Abstract
This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall′s inequality, and Young′s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of V‐function technique, M‐matrix method, and Chebyshev′s inequality. Finally, an example is given to illustrate the main results.
Highlights
The delay differential equation dx t x t a − bx t cx t − τ has been used to model the population growth of certain species, known as the delay logistic equation
The stochastic delay logistic 1.3 in random environments can be described by the following stochastic model with regime switching: dx txtart − brtxtcrtxt − τ dt σ r t dw t
The arguments above can give an alternative result which we describe as a theorem as below
Summary
1.1 dt has been used to model the population growth of certain species, known as the delay logistic equation. The stochastic delay logistic 1.3 in random environments can be described by the following stochastic model with regime switching: dx txtart − brtxtcrtxt − τ dt σ r t dw t. The study of stochastic permanence and extinction of a logistic model under regime switching was considered in 18 , a new singlespecies model disturbed by both white noise and colored noise in a polluted environment was developed and analyzed in 22 , a general stochastic logistic system under regime switching was proposed and was treated in 23. Since 1.4 describes a stochastic population dynamics, it is critical to find out whether or not the solutions will remain positive or never become negative, will not explode to infinity in a finite time, will be bounded, will be stochastically permanent, will become extinct, or have good asymptotic properties. An example is given to illustrate our main results
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