2014, 11(5): 1167-1174. doi: 10.3934/mbe.2014.11.1167

Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population

1. 

Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str., 163-a, Donetsk, 83015

Received  February 2014 Revised  March 2014 Published  June 2014

The known nonlinear mathematical model of the Glassy-winged Sharpshooter is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the positive equilibrium point. A necessary and sufficient condition for asymptotic mean square stability of the equilibrium point of the linear part of the considered stochastic differential equation is obtained. This condition is at the same time a sufficient one for stability in probability of the equilibrium point of the initial nonlinear equation. Numerical calculations and figures illustrate the obtained results.
Citation: Leonid Shaikhet. Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1167-1174. doi: 10.3934/mbe.2014.11.1167
References:
[1]

M. Bandyopadhyay and J. Chattopadhyay, Ratio dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022.

[2]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation (Special Issue "Delay Systems"), 45 (1998), 269-277. doi: 10.1016/S0378-4754(97)00106-7.

[3]

N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dynamics in Nature and Society, 2007 (2007), 25 pp. doi: 10.1155/2007/92959.

[4]

M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117-131. doi: 10.1016/S0025-5564(01)00089-X.

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I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.

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M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228. doi: 10.1016/j.apm.2011.11.087.

[7]

B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virus-tumor-immune system interaction: Deterministic and stochastic analysis, Stochastic Analysis and Applications, 27 (2009), 409-429. doi: 10.1080/07362990802679067.

[8]

R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach, Mathematical Biosciences, 196 (2005), 65-81. doi: 10.1016/j.mbs.2005.04.001.

[9]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, Dordrecht, Heidelberg, New York, 2011. doi: 10.1007/978-0-85729-685-6.

[10]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Dordrecht, Heidelberg, New York, London, 2013. doi: 10.1007/978-3-319-00101-2.

[11]

J. Yoon, V. Hrynkiv, L. Morano A. Nguyen, S. Wilder and F. Mitchell, Mathematical modeling of Glassy-winged sharpshooter population, Mathematical Biosciences and Engineering, 11 (2014), 667-677. doi: 10.3934/mbe.2014.11.667.

show all references

References:
[1]

M. Bandyopadhyay and J. Chattopadhyay, Ratio dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022.

[2]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation (Special Issue "Delay Systems"), 45 (1998), 269-277. doi: 10.1016/S0378-4754(97)00106-7.

[3]

N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dynamics in Nature and Society, 2007 (2007), 25 pp. doi: 10.1155/2007/92959.

[4]

M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117-131. doi: 10.1016/S0025-5564(01)00089-X.

[5]

I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.

[6]

M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228. doi: 10.1016/j.apm.2011.11.087.

[7]

B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virus-tumor-immune system interaction: Deterministic and stochastic analysis, Stochastic Analysis and Applications, 27 (2009), 409-429. doi: 10.1080/07362990802679067.

[8]

R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach, Mathematical Biosciences, 196 (2005), 65-81. doi: 10.1016/j.mbs.2005.04.001.

[9]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, Dordrecht, Heidelberg, New York, 2011. doi: 10.1007/978-0-85729-685-6.

[10]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer, Dordrecht, Heidelberg, New York, London, 2013. doi: 10.1007/978-3-319-00101-2.

[11]

J. Yoon, V. Hrynkiv, L. Morano A. Nguyen, S. Wilder and F. Mitchell, Mathematical modeling of Glassy-winged sharpshooter population, Mathematical Biosciences and Engineering, 11 (2014), 667-677. doi: 10.3934/mbe.2014.11.667.

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