Modelling and Simulation of Complex Biological SystemsView this Special Issue
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Haokun Qi, Hua Guo, "Dynamics Analysis of a Stochastic Hybrid Logistic Model with Delay and Two-Pulse Perturbations", Complexity, vol. 2020, Article ID 5024830, 24 pages, 2020. https://doi.org/10.1155/2020/5024830
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. 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.
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 [1–8]). The classical logistic equation is expressed bywhere denotes the population size and and 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 [9–16]) can better reflect the actual situation. Based on model (1), we obtain the following stochastic model:where and are non-negative constants. is the time delay. stands for a probability measure on . is the independent standard Brownian motion defined on a complete probability space and is the intensity of the white noise. is the left limit of . is a Poisson random measure with characteristic measure λ on a measurable bounded below subset of with , is a Lévy measure such that .
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  and which is different due to rainfall, nutrition, and other factors [18, 19]. Thus, a series of articles about Markovian switching have been investigated (see [20–28]). We focus on the stochastic logistic model with Markovian switching:where is a continuous-time Markov chain with values in finite state space . We assume that and 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. The 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. Therefore, these phenomena can be described more accurately by the stochastic models with impulsive effects (see [29–34]).
On the other hand, human activities not only have a direct impact on the population but also have an indirect impact. The 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. Therefore, it has become an inevitable trend to consider the influence of environmental toxins on the population (see [35–38]).
Based on the above discussion, we first consider the following stochastic hybrid logistic model with two-pulse perturbations:where and represent the concentration of toxins in organism and in environment at time , respectively. is the decreasing rate of the growth rate associated with the uptake of the toxins, stands for the uptake rate of toxicant in the environment, and are the excretion rate and depuration rate, respectively, is the loss rate of toxicant in the environment, and stands for the toxin input amount at every time. Let , where is the set of positive integers. When , the impulsive effects imply releasing population, while if , the impulsive effects indicate harvesting for population. In this paper, we always suppose that for all .
The rest of the paper is organized as follows. In Section 2, we give some preliminaries. The existence and uniqueness of the global positive solution of the model are given in Section 3. The 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.
Denote the generator of the Markov chain given bywhere if while . When is irreducible, then has a unique stationary distribution which is the solution of
Consider a stochastic differential delay equation (SDDE) with Markovian switching and Lévy noise (see ) as follows:where is the state vector and is the delayed state vector. . The time-varying delay is a Borel measurable function. is the drift coefficient vector, is the diffusion coefficient matrix, and .
For each , let be any twice continuously differentiable function; the operator can be defined bywhere
Then, one has the generalized Itô’s formula:
Hypothesis 1. (locally Lipschitz condition). For any integer , there exists a constant such thatfor those with and any .
Hypothesis 2. (linear growth condition). There is a constant such thatfor any .
Hypothesis 3. For each , there is a constant that depends on such thatwith .
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 notationsIn order to give the proof in this paper, we provide some assumptions. : there exists a constant such that : , and there exists constant such that : let the initial value be positive and (see [31, 39]), which is defined by There exists a probability measure and a constant such that : . : there exist two positive constants and such that : .We give some useful inequality in .(1)(Exponential martingale inequality) Let , , be any positive numbers. Then,(2)(Chebyshev’s inequality) If , , which is the family of -valued random variables with . Then,Next, we consider the following subsystem of system (4):From , we can get the following lemma.
3. Positive and Global Solutions
Theorem 1. For any initial data , system (25) has a unique positive solution with probability one
Proof. Consider the following SDDEs with Markovian switching and without impulses:with initial value . By the theory of SDDEs with Markovian switching and Lévy jump, we refer the reader to . System (26) has a unique global positive solution .
Let with initial value .
Since is continuous on each interval , thenfor , .
And for , we getMoreover,Thus, system (25) has the unique global positive solution .
4. Extinction and Persistence
Theorem 2. When – hold, if , then
Namely, the population of system (25) is extinct.
Proof. Applying Itô’s formula to system (26), we haveIntegrating both sides of (31) from 0 to yieldsowing toThen, we havewhereSince and are local martingales, the quadratic variations areMaking use of the strong law of large numbers for local martingales (see ) yieldsFrom (34), we can get thatThus,Taking superior limit on both sides of (39) and applying the ergodicity of and (37), we obtain
Theorem 3. When – hold, if , thenParticularly, if , then , that is, the population of system (25) is nonpersistent in the mean.
Proof. For , there exists a constant , for all , such thatSubstituting above inequalities into (39), we haveDenote ; then, we have . Taking exponent on both sides of (43) yieldsIntegrating (44) from to , we can show thatTaking logarithm of (45) yieldsTaking superior limit on (46) elicits thatUtilizing L’Hospital’s rule results in
Theorem 4. When – hold, if , then the population of system (25) is weakly persistent a.s.
Remark 1. Through Theorems 2–4, we find an interesting biological phenomenon: when , the population is weakly persistent; when , the population goes to extinction, which means that the persistence and extinction of depend on and the absorption intensity of toxins .
Theorem 5. When – hold, if , then
That is, the population of system (25) is persistent in the mean a.s.
Proof. Applying Itô’s formula to (26) yields