IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i2p273-d1319009.html
   My bibliography  Save this article

The Equilibrium Solutions for a Nonlinear Separable Population Model

Author

Listed:
  • Dragos-Patru Covei

    (The Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st District, 010374 București, Romania
    These authors contributed equally to this work.)

  • Traian A. Pirvu

    (The Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4K1, Canada
    These authors contributed equally to this work.)

  • Catalin Sterbeti

    (The Department of Applied Mathematics, University of Craiova, 13, A.I. Cuza Street, 200585 Craiova, Dolj, Romania
    These authors contributed equally to this work.)

Abstract

The paper investigates a nonlinear model that describes population dynamics with an age structure. The fertility rate, which varies with age, follows a nonconstant pattern. The model exhibits a multiplicative structure for both fertility and mortality rates. Remarkably, this multiplicative structure renders the model separable. In this setting, it is shown that the number of births in unit time can be expressed using a system of nonlinear ordinary differential equations. The asymptotic behavior of solutions to this system has been established for a specific case. This result is significant because it provides a mathematical framework for understanding the dynamics of birth rates in certain settings. Furthermore, this paper explicitly identifies the steady-state solution and the equilibrium solution. As in any research paper, new directions of study remain open.

Suggested Citation

  • Dragos-Patru Covei & Traian A. Pirvu & Catalin Sterbeti, 2024. "The Equilibrium Solutions for a Nonlinear Separable Population Model," Mathematics, MDPI, vol. 12(2), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:273-:d:1319009
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/2/273/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/2/273/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Daniel Dufresne, 2007. "Fitting combinations of exponentials to probability distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 23-48, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    2. Kyng, T. & Konstandatos, O. & Bienek, T., 2016. "Valuation of employee stock options using the exercise multiple approach and life tables," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 17-26.
    3. Zhang, Zhehao, 2018. "Renewal sums under mixtures of exponentials," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 281-301.
    4. Li, Shu & Landriault, David & Lemieux, Christiane, 2015. "A risk model with varying premiums: Its risk management implications," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 38-46.
    5. Okhli, Kheirolah & Jabbari Nooghabi, Mehdi, 2023. "On the three-component mixture of exponential distributions: A Bayesian framework to model data with multiple lower and upper outliers," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 480-500.
    6. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2015. "Geometric stopping of a random walk and its applications to valuing equity-linked death benefits," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 313-325.
    7. Dan Pirjol & Lingjiong Zhu, 2016. "Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options," Papers 1609.07558, arXiv.org.
    8. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    9. Cheung, Eric C.K., 2013. "Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 343-354.
    10. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    11. Zhou, Jiang & Wu, Lan, 2015. "The time of deducting fees for variable annuities under the state-dependent fee structure," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 125-134.
    12. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.
    13. Liang, Xiaoqing & Tsai, Cary Chi-Liang & Lu, Yi, 2016. "Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 150-161.
    14. Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2008. "Methods for estimating the optimal dividend barrier and the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 243-254, February.
    15. Pirjol, Dan & Zhu, Lingjiong, 2016. "Discrete sums of geometric Brownian motions, annuities and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 19-37.
    16. Zhou, Jiang & Wu, Lan, 2015. "Valuing equity-linked death benefits with a threshold expense strategy," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 79-90.
    17. Shang, Zhaoning & Goovaerts, Marc & Dhaene, Jan, 2011. "A recursive approach to mortality-linked derivative pricing," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 240-248, September.
    18. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.
    19. Cheung, Eric C.K. & Liu, Haibo & Willmot, Gordon E., 2018. "Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 358-377.
    20. Hansjörg Albrecher & Dina Finger & Pierre-Olivier Goffard, 2022. "Blockchain mining in pools: Analyzing the trade-off between profitability and ruin," Working Papers hal-03336851, HAL.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:273-:d:1319009. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.