Abstract
In this chapter we set up the modeling framework for a study of cannibalism by means of low-dimensional structured population models that distinguish only between cannibals and victims. Our specific interest is on adult cannibalism of juveniles . In Sect. 11.2 we use matrix modeling methodology [5, 7, 8] to develop and analyze a general discrete-time juvenile-adult matrix model.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abrams PA (2001) Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: an assessment of three methods. Ecol Lett 4:166–175
Abrams PA (2006) Adaptive change in the resource-exploitation traits of a generalist consumer: the evolution and coexistence of generalists and specialists. Evolution 60:427–439
Abrams PA, Matsuda H, Harada Y (1993) Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol Ecol 7:465–487
Barfield J, Holt RD, Gomulkiewicz R (2011) Evolution in stage-structured populations. Am Nat 177(4):397–408
Caswell H (2001) Matrix Population Models: Construction, Analysis and Interpretation, 2nd edn. Sinauer Associates, Inc. Publishers, Sunderland
Courchamp F, Berec L, Gascoigne J (2008) Allee Effects in Ecology and Conservation. Oxford University Press, Oxford
Cushing JM (1998) An Introduction to Structured Population Dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics, vol 71. SIAM, Philadelphia
Cushing JM (2009) Matrix models and population dynamics. In: Lewis M, Chaplain AJ, Keener JP, Maini PK (eds) Mathematical Biology, IAS/Park City Mathematics Series, vol 14. American Mathematical Society, Providence, pp 47–150
Cushing JM (2010) On the dynamics of a class of Darwinian matrix models. Nonlin Dyn Syst Theory 10(2):103–116
Cushing JM (2010) A bifurcation theorem for Darwinian matrix models. Nonlin Stud 17(1):1–13
Cushing JM (2011) On the relationship between r and R0 and its role in the bifurcation of equilibria of Darwinian matrix models. J Biol Dyn 5:277–297
Cushing JM (2014) Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations. J Biol Dyn 8:57–73
Cushing JM (2021) A bifurcation theorem for Darwinian matrix models with an application to the evolution of reproductive life history strategies. J Biol Dyn 15(Sup1):S190–S213. https://doi.org/10.1080/17513758.2020.1858196
Cushing JM, Diekmann O (2016) The many guises of R0 (a didactic note). J Theor Biol 404:295–302
Cushing JM, Farrell AP (2019) A bifurcation theorem for nonlinear matrix models of population dynamics. J Diff Equ Appl 26:25–44. http://dx.doi.org/10.1080/10236198.2019.1699916
Cushing JM, MacCracken-Stump S (2010) Darwinian dynamics of a juvenile-adult model. Math Biosci Eng 10(4):1017–1044
Cushing JM, Zhou Y (1994) The net reproductive value and stability in structured population models. Nat Resour Model 8:1–37
Cushing JM, Henson SM, Hayward JL (2015) An evolutionary game theoretic model of cannibalism. Nat Resour Model 28(4):497–521. https://doi.org/10.1111/nrm.12079
Cushing JM, Martins F, Pinto AA, Veprauskas A (2017) A bifurcation theorem for evolutionary matrix models with multiple traits. J Math Biol 75(1):491–520. https://doi.org/10.1007/s00285-016-1091-4
Dercole F, Rinaldi S (2008) Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications. Princeton University Press, Princeton
Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34:569–612
Elaydi SN (2005) An Introduction to Difference Equations, 3rd edn. Springer, New York
Kielhöfer H (2004) Bifurcation Theory: An Introduction with Applications to PDEs. Applied Mathematical Sciences 156, Springer, New York
Kon R, Saito Y, Takeuchi Y (2004) Permanence of single-species stage-structured models. J Math Biol 48:515–528
Lande R (1976) Natural selection and random genetic drift in phenotypic evolution. Evolution 30:314–334
Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33:183–212
Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35:213–245
Li C-K, Schneider H (2002) Applications of Perron-Frobenius theory to population dynamics. J Math Biol 44:450–462
McGill BM, Brown JS (2007) Evolutionary game theory and adaptive dynamics of continuous traits. Ann Rev Ecol Evol Syst 38:403–435
Meissen EP, Salau KR, Cushing JM (2016) A global bifurcation theorem for Darwinian matrix models. J Diff Equ Appl 22:1114–1136. https://doi.org/10.1080/10236198.2016.1177522
Rabinowitz PH (1971) Some global results for nonlinear eigenvalue problems. J Funct Anal 7:487–513
Vincent TL, Brown JS (2005) Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Cushing, J.M., Henson, S.M., Hayward, J.L. (2023). Models of Adult-on-Juvenile Cannibalism. In: Modeling Behavior and Population Dynamics. Interdisciplinary Applied Mathematics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-031-34283-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-031-34283-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-34282-0
Online ISBN: 978-3-031-34283-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)