STABILITY OF A POSITIVE POINT OF EQUILIBRIUM OF ONE NONLINEAR SYSTEM WITH AFTEREFFECT AND STOCHASTIC PERTURBATIONS

The aim of the paper is to show one useful way for stability investigation of the positive point of equilibrium of some nonlinear system with aftereect and stochastic perturbations. Obtained results are applied for stability investigation of some mathematical predator-prey models. AMS (MOS) Subject Classication. 34D20.

[1]  Wan-Tong Li,et al.  Existence and global stability of positive periodic solutions of a predator-prey system with delays , 2003, Appl. Math. Comput..

[2]  A. V. Skorohod,et al.  The theory of stochastic processes , 1974 .

[3]  V. B. Kolmanovskii,et al.  Construction of lyapunov functionals for stochastic hereditary systems: a survey of some recent results , 2002 .

[4]  V. B. Kolmanovskii,et al.  About one application of the general method of Lyapunov functionals construction , 2003 .

[5]  Shigui Ruan,et al.  Global Analysis in a Predator-Prey System with Nonmonotonic Functional Response , 2001, SIAM J. Appl. Math..

[6]  Yasuhiro Takeuchi,et al.  Global stability of an SIR epidemic model with time delays , 1995, Journal of mathematical biology.

[7]  Ke Wang,et al.  Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses ✩ , 2003 .

[8]  M. Carletti,et al.  On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. , 2002, Mathematical biosciences.

[9]  Wan-Tong Li,et al.  Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response , 2004 .

[10]  Alan Hastings,et al.  Age-dependent predation is not a simple process. I. Continuous time models , 1983 .

[11]  Fengde Chen.  Periodicity in a ratio-dependent predator-prey system with stage structure for predator , 2005 .

[12]  Yasuhiro Takeuchi,et al.  QUALITATIVE PROPERTIES OF CHEMOSTAT EQUATIONS WITH TIME DELAYS , 1995 .

[13]  John B. Collings,et al.  The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model , 1997 .

[14]  A Hastings,et al.  Delays in recruitment at different trophic levels: Effects on stability , 1984, Journal of mathematical biology.

[15]  Wan-Tong Li,et al.  Periodic solutions of delayed ratio-dependent predator–prey models with monotonic or nonmonotonic functional responses , 2004 .

[16]  Joydev Chattopadhyay,et al.  Ratio-dependent predator–prey model: effect of environmental fluctuation and stability , 2005 .

[17]  A. Bush,et al.  The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. , 1976, Journal of theoretical biology.

[18]  Lansun Chen,et al.  The stage-structured predator-prey model and optimal harvesting policy. , 2000, Mathematical biosciences.

[19]  Yang Kuang,et al.  Periodic Solutions of Periodic Delay Lotka–Volterra Equations and Systems☆ , 2001 .

[20]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[21]  V. B. Kolmanovskii,et al.  Stability of epidemic model with time delays influenced by stochastic perturbations 1 This paper was , 1998 .

[22]  I. I. Gikhman.  Theory of stochastic processes , 1974 .

[23]  Leonid Shaikhet,et al.  Stability in probability of nonlinear stochastic systems with delay , 1995 .

[24]  Leonid E. Shaikhet,et al.  Some peculiarities of the general method of Lyapunov functionals construction , 2002, Appl. Math. Lett..

[25]  Wan-Tong Li,et al.  Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response , 2004 .

[26]  Y. Kuang,et al.  Global analyses in some delayed ratio-dependent predator-prey systems , 1998 .

[27]  Dongmei Xiao,et al.  Multiple Bifurcations in a Delayed Predator–prey System with Nonmonotonic Functional Response , 2022 .