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 ofglassy-winged sharpshooter population[J]. Mathematical Biosciences and Engineering, 2014, 11(5): 1167-1174. doi: 10.3934/mbe.2014.11.1167
Abstract
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.
References
[1]
|
Nonlinearity, 18 (2005), 913-936.
|
[2]
|
Mathematics and Computers in Simulation (Special Issue "Delay Systems"), 45 (1998), 269-277.
|
[3]
|
Discrete Dynamics in Nature and Society, 2007 (2007), 25 pp.
|
[4]
|
Mathematical Biosciences, 175 (2002), 117-131.
|
[5]
|
Springer-Verlag, Berlin, 1972.
|
[6]
|
Applied Mathematical Modelling, 36 (2012), 5214-5228.
|
[7]
|
Stochastic Analysis and Applications, 27 (2009), 409-429.
|
[8]
|
Mathematical Biosciences, 196 (2005), 65-81.
|
[9]
|
Springer, London, Dordrecht, Heidelberg, New York, 2011.
|
[10]
|
Springer, Dordrecht, Heidelberg, New York, London, 2013.
|
[11]
|
Mathematical Biosciences and Engineering, 11 (2014), 667-677.
|