Abstract
In this article, a nonautonomous stochastic modified Bazykin model is introduced. The positive solution is proved to be unique and global for any initial data via stochastic comparison theorem and Itô formula. Stochastic ultimate boundedness and stochastic permanence are also considered. Finally, some simulations are performed to verify the validity of results.
1. Introduction
The study on population systems is one of the hot topic in applied mathematics and ecology. Mathematical modelling and qualitative analysis have received much concern, such as the famous Lotka–Volterra model [1]. For more than a decade, predator-prey models with ration-dependent response function have been discussed extensively [2–4]. Today, most predator-prey models assume that the functional response function is a function of prey population density [5–7].
Considering the classical assumptions is not always appropriate which assume that predators encounter prey at random, and the functional response only depends on prey abundance. In 1998, Alexeev and Bazykin first proposed the following Bazykin’s model [2].where and denote the number (or density) of prey species and predator species. The parameters , and m are greater than zero with their usual ecological meanings (for details refer [2]). Many research studies about system (1) have been received, more about its generalization [8–11].
Taking the ratio-dependent of predator into account (more predators are good for hunting in serious situations), M. Haque remodeled a modified Bazykin model according to system (1)in which the analytical behavior of the system near origin is observed [1].
In the real environment, there is the interference of random factors everywhere, such as emergencies, weather changes, and disease spread, and then the actual observed data have some deviation from the results of the deterministic model. Therefore, the influence of the real random factors on the populations should be considered, and the stochastic model can describe the real situation better [12–16].
Considering that the parameters a and are affected by white noise, the rates a and can been replaced by , and , respectively, where and are Wiener process (or Brownian motion), and they are independent. and indicate the intensities of white noise. The following autonomous stochastic model was built by Lv et al. [13]:and has been investigated, and the system (3) is persistent in mean or die out under some conditions [13].
In this paper, we will consider the nonautonomous stochastic modified Bazykin modelwhere , and are assumed to be continuous and bounded.
It is important for population systems to keep permanence, which means that the species will not extinct in the future. Some definitions and conditions which guarantee that system (4) is stochastic permanent and boundedness are given as follows.
Without specification, is an initial data by default in the full text, and is the solution of system (4) with an initial .
Definition 1 (see [15]). Assume any , then and , when following is hold:so solution for system (4) is called stochastically permanent.
Definition 2 (see [15]). Assume any , then , when following is holdso solution for system (4) is called stochastically ultimately bounded.
In Section 2, main results about the existence and uniqueness of global positive solution and the properties of asymptotic bounded were introduced and then shows us that the solution of system (4) is stochastically ultimate bounded and stochastically permanent. In Section 3, we briefly give conclusions and remark. In Section 4, numerical simulations are carried out to illustrate our results. In the end, some lemmas and proofs of main results are given.
2. Main Results
2.1. Existence and Globality
Theorem 1. Let be initial value, then the positive solution to system (4) exists globally a.s.
Corollary 1. The solution of system (4) must stay in a.s., and thatwhere upper-lower solutions are defined later in (20)–(22) and (51).
2.2. Dynamical Behavior of System (4)
In the following, we denote
The functions above are bounded and continuous on .
Theorem 2. If and , the solution to system (4) satisfiesFurthermore,when .
Theorem 3. The solutions of system (4) are stochastically ultimately bounded when , and .
Theorem 4. The solutions of system (4) are stochastically permanent, if , and .
Theorem 5. The solutions of system (4) are stochastically permanent, if , and .
3. Conclusion
Theorem 5 shows that predator species is permanent a.s., if , and . It is clear that prey species is also permanent a.s., i.e., predator species and prey species are both stochastically permanent. Furthermore, under assumption of Theorem 5, system (4) is persistent according to Theorem 7 in [13]. By our Theorem 5, we can conclude that prey species and predator species in equation (4) are both stochastically permanent. Our work is based on the result of Lv et al. [13] and Wu et al. [15]. At the same time, we have achieved some results about the stochastic model with Markovian switching and Lévy jumps [17, 18].
4. Simulations
In this part, some numerical experiments are given to show that our main results as true. In Figure 1, we choose
(a)
(b)
We notice that , and then the solution of system (4) will be stochastically permanent by virtue of Theorem 5.
In Figure 2, we choosewhere , and , but . According to Theorem 4, we know that the solution of system (4) is also stochastically permanent.
(a)
(b)
5. Proofs of Main Results
Lemma 1 (see [16]). Let be a solution of equation (21). If , then
Lemma 2 (see [16]).
Lemma 3. If , then
Lemma 4. If and , for every , there exists ; then,
Lemma 5. If and , then
Proof of Theorem 1 and Corollary 1. The proof process is a standard operation and is omitted here (see [12, 13, 15]).
The following system is obtained by comparisonThey have solutions as follows:Consequently
Proof of Theorem 2. From , and Lemma 1, one obtain that LetThe derivative of along system (4) can be obtained as follows:Through calculating the integral, we haveThen, we calculate the expectationThe following formula is easy to be obtained:Note that inequality , and thenTherefore, by basic theories of ordinary differential equation, we haveAccording to the above equation, the following is obtained:
Proof of Theorem 3. By Theorem 2,Now, for any , let and . Then, the following are obtained through the Chebyshev inequalityTaking , it yieldsThen,
Proof of Lemma 3. Combing (21) with Lemma 2, we haveFrom Corollary 1, it hasTherefore, .
Proof of Lemma 4. By Theorem 2, there exists s.t. . For each , denote ; hence, we haveThen,By Lemma 3, we haveTherefore, for any , let , we havewhich yieldsThis implies
Proof of Theorem 4. According to the arguments in Lemma 4, we know that for any ϵ, existing and the following is true:Then, we getSimilarly, we haveFrom Definition 1, we know the solution of system (4) is stochastically permanent.
Proof of Lemma 5. By Corollary 1, it is easy to haveSo,By Lemma 2 and Hölder inequality, one hasCombiing inequality with (21), one can getMeanwhile,Furthermore,Substituting (53) into (52), we haveTherefore, by (51) and (54), one can getFinally, we obtain the required assertion through substituting (55) into (51).
In summary, Theorem 5 has been obtained.
Remark 1. It is clear that stochastic differential equation which is stochastically permanent implies that its solutions are stochastically ultimately bounded. Lemma 4 shows that the prey species will be permanent if and , or it will be extinct. Theorem 4 shows that if and , system (4) is permanent in probability 1, i.e., the number (or density) is bounded a.s.
Data Availability
The data used to support this study are provided in our paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of China (no. 11771001), Provincial Natural Science Research Project of Anhui Colleges (nos. KJ2019A0672 and KJ2019A0666), Program for Excellent Young Talents in University of Anhui Province (no. gxyq2017092), and Key Teaching Research Project of Suzhou University (no. szxy2018jyxm12).