# Dynamic Behavior Analysis of Stochastic Nonautonomous Logistic Model with Limited Resources and Lévy Jump

Authors: Yonggang Ma; Junmei Liu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## 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 ofp-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.


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## Body

## 1. Introduction

The logistic population model, an ordinary differential equation depicting the self-limiting growth of a population of sizext, is exploited by Verhulst (1845)(1)x′t=xtr−axt.The model describes a population growing from the initial sizex0 with an intrinsic growth rate r, 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 r/a (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:(2)x′t=xtr−axt−hx,where a,hx stand competing rate and predation, respectively. In [2], Murray takes the form for hx, that is, cx2t/b+x2t. Then, the dynamics of xt is governed by(3)x′t=xtr−axt−cx2tb+x2t.In [3], a logistic differential equation with stochastic perturbation is a system of the form(4)dxt=xt1−xtKrdt+σdBt,where Bt is a standard Brown motion, and σ 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:(5)dxt=xt1−xtKrtdt+σtdBt,where rt and σt are both continuous functions on 0,+∞. In [10], Jiang et al. discuss a randomized logistic equation with food-limited effect of the form(6)dxt=xtK−xtK+Cxtrdt+σdBt,where C 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(7)dxt=xtK−xtK+Cxtrtdt+σtdBt.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:(8)dxt=xtK−xtK+Cxtrt−atxtdt+σtdBt+∫YγuN˜dt,du,where N is a Poisson counting measure with characteristic measure λ on a measurable subset Y of 0,+∞, with λY<∞, and N˜dt,du:=Ndt,du−λdudt. Throughout the paper, we assume that B is independent of N.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 Section2, we propose some assumptions and give the necessary lemmas. In Section 3, we firstly prove the persistence and extinction in the sense of p-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)
rt,at,σt,γu are continuous and bounded functions on 0,+∞ and σt<σ¯,γu<γ¯(b)
For everym>0, there is a constant Lm, such that ∫YHx,u−Hy,u2λdu≤Lmx−y2, where Hx,u=xtK−xtγu/K+Cxt(c)
There is a positive constantm, such that ln1+γu/1−γu/1+CC+1≤m,γu>−1In the following, we prove a lemma to show that the solution will remain in0,K with an initial value x0.Lemma 1.
For any initial value0<x0<K, there is a unique uniformly continuous solution xt to model (8) on t≥0, and xt will remain in 0,K with probability one.Proof.
Since the coefficients of the equation are local Lipschitz continuous, for any initial valuex0>0, there exists a unique local solution xt on t∈0,τe¯, where τe¯ is the explosion time.
Define a stopping timeτe=inft∈R+:xt∉0,K, then τe≤τe¯ a.s. we need only to show τe=∞ a.s. Let k0∈Z+ be sufficiently large such that 1/k0≤x0≤K−1/k0. For any integer k>k0, define the stopping time(9)τk=inft∈0,τe:xt∉1k,K−1k,where we set infø=∞. Clearly, τk is increasing as k⟶∞. Set τ∞=limk⟶∞τk, then τ∞≤τe a.s. If we can show that τ∞=∞, then τe=∞ a.s. We assume that τe=∞ a.s. does not hold, there has to be a pair of constants T>0, and ε∈0,1 such that Pτk≤T≥ε. For 0≤t≤τk∧T, we define a C2-function V:R+⟶R+ by(10)Vx=1−lnxK−lnK−xK.
Fork>k0 and 0≤t≤τk∧T, we use Itô’s formula for model (8)(11)dVx=2xt−KxtK−xtdxt+x2t+K−xt22x2tK−xt2dxt2,=LVxtdt+2xt−KK+CxtdBt,+∫Yln11+K−xtγu/K+Cxt+ln11−xtγu/K+CxtN˜dt,du,where(12)LVx=2xt−KK+Cxtrt−atxt,+∫Yln11+K−xtγu/K+Cxt+ln11−xtγu/K+Cxt−2xt−KγuK+Cxtλdu,+σ2tx2t+K−xt22K+Cxt2.
Both sides of equation (11) are integrated from 0 to τk∧T and take the expectation(13)EVxτk∧T=Vx0+E∫0τk∧TLVxtds.
For equation (12), we use hypothesis (A) and function monotonicity to simplify(14)LVx≤2xt−KK+Cxtrt+σ2t+∫Yγuλdu,≤2xtrtK+Cxt+σ2t+γuλY,≤2r¯+σ¯2+γ¯λY=≕M<∞.
According to equations (13) and (14),(15)EVxτk∧T≤Vx0+MT.
LetΩk=τk≤T,k≥k1, then PΩk≥ε. Notice that for arbitrary ω∈Ωk,xτk,ω=k or 1/k,(16)EVxτk∧T≥EIΩkω∨xτk,=1−ln1/kK−lnK−1/kK∧1−lnkK−lnK−kKPτk≤T,≥ε1−ln1/kK−lnK−1/kK∧1−lnkK−lnK−kK,where IΩk is the indicator function. Letting k⟶∞ results in the contradiction(17)+∞>EVxτk∧T>+∞.
Thus, we obtain thatτ∞=∞ a.s.
Therefore, the solutionxt of the model (8) satisfies 0<xt<K. 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 xt is uniformly continuous on t>0. The proof is complete.Lemma 2 see ([16]).
LetM=Mtt≥0 be a local martingale and for each t≥0 define(18)ρMt≔=∫0tdM,Ms1+s2,where M,Ms is Meyer’s angle bracket process. If Plimt⟶∞ρMt<∞=1, then(19)Plimt⟶∞Mtt=0=1.Lemma 3 see ([7]).
Letf be a nonnegative function defined on 0,∞ such that f is integrable on 0,∞ and is uniformly continuous on 0,∞, then limt⟶∞ft=0.


## 3. Main Results

In this section, we discuss the equilibrium stability under the influence of various factors.Theorem 1.
Assumptionγ¯≤0, let xt be a continuous positive solution to equation (8) for any initial value x0=x0 with 0<x0<K.(1)
If there is a constantθ>0 such that(20)rt−σ2t2−Kat≥θ,then(21)limt⟶∞Ext−K2=0.(2)
If there is a constantη>0 such that(22)rt+σ2t2≤−η,then(23)limt⟶∞Ext2=0.Proof.
The certification process is detailed in Appendix A.Remark 1.
Theorem 1 can be extended top-moment persistence and extinction. Specifically, when 0<p<2, by the moment inequality(24)EK−xtp≤K−xt2p/2.
Forp≥2,xt∈0,K(25)EK−xtp=KpEK−xtKp≤Kp−2EK−xt2.
Therefore, for anyp>0,limt⟶∞Ext−Kp=0.Remark 2.
Similarly, we can deducelimt⟶∞Expt=0.
Next, we prove that model (8) is almost surely asymptotically stable under some conditions.Theorem 2.
Assume conditions (A–C) hold, letxt∈0,K be a solution of equation (8) with x0∈0,K. If μ<0, then the zero solution of model (8) is almost surely asymptotically stable, where(26)μ=limsupt⟶∞1t∫0trs+σ2s21+C+∫Yln1+γu1−γu/1+CC+1λduds.Proof.
See Appendix B for details.Remark 3.
Obviously model (4) in [8] is a special case of our model. When γu=0, the condition is simplified to μ=limsupt⟶∞1/t∫0trs+σ2s/21+Cds. 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, ifν>0, then the positive equilibrium K of model (8) is almost surely asymptotically stable. Where(27)ν=liminft⟶∞1t∫0trs−σ2s2−Kas−∫Yγuλduds.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 K value. In particular, when at=ct=γu≡0, our conclusion is the same as in [8]. It can be seen that our results promote existing conclusions.Remark 5.
Letr1t=−σ2s/21+C−∫Yln1+γu/1−γu/1+CC+1λdu,r2t=σ2s/2+Kas+∫Yγuλdu. When rt<r1t, Theorem 2 shows that the population scale is asymptotically stable at zero. When rt>r2t, Theorem 3 shows that the population size is asymptotically stable at K. When r2t<rt<r1t, 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 rt=−0.01+0.2sint,K=1,Y=0,+∞,λY=1, initial data x0=0.4, and step size Δt=0.001, and other parameter values are shown in detail in the subscript of the figure. For graphs (a)–(h), it is easy to obtain <0, which satisfies Theorem 2. We only calculated (h) as an example μ=limsupt⟶∞1/t∫0t−0.01+0.2sint+ln0.95/1+0.0252ds≈−0.11<0. Specifically, Figure (a) and Figure (b) differ only in the value of at. 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.Figure 1
The lines represent the solution of model (8) for rt=−0.01+0.2sint,K=1, initial data x0=0.4, step size Δt=0.001. (a). C=0,at=0.5,σt=γu=0; (b). C=0,at=1,σt=γu=0.
(a)(b)Figure 2
The lines represent the solution of model (8) for rt=−0.01+0.2sint,K=1, initial data x0=0.4, step size Δt=0.001. (a) C=2,at=0.5,σt=γu=0; (b) C=6,at=0.5,σt=γu=0.
(a)(b)Figure 3
The lines represent the solution of model (8) for rt=−0.01+0.2sint,K=1, initial data x0=0.4, step size Δt=0.001. (a) C=0,at=0,σt=1,γu=0; (b) C=0,at=0,σt=0,γu=−0.05.
(a)(b)Figure 4
The lines represent the solution of model (8) for rt=−0.01+0.2sint,K=1, initial data x0=0.4, step size Δt=0.001. (a) C=0,at=0,σt=1,γu=−0.05; (b) C=1,at=0.1,σt=1,γu=−0.05.
(a)(b)In Figures5 and 6, we choose rt=0.4+0.5sint,K=1,C=1.2,Y=0,+∞,λY=1, initial data x0=0.3, and step size Δt=0.001. Other parameters are shown in the analog graphic subscript. It is also easy to obtain ν>0 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.Figure 5
The lines represent the solution of model (8) for rt=0.4+0.5sint,K=1,C=1.2, initial data x0=0.3, step size Δt=0.001. (a) at=0.1,σt=0,γu=0; (b) at=0.1,σt=0.2,γu=0.
(a)(b)Figure 6
The lines represent the solution of model (8) for rt=0.4+0.5sint,K=1,C=1.2, initial data x0=0.3, step size Δt=0.001. (a) at=0.1,σt=0,γu=−0.3; (b) at=0.1,σt=0.2,γu=−0.3.
(a)(b)


## 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 K, 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.



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