Advertisement

Bifurcations of cycles in nonlinear semelparous Leslie matrix models

  • 17 Accesses

Abstract

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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

References

  1. Bulmer MG (1977) Periodical insects. Am Nat 111:1099–1117

  2. Cushing JM (2006) Nonlinear semelparous Leslie models. Math Biosci Eng 3:17–36

  3. Cushing JM (2009) Three stage semelparous Leslie models. J Math Biol 59(1):75–104

  4. Cushing JM, Henson SM (2012) Stable bifurcations in semelparous Leslie models. J Biol Dyn 6:80–102

  5. Cushing JM, Li J (1989) On Ebenman’s model for the dynamics of a population with competing juveniles and adults. Bull Math Biol 51(6):687–713

  6. Cushing JM, Yicang Z (1994) The net reproductive value and stability in matrix population models. Natur Resour Model 8:297–333

  7. Davis P (1979) Circulant matrices. Wiley, New York

  8. Davydova NV, Diekmann O, van Gils SA (2003) Year class coexistence or competitive exclusion for strict biennials? J Math Biol 46(2):95–131

  9. Diekmann O, Planque R (2019) The winner takes it all: how semelparous insects can become periodical. bioRxiv

  10. Diekmann O, van Gils SA (2009) On the cyclic replicator equation and the dynamics of semelparous populations. SIAM J Appl Dyn Syst 8:1160–1189

  11. Ebenman B (1988) Competition between age classes and population dynamics. J Theor Biol 131(4):389–400

  12. Hassell M (2000) The spatial and temporal dynamics of host-parasitoid interactions. Oxford University Press, Oxford

  13. Hofbauer J (1981) On the occurrence of limit cycles in the Volterra–Lotka equation. Nonlinear Anal Theory Methods Appl 5(9):1003–1007

  14. Hofbauer J, Sigmund K (1988) The theory of evolution and dynamical systems: mathematical aspects of selection. Cambridge University Press, Cambridge

  15. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

  16. Hoppensteadt FC, Keller JB (1976) Synchronization of periodical cicada emergences. Science 194(4262):335–337

  17. Kon R (2006) Invasibility of missing year-classes in Leslie matrix models for a semelparous biennial population. In: Proceedings of Czech-Japanese seminar in applied mathematics 2005, vol 3 of COE lectures note, pp 77–87. Kyushu University The 21 Century COE Program, Fukuoka

  18. Kon R (2007) Competitive exclusion between year-classes in a semelparous biennial population. In: Deutsch A, de la Parra RB, de Boer RJ, Diekmann O, Jagers P, Kisdi E, Kretzschmar M, Lansky P, Metz H (eds) Mathematical modeling of biological systems, vol II. Birkhäuser, Boston, pp 79–90

  19. Kon R (2011) Age-structured Lotka–Volterra equations for multiple semelparous populations. SIAM J Appl Math 71(3):694–713

  20. Kon R (2012) Permanence induced by life-cycle resonances: the periodical cicada problem. J Biol Dyn 6(2):855–890

  21. Kon R (2017a) Stable bifurcations in multi-species semelparous population models. In: Advances in difference equations and discrete dynamical systems, Springer proceedings in mathematics and statistics, vol 212, pp 3–25. Springer, Singapore

  22. Kon R (2017b) Non-synchronous oscillations in four-dimensional nonlinear semelparous Leslie matrix models. J Differ Equ Appl 23(10):1747–1759

  23. Kon R, Iwasa Y (2007) Single-class orbits in nonlinear Leslie matrix models for semelparous populations. J Math Biol 55(5–6):781–802

  24. La Salle JP (1976) The stability of dynamical systems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia

  25. Li C-K, Schneider H (2002) Applications of perron-frobenius theory to population dynamics. J Math Biol 44(5):450–462

  26. Machta J, Blackwood JC, Noble A, Liebhold AM, Hastings A (2018) A hybrid model for the population dynamics of periodical cicadas. Bull Math Biol 81:1122–1142

  27. May RM (1973) Stability and complexity in model ecosystems. Princeton University Press, Princeton

  28. May RM (1979) Periodical cicadas. Nature 277:347–349

  29. May RM, Leonard WJ (1975) Nonlinear aspects of competition between three species. SIAM J Appl Math 29(2):243–253

  30. Mjølhus E, Wikan A, Solberg T (2005) On synchronization in semelparous populations. J Math Biol 50(1):1–21

  31. Schuster P, Sigmund K, Wolff R (1979) On \(\omega \)-limits for competition between three species. SIAM J Appl Math 37(1):49–54

Download references

Acknowledgements

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

Author information

Correspondence to Ryusuke Kon.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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)

    \(\hat{\varvec{p}}_0(0)=\tilde{\varvec{q}}_0(0)=\varvec{0}\),

  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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kon, R. Bifurcations of cycles in nonlinear semelparous Leslie matrix models. J. Math. Biol. (2020). https://doi.org/10.1007/s00285-019-01459-9

Download citation

Keywords

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

Mathematics Subject Classification

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