Dynamics Analysis of a Stochastic Hybrid Logistic Model with Delay and Two-Pulse Perturbations

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. (en, 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.


Introduction
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 (see [1][2][3][4][5][6][7][8]). e classical logistic equation is expressed by where X denotes the population size and r and a 1 stand for the intrinsic growth rate and the intraspecific competition rate, respectively. With the improvement of the understanding of biological mathematical models, some factors have been considered, such as random interference, time delay, and so on. Compared with the classical original model, stochastic models (see [9][10][11][12][13][14][15][16]) can better reflect the actual situation. Based on model (1), we obtain the following stochastic model: where r and a i (i � 1, 2, 3) are non-negative constants. τ ≥ 0 is the time delay. ς(θ) stands for a probability measure on (− ∞, 0]. B(t) is the independent standard Brownian motion defined on a complete probability space (Ω, F, F t t≥0 , P) and σ ≥ 0 is the intensity of the white noise. X(t − ) is the left limit of X(t). N(dt, dv) is a Poisson random measure with characteristic measure λ on a measurable bounded below subset Y of R\0 with λ(Y ) < ∞, c is a Lévy measure such that N(dt, dv) � N(dt, dv) − λ(dv)dt.
In addition to the white noise and Lévy noise mentioned above, there are other noises in nature, such as telegraph noise, which can be expressed by continuous-time Markov chain that mainly describes the random switching between two or more environment states [17] and which is different due to rainfall, nutrition, and other factors [18,19]. us, a series of articles about Markovian switching have been investigated (see [20][21][22][23][24][25][26][27][28]). We focus on the stochastic logistic model with Markovian switching: dX(t) � X(t) r(ξ(t)) − a 1 (ξ(t))X(t) + a 2 (ξ(t))x(t − τ) + a 3 where ξ(t) is a continuous-time Markov chain with values in finite state space M � 1, 2, . . . , N { }. We assume that B(t) and ξ(t) are independent.
As we all know human activities will have a significant impact on the population system, we must pay attention to the growing influence of human beings on population systems. e main manifestation of human activities is the regular harvesting of species or the regular stocking for the protection of endangered species, which cannot be considered continuously. erefore, these phenomena can be described more accurately by the stochastic models with impulsive effects (see [29][30][31][32][33][34]).
On the other hand, human activities not only have a direct impact on the population but also have an indirect impact. e toxin produced by environmental pollution has an indirect impact on the species. Environmental pollution caused by human activities has become an important issue that the world has to consider. Environmental pollution not only pollutes the atmosphere but also 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. And these toxins will also accumulate in animals and plants. People transfer toxins in their bodies by eating the animals and plants, which can cause harm to human health. erefore, it has become an inevitable trend to consider the influence of environmental toxins on the population (see [35][36][37][38]).
Based on the above discussion, we first consider the following stochastic hybrid logistic model with two-pulse perturbations: where C 0 (t) and C e (t) represent the concentration of toxins in organism and in environment at time t, respectively. α ≥ 0 is the decreasing rate of the growth rate associated with the uptake of the toxins, k > 0 stands for the uptake rate of toxicant in the environment, g > 0 and m > 0 are the excretion rate and depuration rate, respectively, h > 0 is the loss rate of toxicant in the environment, and u stands for the toxin input amount at every time. Let 0 < t 1 < t 2 < · · · , lim n⟶+∞ t n � +∞, n ∈ Z + , where Z + is the set of positive integers. When δ n > 0, the impulsive effects imply releasing population, while if δ n < 0, the impulsive effects indicate harvesting for population. In this paper, we always suppose that 1 + δ n > 0 for all n ∈ Z + . e rest of the paper is organized as follows. In Section 2, we give some preliminaries. e existence and uniqueness of the global positive solution of the model are given in Section 3. e sufficient conditions for the stochastic permanence and extinction are studied in Section 4. Some asymptotic properties of the solution are proved in Section 5. Finally, we give some numerical simulations to illustrate our results.
Hypothesis 2 (linear growth condition). ere is a constant L > 0 such that for any (x, y, t, k) ∈ R n × R n × R + × M.
Here, Hypotheses 1-3 are the conservative conditions to check the existence and uniqueness of the global solution of (7). In this paper, Hypotheses 1-3 are always satisfied.
For simplicity, denote some notations In order to give the proof in this paper, we provide some assumptions.
A.1: there exists a constant K 1 > 0 such that A.2: 1 + c((ξ(t)), v) > 0, and there exists constant A.3: let the initial value X � β be positive and β ∈ C g (see [31,39]), which is defined by ere exists a probability measure ρ and a constant r > 0 such that A. 6: We give some useful inequality in [40].

Positive and Global Solutions
Proof. Consider the following SDDEs with Markovian switching and without impulses: with initial value (Y(0), ξ(0)). By the theory of SDDEs with Markovian switching and Lévy jump, we refer the reader to [12]. System (26) has a unique global positive solution (Y(t), ξ(t)).

Extinction and Persistence
Namely, the population X(t) of system (25) is extinct.
Proof. Applying Itô's formula to system (26), we have Integrating both sides of (31) from 0 to t yields Complexity en, we have Since M 1 (t) and M 2 (t) are local martingales, the quadratic variations are Making use of the strong law of large numbers for local martingales (see [41]) yields From (34), we can get that 0<t n < t us, Taking superior limit on both sides of (39) and applying the ergodicity of ξ(·) and (37), we obtain Particularly, if η * � αC 0 , then lim t⟶+∞ (1/t) t 0 X(s)ds � 0, that is, the population X(t) of system (25) is nonpersistent in the mean.
Taking exponent on both sides of (43) yields Integrating (44) from T to t, we can show that Taking logarithm of (45) yields Taking superior limit on (46) elicits that Utilizing L'Hospital's rule results in For ∀ω ∈ L, we have lim t⟶+∞ X(t, ω) � 0. As a result, which is a contradiction.

Theorem 5. When
at is, the population X(t) of system (25) is persistent in the mean a.s.

(77)
On the one hand, Taking expectations on both sides of (79) yields is leads to en, for ∀ε > 0, let b 2 � (MG/ε), and we have at is, From (71) and (83), X(t) of system (25) is stochastically permanent. A.1-A.3, A.5, and A.6 hold and any solution X(t) of system (25) have the property that
en, we obtain η * � 0.8771 > 0.4208 � αC 0 . According to eorem 4, we know that X(t) is weakly persistent (see Figure 1(a)). It can be seen from Figures 1(b) and 1(c) that the toxins in the organism C 0 (t) and the toxins in the environment C e (t) have a periodic solution.
6.1. Effect of the White Noise σ. Choosing the same parameters as in Example 1 but σ: by comparing Figure 1          Complexity with the intensity of white noise. By comparing Figures 2(a)-2(c), the population will go extinct with increasing intensity of white noise. When the intensity of white noise is less than a certain level, the population still be persistent, while the white noise with large intensity may cause population extinction.

Effect of the Lévy Noise c.
Choosing the same parameters as in Example 1 but σ(1) � 0.4, σ(2) � 0.4, and c: from Figure 3, it is concluded that the Lévy noise has a large impact on the persistence of population. It is shown that the number of the population will decrease when the intensity of the Lévy noise increases. When the intensity of the Lévy noise became larger, the population became extinct. erefore, Lévy noise will not only reduce the number of species but also lead to population extinction.

Effect of Telegraph Noise.
Choosing the same parameters as in Example 1 but δ n � 0: from Figure 4, we find that all parameter values are the same except the Markov chain. Since telegraph noise is described by a Markov chain, choosing a different Markov chain will produce different results.

Effect of the Exogenous Total Toxicant Input u.
Choosing the same parameters as in Example 1 but u: from Figure 5, we find that the population can still be persistent when the total input of exogenous toxicant is small (see Figures 5(a) and 5(b)), while the population will go extinct when the total input of exogenous toxicant is strong (see Figure 5(c)).
6.5. Effect of Impulse δ n . Choosing the same parameters as in Example 1 but δ n : in Figure 6, we describe the impact of pulse harvest on the population. By comparing Figures 6(a)-6(c), we find that with the increase of the harvest, the population will gradually decrease until it becomes completely extinct.

Conclusion
In this paper, we explore the dynamics of a stochastic delay hybrid logistic model with two-pulse perturbations. First, by using Itô's formula, exponential martingale inequality, Chebyshev's inequality, and other mathematical skills, we establish some sufficient conditions for extinction, nonpersistence in the mean, weak persistence, persistence in the mean, and stochastic permanence. en, the asymptotic properties of the lower growth rate and the upper growth rate of the solution are estimated. Now, we give the key results as follows: (I) (1) If η * < αC 0 , then the population X(t) is extinct.
(4) If η * > � αC 0 , then the population X(t) is persistent in the mean. (5) If N k�1 π k b(k) > 0, then the population X(t) is stochastically permanent. (II) e solution X(t) obeys From our results and analysis, we can obtain the following conclusions. (1) Both white noise and Lévy noise tend to have negative effects on the persistence of population. However, the Lévy jump may have a greater influence than white noise on the persistence of population. (2) If the choice of telegraph noise is different, the persistence of the population will produce different results. (3) e total input of exogenous toxicant has a great disadvantaged influence on persistence of the population. With the increase of the total input of exogenous toxicant, the number of population will decrease, which enlightens us to reduce pollutant emissions to protect the ecological system. (4) e pulse release of the population is beneficial to the growth of the population; the pulse harvest of the population is beneficial to the growth of the population under reasonable conditions, but once overharvested, it will cause the population to become extinct. ese give us significant enlightenment: (1) when fishermen are fishing, they must maintain reasonable fishing to avoid overfishing that can lead to population extinction; (2) the discharge of factory sewage, exhaust gas, domestic sewage, etc., should be strictly controlled, and we should do our best to reduce the discharge of polluted toxins; and (3) humans should reduce interference with populations.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.