ANALYSIS OF RANDOM DISCRETE TIME LOGISTIC MODEL

In this study, the behavior of the logistic difference model is investigated under random conditions using discrete probability distributions. The logistic difference model consists of parameters that depend on the population models to be used. For the study of random difference equation population models, the parameters are treated as random variables which constitutes the basis of the study. Random models were created using Uniform, Bernouilli, Binom, Negative Binomial (or Pascal), Geometric, Hypergeometric, Poisson distributions and their numerical characteristics are obtained through their simulations. Then, the results showing random numerical characteristics such as expected value, variance, standard deviation, coefficient of variation and confidence intervals were obtained with MATLAB package program. Analysis of random logistic difference model is given with the help of graphics and tables.

Keywords:

Logistic difference equation, probability distributions expected value, variance,

PDF

___

[1] Akdeniz, F., Olasılık ve İstatistik, Akademiyen Kitabevi, Ankara, Türkiye, 2014.
[2] Feller W., 1968. An Introduction to Probability Theory and Its Applications, volume 1, 3rd edition. New York: John Wiley & Sons.
[3] Appleby, J.A.D., Kelly, C.,(2006), Oscillation of solutions of a nonuniform discretisation of linear stochastic differential equations with vanishing delay, Dy Contin Discret Impuls Syst A 13B:535-550.
[4] Appleby, J.A.D., Berkolaiko, G., Rodkina, A., (2009a),Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise ,Stochastic 81(2):99-127.
[5] Appleby,J.A.D., Kelly, C., Mao,X., Rodkina, A.,(2010), On the local dynamic of polynomial difference equations with fading stochastic perturbations,Dy Contin Discret Impuls Syst A 17(3):401-430.
[6] Appleby, J.A.D.,Rodkina A.,Schurz,H.,(2010),Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise,J Differ Equ Appl 6(7):807-830.
[7] Fakharzadeh, J., Hesamaeddini, E. and Soleimanivareki, M., 2015. Multi-step Stochastic Differential Transformation Method for solving Some Class of Random Differential Equations. Applied Mathematics in Engineering, Management and Technology, 3(3), 115–123.
[8] Kloeden, P., E. ve Platen, E., Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, Berlin, 1995.
[9] Merdan, M., Bekiryazici, Z., Kesemen, T. ve Khaniyev, T., Comparison of Stochastic and Random Models for Bacterial Resistance, Advances in Difference Equations, 2017, 133 (2017).
[10] Calbo, G., Cortés, J.C. and Jódar, L., 2010. Mean Square Power Series Solution Of Random Linear Differential Equations. Computers And Mathematics With Applications, 59, 559-572.
[11] Cortés, J.C., Jódar, L. and Villafuerte, L., 2007. Mean Square Numerical Solution Of Random Differential Equations: Facts And Possibilities. Computers And Mathematics With Applications, 53, 1098-1106.
[12] Cortés, J.C., Jódar, L. and Villafuerte, L., 2007. Numerical Solution Of Random Differential Equations: A Mean Square Approach. Mathematical And Computer Modelling, 45, 757-765.
[13] Cortes, J.C., Jodar, L. and Villafuerte, L., 2009. Random Linear-Quadratic Mathematical Models: Computing Explicit Solutions and Applications. Mathematics and Computers in Simulation, 79, 2076-2090.
[14] Cortés, J.C., Jódar, L., Villafuerte, L. and Company, R., 2011. Numerical Solution Of Random Differential Models. Mathematical And Computer Modelling, 54, 1846-1851.
[15] Cortés, J.C., Jódar, L., Villanueva, R.-J. and Villafuerte, L., 2010. Mean Square Convergent Numerical Methods For Nonlinear Random Differential Equations. Lecture Notes In Computer Science.
[16] Golmankhaneh, A.K., Porghoveh, N.A. and Baleanu, D., 2013. Mean Square Solutions of Second-Order Random Differential Equations by Using Homotopy Analysis Method. Romanian Reports in Physics, 65(2), 350–362.
[17] Khalaf, S.L., 2011. Mean Square Solutions of Second-Order Random Differential Equations by Using Homotopy Perturbation Method. International Mathematical Forum, 6, 2361-2370.
[18] Merdan, M., Anac, H., Bekiryazici, Z. and Kesemen, T.(2019). Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, Gumushane Universitesi Fen Bilimleri Enstitusu Dergisi, 9(1), 108-118.
[19] Soong, T. T., Random Differential Equations in Science and Engineering, Academic Press, New York, 1973.
[20] Appleby, J.A.D., Berkolaiko, G.,Rodkina, A.,(2008),On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J Differ Equ Appl 14(9):923-951.
[21] Elaydi S., 2005. An Introduction to Difference Equations third edition, Springer New York USA
[22] Hırsch, M. ve Smale. S. (Academic Press, New York, 1974). Differential Equations, Dynamical Systems, and Linear Algebra.
[23] Kelly W., 2003. “Theory of diference equations numerical methods and applications, 2nd ed., by V. Lakshmikantham and Donato Trigiante,Marcel Dekker, Inc., New York, 2002,”Bulletin (New Series) of the american mathematical society, 40(2): 259262.
[24] Kelly,C., Rodkina,A.,(2009) Constrained stability and instability of polynomial difference equations with state-dependent noise,Discret Contin Dyn Syst B11(4):913-933.
[25] Kelly G. W., Peterson, C. A., 2001. Difference Equations: An Introduction with Applications, Academic Press, San Diego
[26] Kulenovic, M. R. S., Kalabusic, S., 2000. Projects For The History of Difference Equations and Recursive Relations, University of Rhode Island,http://hypatia.math.uri.edu/~kulenm/diffequaturi/m381f00fp/m381f00mp.htm.
[27] Kulenovic M. R. S and Ladas G. (2001). Dynamics of Second Order Rational Difference Equations, With Open Problems and Conjecture. Chapman & Hall,Boca Raton, Fla, USA.
[28] Bartlett, M.S., Stochastic population models in ecology and epidemiology, London: Methuen, 1960.
[29] Brännström, A., Sumpter, D.J.T., (, 2006.).Stochastic Analogues of Deterministic Single-Species Population Models, Theor. Popul. Biol., 69: 442-451.
[30] Geritz, S., www.wiki.helsinki.fi/download/attachments/63734499/SPM+ Intro.pdf.
[31] Juliano, S.A., Population Dynamics, J Am Mosq Control Assoc., 23 (2): 265-275, 2007.
[32] Verhults, F. (1990). Nonlinear Differential Equations and Dynamical Systems, Springer-Werlag, New York.
[33] http://en.wikipedia.org/wiki/Population_dynamics
[34] http://www.math.epn.edu.ec/emalca2010/files/material/Biomatematica/ CUP101310.pdf