Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the <em>r</em>-Lambert <em>W</em> function

Cristeta U. Jamilla; Renier G. Mendoza; Victoria May P. Mendoza; Cristeta U. Jamilla; Renier G. Mendoza; Victoria May P. Mendoza

doi:10.3934/mbe.2020306

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5686-5708. doi: 10.3934/mbe.2020306

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Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the r-Lambert W function

Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines

Received: 25 June 2020 Accepted: 03 August 2020 Published: 28 August 2020

Structured population models, which account for the state of individuals given features such as age, gender, and size, are widely used in the fields of ecology and biology. In this paper, we consider an age-structured population model describing the population of adults and juveniles. The model consists of a system of ordinary and neutral delay differential equations. We present an explicit solution to the model using a generalization of the Lambert W function called the r-Lambert W function. Numerical simulations with varying parameters and initial conditions are done to illustrate the obtained solution. The proposed method is also applied to an insect population model with long larval and short adult phases.
- age-structured population model,
- r-Lambert W function,
- neutral delay differential equation,
- explicit solution,
- Lotka-Sharpe-McKendrick system,
- insect population model
Citation: Cristeta U. Jamilla, Renier G. Mendoza, Victoria May P. Mendoza. Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the r-Lambert W function[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5686-5708. doi: 10.3934/mbe.2020306

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Abstract

Structured population models, which account for the state of individuals given features such as age, gender, and size, are widely used in the fields of ecology and biology. In this paper, we consider an age-structured population model describing the population of adults and juveniles. The model consists of a system of ordinary and neutral delay differential equations. We present an explicit solution to the model using a generalization of the Lambert W function called the r-Lambert W function. Numerical simulations with varying parameters and initial conditions are done to illustrate the obtained solution. The proposed method is also applied to an insect population model with long larval and short adult phases.

References

[1]	F. M. Asl, G. A. Ulsoy, Analysis of a system of linear delay differential equations, ASME J. Dyn. Sys., Meas., Control, 125 (2003), 215-223.
[2]	S. Yi, P. W. Nelson, A. G. Ulsoy, Solution of systems of linear delay differential equations via Laplace transformation, in Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, (2006), 2535-2540.
[3]	S. Yi, P. W. Nelson, A. G. Ulsoy, Survey on analysis of time delayed systems via the Lambert W function, Differ. Equations, 14 (2007), 296-301.
[4]	S. Yi, A. G. Ulsoy, Solution of a system of linear delay differential equations using the matrix Lambert function, in Proceedings of 2006 American Control Conference, Minneapolis, IEEE, (2006), 2433-2438.
[5]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert w function, Adv. Comput. Math., 5 (1996), 329-359.
[6]	I. Mezö, G. Keady, Some physical applications of generalized Lambert functions, Eur. J. Phys., 37 (2016), 065802.
[7]	I. Mezö, A. Baricz, On the generalization of the lambert W function, Trans. Amer. Math. Soc., 369 (2017), 7917-7934.
[8]	I. Mezö, On the structure of the solution set of a generalized Euler-Lambert equation, J. Math. Anal. Appl., 455, (2017), 538-553.
[9]	C. B. Corcino, R. B. Corcino, An asymptotic formula for r-Bell numbers with real arguments, ISRN Discrete Math., 2013 (2013).
[10]	V. Barsan, Inverses of Langevin, Brillouin and related functions: a status report, Rom. Rep. Phys., 72 (2020).
[11]	C. Ewerhart, G. Z. Sun, Equilibrium in the symmetric two-player Hirshleifer contest: uniqueness and characterization, Econ. Lett., 169 (2018), 51-54.
[12]	L. M. Briceño-Arias, G. Chierchia, E. Chouzenoux, J. C. Pesquet, A random block-coordinate Douglas-Rachford splitting method with low computational complexity for binary logistic regression, Comput. Optim. Appl., 72 (2019), 707-726.
[13]	V. Barsan, Simple and accurate approximants of inverse Brillouin functions, J. Magn. Magn. Mater., 473 (2019), 399-402.
[14]	V. Barsan, Siewert solutions of transcendental equations, generalized Lambert functions and physical applications, Open Phys., 16 (2018), 232-242.
[15]	V. Barsan, New results concerning the generalized Lambert functions and their applications to solar energy conversion and nanophysics, CTP-Trieste, Spectroscopy and Dynamics of Photoinduced Electronic Excitations (Workshop Poster), 2017. Available from: https://www.researchgate.net.
[16]	D. Belkić, All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and fox-wright function: illustration for genome multiplicity in survival of irradiated cells, J. Math. Chem., 57 (2019), 59-106.
[17]	N. Bovenzi, Spin-momentum locking in oxide interfaces and in Weyl semimetals, Ph.D. thesis, University of Leiden, 2018.
[18]	N. Bovenzi, M. Breitkreiz, T. E. O'Brien, J. Tworzydło, C. W. J. Beenakker, Twisted fermi surface of a thin-film Weyl semimetal, New J. Phys., 20 (2018), 023023.
[19]	R. M. Digilov, Gravity discharge vessel revisited: an explicit Lambert W function solution, Am. J. Phys., 85 (2017) 510-514.
[20]	J. Guo, Exact procedure for Einstein-Johnson's sidewall correction in open channel flow, J. Hydraul. Eng., 143 (2017), 06016027.
[21]	R. Jedynak, A comprehensive study of the mathematical methods used to approximate the inverse Langevin function, Math. Mech. Solids., 24 (2019), 1992-2016.
[22]	R. Jedynak, New facts concerning the approximation of the inverse Langevin function, J. Nonnewton Fluid Mech., 249 (2017), 8-25.
[23]	I. Lopez-Garcia, C. S. Lopez-Monsalvo, E. Campero-Littlewood, F. Beltran-Carbajal, E. Campero-Littlewood, Alternative modes of operation for wind energy conversion systems and the generalised Lambert W-function, IET Gener. Transm. Distrib., 12 (2018), 3152-3157.
[24]	B. C. Marchi, E. M. Arruda, Generalized error-minimizing, rational inverse Langevin approximations, Math. Mech. Solids., 24 (2019), 1630-1647.
[25]	O. Olendski, Thermodynamic properties of the 1D Robin quantum well, Ann. Phys., 530 (2018).
[26]	S. Rebollo-Perdomo, C. Vidal, Bifurcation of limit cycles for a family of perturbed Kukles differential systems, Am. Inst. Math. Sci. Discrete Contin. Dyn. Syst. A, 38 (2018), 4189-4202.
[27]	H. Vazquez-Leal, M. A. Sandoval-Hernandez, J. L. Garcia-Gervacio, A. L. Herrera-May, U. A. Filobello-Nino, PSEM approximations for both branches of Lambert W function with applications, Discrete Dyn. Nat. Soc., 2019 (2019).
[28]	M. Vono, P. Chainais, Sparse Bayesian binary logistic regression using the split-and-augmented Gibbs sampler, in Proceedings of 2018 IEEE International Workshop on Machine Learning for Signal Processing, IEEE, (2018).
[29]	C. Jamilla, R. Mendoza, I. Mezö, Solutions of neutral delay differential equations using a generalized Lambert W function, Appl. Math. Comput., 382 (2020), 125334.
[30]	G. Bocharov, K. P. Hadeler, Structured population models, conservation Laws, and delay equations, J. Differ. Equations, 168 (2000), 212-237.
[31]	P. J. Gullan, P. S. Cranston, The Insects: An Outline of Entomology, 5^th edition, John Wiley & Sons, Ltd, 2014.
[32]	M. A. Zabek, Understanding population dynamics of feral horses in the Tuan and Toolara State Forest for successful long-term population management, Ph.D. thesis, The University of Queensland, 2015.
[33]	M. A. Zabek, D. M. Berman, S. P. Blomberg, C. W. Collins, J. Wright, Population dynamics of feral horses (Equus caballus) in an exotic coniferous plantation in Australia, Wildlife Res., 43 (2016), 358-367.
[34]	S. A. Gourley, Y. Kuang, Dynamics of a neutral delay equation for an insect population with long larval and short adult phases, J. Differ. Equations, 246 (2009), 4653-4669.
[35]	K. S. Williams, C. Simon, The ecology, behavior and evolution of periodical cicadas, Annu. Rev. Entomol., 40 (1995), 269-295.
[36]	K. Soong, G. F. Chen, J. R. Cao, Life history studies of the flightless marine midges Pontomyia spp. (Diptera: Chironomidae), Zool. Stud., 38 (1999), 466-473.
[37]	J. E. Brittain, M. Sartori, Ephemeroptera:(Mayflies), in Encyclopedia of Insects, Academic Press, (2009), 33-75.
[38]	B. Dorociaková, I. Ilavská, R. Olach, Existence of solutions for an age-structured insect population model with a larval stage, Electron. J. Qual. Theo., 65 (2017), 1-14.

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