Survival and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment

Periodic Solution of a Neutral Delay Leslie Predator-Prey Model and the Effect of Random Perturbation on the Smith Growth Model

“…In [31], Markov semigroup was used to study a stochastic ecological model of plants with infectious diseases. Authors in [32][33][34] focused on the long-term dynamic behavior of ecosystems affected by environmental noise. In particular, conditions for the existence of stationary distribution were obtained.…”

“…(t))e w t− τ 1 (t)() + R(t)e w(t) dt ≤ 2rT (33). On the basis of the mean value theorem of integral calculus, we know that there exists ξ ∈ [0, T], such that (A(ξ) − μ ′ (ξ))e w ξ− τ 1 (ξ)( ) + R(ξ)e w(ξ) ≤ 2r.…”

Conditions for prosperity and depression of a stochastic R&D model under regime switching

“…Capital accumulation may have optimistic asymptotic behavior when stock prices rise, and capital accumulation may have another asymptotic behavior if stock prices decrease; additionally, the company may enter another development level under the state of talent introduction. Meanwhile, a continuous time Markov chain, namely, color noise, can delineate the system switch from one environmental regime to another in a population system [11][12][13][14][15][16][17]. For instance, Liu et al [13] showed that an ergodic Markov chain could accurately describe the stochastic phenomena of a population system in practice.…”

Asymptotic Analysis of Impulsive Dispersal Predator-Prey Systems with Markov Switching on Finite-State Space

“…In view of the environmental pollution, we consider a dispersal predator-prey system with impulsive toxicant input [13][14][15][16][17], which can be described as follows:…”