Asymptotic Behavior of Discrete Time Fuzzy Single Species Model

<jats:p>This work is concerned with the qualitative behavior of discrete time single species model with fuzzy environment <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="normal">exp</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>A</mml:mi><mml:mo>-</mml:mo><mml:mi>B</mml:mi><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo>,</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> denotes the number of individuals of generation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math> is the intrinsic growth rate, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:math> is interpreted as the carrying capacity of the surrounding environment. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mfenced open="{" close="}" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math> is a sequence of positive fuzzy number. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:math> and the initial value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> are positive fuzzy numbers. Applying difference of Hukuhara (H-difference), the existence, uniqueness of the positive solution, and global asymptotic behavior of all positive solution with the model are obtained. Moreover a numerical example is presented to show the effectiveness of theoretic results obtained.</jats:p>

[1]  R. Bassanezi,et al.  Fuzzy modelling in population dynamics , 2000 .

[2]  Arie Tzvieli.  Possibility theory: An approach to computerized processing of uncertainty , 1990, J. Am. Soc. Inf. Sci..

[3]  Q. Din.  Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation , 2015 .

[4]  Garyfalos Papaschinopoulos,et al.  On the fuzzy difference equation xn+1 = A+B/xn , 2002, Soft Comput..

[5]  Alireza Khastan,et al.  On the new solutions to the fuzzy difference equation xn+1=A+Bxn , 2019, Fuzzy Sets Syst..

[6]  W. Congxin,et al.  Embedding problem of fuzzy number space: part II , 1992 .

[7]  Rina Suzuki,et al.  Classification of global behavior of a system of rational difference equations , 2018, Appl. Math. Lett..

[8]  Eduardo Liz,et al.  Chaos in Discrete Structured Population Models , 2012, SIAM J. Appl. Dyn. Syst..

[9]  Ö. Gümüs.  Global and local stability analysis in a nonlinear discrete-time population model , 2014 .

[10]  Garyfalos Papaschinopoulos,et al.  On the fuzzy difference equations of finance , 2008, Fuzzy Sets Syst..

[11]  A. de Korvin,et al.  Analysis by fuzzy difference equations of a model of CO2 level in the blood , 1999 .

[12]  Fubiao Lin,et al.  On Dynamical Behavior of Discrete Time Fuzzy Logistic Equation , 2018, Discrete Dynamics in Nature and Society.

[13]  Ioan Rasa,et al.  Approximate solutions of the logistic equation and Ulam stability , 2018, Appl. Math. Lett..

[14]  Lihui Yang,et al.  On first order fuzzy Ricatti difference equation , 2014, Inf. Sci..

[15]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[16]  Qamar Din,et al.  Qualitative behavior of a second-order fuzzy difference equation , 2018, J. Intell. Fuzzy Syst..

[17]  W. Ricker.  Stock and Recruitment , 1954 .

[18]  Ravi P. Agarwal,et al.  Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (V. L. Kocic and G. Ladas) , 1995, SIAM Rev..

[19]  Apu Kumar Saha,et al.  Solution of second order linear fuzzy difference equation by Lagrange's multiplier method , 2016 .

[20]  Alireza Khastan,et al.  New solutions for first order linear fuzzy difference equations , 2017, J. Comput. Appl. Math..

[22]  Dynamics of a system of rational third-order difference equation , 2012 .

[23]  M. Hassell,et al.  Discrete time models for two-species competition. , 1976, Theoretical population biology.

[25]  C. J. Schinas,et al.  On an Exponential-Type Fuzzy Difference Equation , 2010 .

[26]  R. Memarbashi,et al.  Fuzzy difference equations of Volterra type , 2013 .

[27]  Zongchen Li,et al.  Chaos in a Discrete Delay Population Model , 2012 .

[28]  M. el-Dessoky,et al.  Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation , 2018, Discrete Dynamics in Nature and Society.

[29]  Rui Li,et al.  On the dynamics of a five-order fuzzy difference equation , 2017 .

[30]  Explicit Solutions and Bifurcations for a System of Rational Difference Equations , 2019, Mathematics.

[31]  A. Khastan.  FUZZY LOGISTIC DIFFERENCE EQUATION , 2017 .