A Darwinian version of the Leslie logistic model for age-structured populations

Jim M. Cushing; Jim M. Cushing

doi:10.3934/mbe.2025047

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 6: 1263-1279. doi: 10.3934/mbe.2025047

Previous Article Next Article

Research article Special Issues

A Darwinian version of the Leslie logistic model for age-structured populations

Jim M. Cushing ^,

Department of Mathematics, University of Arizona, Tucson 85721, USA

Received: 11 December 2024 Revised: 12 February 2025 Accepted: 19 March 2025 Published: 21 April 2025

The known global dynamics of the classic Leslie logistic model for the dynamics of an age-structured population are extended to a Darwinian dynamic version of the model for a single phenotypic trait (that is subject to natural selection). This is done under the assumption that the speed of evolution does not exceed an upper bound and that the maximum intraspecific competition intensity experienced by an individual occurs when its inherited trait equals that of the population mean trait. An example is given that applies the results to a model in which age-specific birth rates are subject to natural selection and that illustrates conditions under which evolution favors an iteroparous-type or a semelparous-type of life history strategy.
- Leslie matrix,
- logistic growth,
- age-structured population dynamics,
- Darwinian dynamics
Citation: Jim M. Cushing. A Darwinian version of the Leslie logistic model for age-structured populations[J]. Mathematical Biosciences and Engineering, 2025, 22(6): 1263-1279. doi: 10.3934/mbe.2025047

Related Papers:

Abstract

The known global dynamics of the classic Leslie logistic model for the dynamics of an age-structured population are extended to a Darwinian dynamic version of the model for a single phenotypic trait (that is subject to natural selection). This is done under the assumption that the speed of evolution does not exceed an upper bound and that the maximum intraspecific competition intensity experienced by an individual occurs when its inherited trait equals that of the population mean trait. An example is given that applies the results to a model in which age-specific birth rates are subject to natural selection and that illustrates conditions under which evolution favors an iteroparous-type or a semelparous-type of life history strategy.

References

[1]	P. H. Leslie, On the use of matrices in certain population mathematics, Biometrika, 33 (1945), 183–212. https://doi.org/10.2307/2332297 doi: 10.2307/2332297
[2]	H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Second Edition, Sinauer Associates, Inc. (an imprint of Oxford University Press), 2001. ISBN 9780878931217
[3]	J. M. Cushing, Matrix models and population dynamics, in Mathematical Biology (eds. M. Lewis, A. J. Chaplain, J. P. Keener, P. K. Maini), IAS/Park City Mathematics Series Vol 14, American Mathematical Society, Providence, RI, (2009), 47–150. ISBN 9780821847657
[4]	P. H. Leslie, Some further notes on the use of matrices in opulation mathematics, Biometrika, 35 (1948), 213–245. https://doi.org/10.2307/2332342 doi: 10.2307/2332342
[5]	M. Barfield, R. D. Holt, R. Gomulkiewicz, Evolution in stage-structured populations, Am. Nat., 177 (2011), 397–409. https://doi.org/10.1086/658903 doi: 10.1086/658903
[6]	S. N. Elaydi, J. M. Cushing, Discrete Mathematical Models in Population Biology: Ecological, Epidemic, and Evolutionary Dynamics, Springer Undergraduate Texts in Mathematics and Technology, Springer Nature Switzerland AG, 2024. https://doi.org/10.1007/978-3-031-64795-6
[7]	B. J. McGill, J. S. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Annu. Rev. Ecol. Evol. S., 38 (2007), 403–435. https://doi.org/10.1146/annurev.ecolsys.36.091704.175517 doi: 10.1146/annurev.ecolsys.36.091704.175517
[8]	T. L. Vincent, J. S. Brown, Evolutionary Game theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, United Kingdom, 2005. https://doi.org/10.1007/s10914-006-9013-7
[9]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1994. https://doi.org/10.1137/1.9781611971262
[10]	J. Impagliazzo, Deterministic Aspects of Mathematical Demography, Biomathematics Vol. 13, Springer-Verlag, Berlin, 1985. https://doi.org/10.007/978-3-642-82319-0
[11]	J. M. Cushing, A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured populations, Nat. Resour. Model., 3 (1989), 331–357. https://doi.org/10.1111/j.1939-7445.1989.tb00085.x doi: 10.1111/j.1939-7445.1989.tb00085.x
[12]	J. M. Cushing, Z. Yicang, The net reproductive value and stability in matrix population models, Nat. Resour. Model., 8 (1994), 297–333. https://doi.org/10.1111/j.1939-7445.1994.tb00188.x doi: 10.1111/j.1939-7445.1994.tb00188.x
[13]	C-K. Li, H. Schneider, Applications of Perron-Frobenius theory to population dynamics, J. Math. Biol., 44 (2022), 450–462. https://doi.org/10.1007/s002850100132 doi: 10.1007/s002850100132
[14]	J. M. Cushing, Matrix Models for Population, Disease, and Evolutionary Dynamics, Student Mathematical Library, Volume 106, American Mathematical Society, 2024. ISBN 978-1-4704-7334-1
[15]	M. Ch-Chaoui, K. Mokni, A discrete evolutionary Beverton–Holt population model, Int. J. Dyn. Contr., 11 (2023), 1060–1075. https://doi.org/10.1007/s40435-022-01035-y doi: 10.1007/s40435-022-01035-y
[16]	J. M. Cushing, A Darwinian Ricker equation, Progress on Difference Equations and Discrete Dynamical Systems (S. Baigent, M. Bohner, S. Elaydi, editors), Springer Proc. Math. Stat., vol. 341, Springer, Cham, 2020, pp. 231–243. https://doi.org/10.1007/978-3-030-60107-2_10. MR4219152
[17]	J. M. Cushing, K. Stefanko, A Darwinian dynamic model for the evolution of post-reproduction survival, J. Biol. Syst., 29 (2021), 433–450. https://doi.org/10.1142/S0218339021400088 doi: 10.1142/S0218339021400088
[18]	R. Luís, Open problems and conjectures in the evolutionary periodic Ricker competition model. Axioms, 13 (2024), 246. https://doi.org/10.3390/axioms13040246 doi: 10.3390/axioms13040246
[19]	K. Mokni, S. Elaydi, M. CH-Chaoui, A. Eladdadi, Discrete evolutionary population models: a new approach, J. Biol. Dyn., 14 (2020), 454–478. https://doi.org/10.1080/17513758.2020.1772997 doi: 10.1080/17513758.2020.1772997
[20]	S. Elaydi, Y. Kang, R Luís, The effects of evolution on the stability of competing species, J. Biol. Dynam., 16 (2022), 816–839. https://doi.org/10.1080/17513758.2022.2154860 doi: 10.1080/17513758.2022.2154860
[21]	J. M. Cushing, On the relationship between $r$ and $R_{0}$ and its role in the bifurcation of equilibria of Darwinian matrix models, J. Biol. Dynam., 5 (2011), 277–297. https://doi.org/10.1080/17513758.2010.491583 doi: 10.1080/17513758.2010.491583
[22]	D. A. Roff, Evolution of Life Histories: Theory and Analysis, Springer New York, New York, USA, 1993. ISBN 978-0-412-02391-0
[23]	P. W. Hughes, Between semelparity and iteroparity: empirical evidence for a continuum of modes of parity, Ecol. Evol., 7 (2017), 8232–8261. https://doi.org/10.1002/ece3.3341 doi: 10.1002/ece3.3341
[24]	J. M. Cushing, A Discrete time Darwinian dynamics and semelparity versus iteroparity, Math. Biosci. Eng., 16 (2019), 1815–1835. https://doi.org/10.3934/mbe.2019088 doi: 10.3934/mbe.2019088
[25]	J. M. Cushing, A bifurcation theorem for Darwinian matrix models with an application to the evolution of reproductive life history strategies, J. Biol. Dynam., 15 (sup1) (2021), S190–S213. https://doi.org/10.1080/17513758.2020.1858196 doi: 10.1080/17513758.2020.1858196
[26]	J. M. Cushing, F. Martins, A. A. Pinto, A. Veprauskas, A bifurcation theorem for evolutionary matrix models with multiple traits, J. Math. Biol., 75 (2017), 491–520. https://doi.org/10.1007/s00285-016-1091-4 doi: 10.1007/s00285-016-1091-4

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)