A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis

"This is the peer reviewed version of the following article: Cortés, J-C, Navarro-Quiles, A, Romero, J-V, Roselló, M-D. A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Comp and Math Methods. 2019; 1:e1013, which has been published in final form at https://doi.org/10.1002/cmm4.1013. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] In this paper a randomized version of the Beverton-Holt type discrete model is proposed. Its solution stochastic process and the random steady state are determined. Its first probability density function and second probability density function are obtained by means of the random variable transformation method, providing a full probabilistic description of the solution. Finally, several numerical examples are shown.This work has been partially supported by the Ministerio de Economía, Industria y Competitividad under grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2019). A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Computational and Mathematical Methods. 1(1):1-12. https://doi.org/10.1002/cmm4.1013S11211Kwasnicki, W. (2013). Logistic growth of the global economy and competitiveness of nations. Technological Forecasting and Social Change, 80(1), 50-76. doi:10.1016/j.techfore.2012.07.007De la Sen, M. (2008). The generalized Beverton–Holt equation and the control of populations. Applied Mathematical Modelling, 32(11), 2312-2328. doi:10.1016/j.apm.2007.09.007Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty. Journal of Computational and Applied Mathematics, 337, 190-208. doi:10.1016/j.cam.2018.01.015Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.01

Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty

[EN] This paper is devoted to construct approximations of the probability density function of the non-autonomous first-order homogeneous linear random diff erential equation, where the initial condition and the diff usion coe fficient are assumed to be a random variable and a stochastic process, respectively. We combine Random Variable Transformation technique and Karhunen-Loève expansion to construct reliable approximations under general conditions. Several numerical examples illustrate our theoretical findings.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2018). Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty. Journal of Computational and Applied Mathematics. 337:190-208. https://doi.org/10.1016/j.cam.2018.01.015S19020833

Improving the approximation of the probability density function of random nonautonomous logistic-type differential equations

[EN] In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P-'(t,omega)=A(t,omega)(1-P(t,omega))P(t,omega), t is an element of[t(0),T], P(t(0),omega)=P-0(omega), where omega is any outcome in the sample space omega. In the recent contribution [Cortes, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121-138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P-0 to approximate the density function f(1)(p,t) of P(t): A(t) is expressed as a Karhunen-Loeve expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P-0, and the density of P-0, fP0, is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L-2([t(0),T]x omega), so that we include other expansions such as random power series. We only require absolute continuity for P-0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For fP0, only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence {f1N(p,t)}N=1 infinity of density functions in terms of expectations that tends to f(1)(p,t) pointwise. Numerical examples illustrate our theoretical results.Secretaria de Estado de Investigacion, Desarrollo e Innovacion, Grant/Award Number: MTM2017-89664-P; Universitat Politecnica de Valencia, Grant/Award Number: Programa de Ayudas de Investigacion y DesarrolloCalatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic-type differential equations. Mathematical Methods in the Applied Sciences. 42(18):7259-7267. https://doi.org/10.1002/mma.5834S725972674218Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2019). Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 72, 121-138. doi:10.1016/j.cnsns.2018.12.013Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Bharucha-Reid, A. T. (1964). On the theory of random equations. Proceedings of Symposia in Applied Mathematics, 40-69. doi:10.1090/psapm/016/0189071Neckel, T., & Rupp, F. (2013). Random Differential Equations in Scientific Computing. doi:10.2478/9788376560267Murray, J. D. (Ed.). (2002). Mathematical Biology. Interdisciplinary Applied Mathematics. doi:10.1007/b98868Licea, J. A., Villafuerte, L., & Chen-Charpentier, B. M. (2013). Analytic and numerical solutions of a Riccati differential equation with random coefficients. Journal of Computational and Applied Mathematics, 239, 208-219. doi:10.1016/j.cam.2012.09.040Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Dorini, F. A., Bobko, N., & Dorini, L. B. (2016). A note on the logistic equation subject to uncertainties in parameters. Computational and Applied Mathematics, 37(2), 1496-1506. doi:10.1007/s40314-016-0409-6Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty. Journal of Computational and Applied Mathematics, 337, 190-208. doi:10.1016/j.cam.2018.01.015Burgos, C., Calatayud, J., Cortés, J.-C., & Navarro-Quiles, A. (2018). A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique. Mathematical Methods in the Applied Sciences, 41(18), 9037-9047. doi:10.1002/mma.4881Lord, G. J., Powell, C. E., & Shardlow, T. (2009). An Introduction to Computational Stochastic PDEs. doi:10.1017/cbo9781139017329CalatayudJ CortésJC JornetM. On the approximation of the probability density function of the randomized non‐autonomous complete linear differential equation. arXiv preprint arXiv:1802.04188;2018.Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Vaart, A. W. van der. (1998). Asymptotic Statistics. doi:10.1017/cbo9780511802256Wolfram Research Inc. Mathematica. Version 11.2 Champaign Illinois;2017