Abstract
In this paper, we discuss the dynamic behavior of the nonautonomous stochastic logistic model under the influence of limited resources and Lévy jump. Firstly, by constructing the Lyapunov function and using the Itô formula, we prove the persistence and extinction of the solution of the model in the sense of -moment. Secondly, with the help of the Itô-Lévy formula, we discuss that the zero solution and positive equilibrium solution of the model are almost asymptotically stable under certain conditions. Finally, we verified the correctness of these conclusions through numerical simulation and explained some specific biological significance.
1. Introduction
The logistic population model, an ordinary differential equation depicting the self-limiting growth of a population of size , is exploited by Verhulst (1845)
The model describes a population growing from the initial size with an intrinsic growth rate , experiencing roughly exponential growth which reduces speed as the availability of some key resource (e.g., nutrients or space) becomes limiting. Eventually, population density saturates at the carrying capacity (maximum achievable population density), once the critical resource is exhausted. Where further flexibility is required, generalized forms of the logistic growth process may be used instead.
In 1978, Ludwig et al. [1] introduce the budworm population dynamics to be modeled by the following equation:where stand competing rate and predation, respectively. In [2], Murray takes the form for , that is, . Then, the dynamics of is governed by
In [3], a logistic differential equation with stochastic perturbation is a system of the formwhere is a standard Brown motion, and is the intensity of the noise. Many authors discuss randomized logistic models [4, 5]. In addition, the impact of random disturbance on the population system can be referred to [6–8]. In [9], Golec and Sathanathan investigate the following nonautonomous randomized population model:where and are both continuous functions on . In [10], Jiang et al. discuss a randomized logistic equation with food-limited effect of the formwhere is a nonnegative constant denoting the delayed effects of the food limited on the growth of the population. In [11], the author investigates the nonautonomous randomized food-limited population equation
Stochastic differential equations with white noise have many advantages in population modeling. However, in reality, biological systems are often attacked by sudden and large-scale disturbances, such as natural disasters: volcanoes, tsunamis, earthquakes, and pandemics (SARS, COVID-19, Ebola virus, etc.). Then, in order to describe these events, it is necessary to add a jump process [12–15] to the model (7). Inspired by the above results, the authors focus on the following logical model affected by various factors:where is a Poisson counting measure with characteristic measure on a measurable subset of , with , and . Throughout the paper, we assume that is independent of .
Compared with the previously mentioned model, the model (8) is more reasonable. Firstly, the population usually shares the same basic requirements and competes for resources, food, habitat, etc. However, the model (7) does not consider this situation. Secondly, the population not only affected by environmental change, such as changes in weather, habitat destruction, exploitation, and other factors, but also may suffer sudden environmental shocks, e.g., earthquakes, hurricanes, and epidemics. To explain these phenomena, we introduce a standard Brownian motion and a jump process, which makes our model closer to the actual population growth process. Compared with the existing literature, the main contribution of this paper is to consider the combined effects of population competition, limited resources, and two kinds of environmental noise.
The rest of this paper is organized as follows. In Section 2, we propose some assumptions and give the necessary lemmas. In Section 3, we firstly prove the persistence and extinction in the sense of -moments. Secondly, we prove that the positive equilibrium and zero equilibrium are almost asymptotically stable. In Section 4, we perform exhaustive numerical simulations for the model in question. We end this paper with a detailed discussion of obtained results in Section 5.
2. Preliminaries
In the first place, we give three hypotheses, which are very important to our conclusions.(a) are continuous and bounded functions on and (b)For every , there is a constant , such that , where (c)There is a positive constant , such that
In the following, we prove a lemma to show that the solution will remain in with an initial value .
Lemma 1. For any initial value , there is a unique uniformly continuous solution to model (8) on , and will remain in with probability one.
Proof. Since the coefficients of the equation are local Lipschitz continuous, for any initial value , there exists a unique local solution on , where is the explosion time.
Define a stopping time , then a.s. we need only to show a.s. Let be sufficiently large such that . For any integer , define the stopping timewhere we set . Clearly, is increasing as . Set , then a.s. If we can show that , then a.s. We assume that a.s. does not hold, there has to be a pair of constants , and such that . For , we define a -function byFor and , we use Itô’s formula for model (8)whereBoth sides of equation (11) are integrated from 0 to and take the expectationFor equation (12), we use hypothesis (A) and function monotonicity to simplifyAccording to equations (13) and (14),Let , then . Notice that for arbitrary or ,where is the indicator function. Letting results in the contradictionThus, we obtain that a.s.
Therefore, the solution of the model (8) satisfies . By Assumption (A) and (B), we can see that the coefficient of model (8) satisfies the local Lipschitz condition and the linear growth condition. By Mao [6], almost every sample path of is uniformly continuous on . The proof is complete.
Lemma 2 see ([16]). Let be a local martingale and for each definewhere is Meyer’s angle bracket process. If , then
Lemma 3 see ([7]). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on , then .
3. Main Results
In this section, we discuss the equilibrium stability under the influence of various factors.
Theorem 1. Assumption , let be a continuous positive solution to equation (8) for any initial value with .(1)If there is a constant such that then(2)If there is a constant such that then
Proof. The certification process is detailed in Appendix A.
Remark 1. Theorem 1 can be extended to -moment persistence and extinction. Specifically, when , by the moment inequalityFor Therefore, for any .
Remark 2. Similarly, we can deduce .
Next, we prove that model (8) is almost surely asymptotically stable under some conditions.
Theorem 2. Assume conditions (A–C) hold, let be a solution of equation (8) with . If , then the zero solution of model (8) is almost surely asymptotically stable, where
Proof. See Appendix B for details.
Remark 3. Obviously model (4) in [8] is a special case of our model. When , the condition is simplified to . This has been the same as the condition in [8]. It can be seen that our model has a wider range of applications.
Theorem 3. Under the condition of Theorem 1, if , then the positive equilibrium of model (8) is almost surely asymptotically stable. Where
Proof. See Appendix C for details of the certificate.
Remark 4. For model (8), when two kinds of noises are controlled within a certain range, the population size is stable around value. In particular, when , our conclusion is the same as in [8]. It can be seen that our results promote existing conclusions.
Remark 5. Let . When , Theorem 2 shows that the population scale is asymptotically stable at zero. When , Theorem 3 shows that the population size is asymptotically stable at . When , the stability problem of model (8) will be our further work.
4. Numerical Simulations
In this section, we apply the Euler–Maruyama method [17] to simulate the theoretical results. In Figures 1–4, we choose , initial data , and step size , and other parameter values are shown in detail in the subscript of the figure. For graphs (a)–(h), it is easy to obtain , which satisfies Theorem 2. We only calculated (h) as an example . Specifically, Figure (a) and Figure (b) differ only in the value of . The trajectory of the graph shows that the internal competition of the population can accelerate the extinction of the population. The trajectories in figures (c) and (d) show that more resources delay the extinction of the population. Figures (e) and (f) show the trajectory of the impact of noise and Lévy jump on the population, respectively. Furthermore, figure (g) shows the trajectory of the joint impact of noise and Levy jump on the population. Figure (h) shows the trajectory of the impact of internal competition, limited resources, noise, and Levy jump on the population.
(a)
(b)
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In Figures 5 and 6, we choose , initial data , and step size . Other parameters are shown in the analog graphic subscript. It is also easy to obtain for (i)–(l), which satisfy Theorem 3. Figure (i) shows the determined trajectory of equation (8). Figure (j) shows the effect of noise on the population at the positive equilibrium point. Figure (k) shows the effect of Lévy jump on the population at the positive equilibrium point. Figure (l) shows the effect of random noise and Lévy jump on the population at the positive equilibrium point.
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5. Conclusion
In this paper, we establish some criteria for the asymptotic behavior of the model (8). Sufficient conditions are obtained for the solution converging to the equilibria: 0 and , which improve Theorem 1.3 obtained in [8]. The results show that limited resources, random perturbations, and lévy jumps have a greater impact on population size among a variety of influencing factors.
Appendix
A The Proof Process of Theorem 1
The authors define a Lyapunov function by
By Itô’s formula,
The authors integrate both sides of the equation (A.2)
Both sides of equation (A.3) take the expectation
By and functional monotonicity
By , inequality (A.5) can be transformed into
By and , the authors simplify the above equation to get the following results:which implies . By Lemma 3, we yield
Similar conclusion (1), the authors define a Lyapunov function by
By Itô’s formula,
The authors integrate both sides of the equation (A.10) and take the expectation
Through and function monotonicity, the above equation can be amplified as follows:
By and , the authors obtainwhich implies that . By Lemma 3,
Thus, the authors complete the proof of Theorem 1.
B The Proof Process of Theorem 2
Applying Itô’s formula to model (8) leads to
The authors substitute model (8) into equation (B.1)
By integrating both sides of equation (B.2) and calculating, the following inequality holds for t that is large enoughwhere and . Therefore,
Define . Under condition (C), the authors obtain
According to Lemma 2,
The authors change the term of inequality (B.3) and divide both sides by
The authors take the limit on both sides of inequality (B.7) and substitute (B.5) and (B.6) into (B.7)where
That is to say, if , then .
C The Proof Process of Theorem 3
The authors rewrite equation (B.2) as fo
Integrating both sides of equation (C.1)where . By equation (A.11),
By equations (A.4), (A.5), and (C.3)where . Then, when , we have .
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
The work was supported by the Yulin University Doctoral Research Foundation (17GK16), the Yulin High-Tech Zone Science and Technology Planning Project (CXY-2021-61, CXY-2021-65), and the Natural Science Foundation of China (12061081).