\hat{\varvec{p}}_0\) and \(\tilde{\varvec{q}}_0\) are equilibria of \(\varvec{\xi }_\epsilon ^{{\tilde{m}}n}\) for all \(\epsilon \in (0,\epsilon _0)\), and

  3. (iii)

    \(\mathrm{supp}(\hat{\varvec{p}}_0(\epsilon ))=\mathrm{supp}(\tilde{\varvec{q}}_0(\epsilon ))=\mathrm{supp}(\varvec{x}^*)\) for all \(\epsilon \in (0,\epsilon _0)\),

the uniqueness property ensured in Theorem 1 implies that \(\hat{\varvec{p}}_0(\epsilon )=\tilde{\varvec{q}}_0(\epsilon )\) for all \(\epsilon \in (0,\epsilon _0)\). This completes the proof.

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Kon, R. Bifurcations of cycles in nonlinear semelparous Leslie matrix models. J. Math. Biol. (2020).

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  • Semelparity
  • Predator satiation
  • Periodical insect
  • Periodical cicada
  • Leslie matrix
  • Lotka–Volterra equation

Mathematics Subject Classification

  • 39A28
  • 39A30
  • 37N25
  • 92D25