Abstract
In this paper, we propose and discuss a stochastic logistic model with delay, Markovian switching, Lévy jump, and two-pulse perturbations. First, sufficient criteria for extinction, nonpersistence in the mean, weak persistence, persistence in the mean, and stochastic permanence of the solution are gained. Then, we investigate the lower (upper) growth rate of the solutions. At last, we make use of Matlab to illustrate the main results and give an explanation of biological implications: the large stochastic disturbances are disadvantageous for the persistence of the population; excessive impulsive harvesting or toxin input can lead to extinction of the population.
Highlights
It is universally known that the logistic model is one of the most significant and classical models in mathematical biology
Many scholars have studied it and achieved fruitful results. e classical logistic equation is expressed by dX(t) X(t)r − a1X(t)dt, (1)
We focus on the stochastic logistic model with Markovian switching: dX(t) X(t)r(ξ(t)) − a1(ξ(t))X(t) + a2(ξ(t))x(t − τ) + a3(ξ(t)) X(t + θ)dς(θ)dt
Summary
It is universally known that the logistic model is one of the most significant and classical models in mathematical biology. We focus on the stochastic logistic model with Markovian switching: dX(t) X(t)r(ξ(t)) − a1(ξ(t))X(t) + a2(ξ(t))x(t − τ) + a3(ξ(t)) X(t + θ)dς(θ)dt (3). Environmental pollution pollutes the atmosphere and produces toxins that can enter into animals and plants, causing unimaginable harm to them; the light ones can make some populations die, and the heavy ones may cause species extinction. These toxins will accumulate in animals and plants. Based on the above discussion, we first consider the following stochastic hybrid logistic model with two-pulse perturbations:. We give some numerical simulations to illustrate our results
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