Stochastic Permanence , Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System with Random Perturbation

and Applied Analysis 3 value, the coefficients of the equation are generally required to satisfy a linear growth condition and a local Lipschitz condition (cf. Mao [13]). However, the coefficients of SDE (5) do not satisfy a linear growth condition, though they are locally Lipschitz continuous. In this section, we will use a method similar to Mao et al. [5, Theorem 2.1] to prove that the solution of SDE (5) is nonnegative and global. Theorem 3. Let Assumption 1 hold. For any given initial value x(0) ∈ R n + , there is a unique positive solution x(t) of system (5), and the solution will remain in R + with probability 1. Proof. Define a C-function V : R + → R + by V (x) = n ∑ i=1 (x i − 1 − logx i ) . (6) The nonnegativity of this function can be observed from a − 1 − log a ≥ 0 on a > 0 with equality holding if and only if a = 1. For x ∈ R + , applying Itô’s formula, we have


Introduction
Spatial factors which play a fundamental role in persistence and evolution of species can be modeled by a diffusion process.We have two typical equations to model the diffusion process.One is semilinear parabolic equations, that is, reaction-diffusion systems, where the populations are continuously spread out in space.The other is discrete diffusion systems, where several species are distributed over an interconnected network of multiple patches and there are population migrations among patches.Allen [1] studied and investigated the logistic nonlinear directed diffusion model ( 2  −    2  ) ,  = 1, 2, . . ., , where   denotes the density dependent growth rate in patch .The constants   (,  = 1, 2, . . ., ,  ̸ = ) are the dispersal rate from the th patch to the th patch, and the nonnegative constant   can be selected to represent different boundary conditions [2].Allen proved that the system (1) has a unique positive solution on a maximal interval (see [3]) and is strongly persistent and the population size can increase without bound or bounded under reversed conditions (see [1]).The fundamental tools to prove these results are the cooperative system theory and the cooperative matrix [1][2][3][4].
For system (1), Lu and Takeuchi [2, Theorem 3] extended Allen's results and obtained the following necessary and sufficient conditions.
(ii) Every solution of the system is unbounded, if the above condition is not satisfied.
) . ( Deterministic models are often subject to stochastic perturbations, and it is useful to reveal how the noise affects the population system.There are many papers which study differential equations with stochastic perturbations (see [5][6][7][8][9][10] and the references therein).Li et al. [7] studied the stochastic logistic populations system under regime switching and analyzed the asymptotic properties of their model.Jiang et al. [8,9] investigated a logistic equation with random perturbation and obtained many results such as global stability and stochastic permanence.More investigations and improvements of these stochastic models can be found in [11,12].There is very little known on the dynamic behavior in the single-species dispersal system with stochastic perturbation.Now we introduce randomly perturbation into the intrinsic growth rate   and assume that parameters   are disturbed to   +   Ḃ  () ,  = 1, 2, . . ., , where   () is mutually independent Brownian motion and   is a positive constant representing the intensity of the white noise.Then the stochastic system takes the form For convenience, let   =   + ∑  =1, ̸ =      and   = 0. Thus, the equation is rewritten as In this paper, we assume that   and   are nonnegative constants, the parameters   ,   are positive constants, and so   > 0.
The rest of the paper is arranged as follows.We will show that there exists a unique positive global solution with any initial positive value in Section 2. In Section 3, we will investigate sufficient conditions for stochastic permanence and persistence in mean which are important in an ecological system.In a deterministic system, the global attractivity of the positive equilibrium is studied, but it is impossible to expect system (5) to tend to a steady state.We investigate the stationary distribution of this system by the Lyapunov functional technique.This can be considered as weak stability, which appears as the solution is fluctuating in a neighborhood of the point.In Section 4, we show that if the white noise is small, there is a stationary distribution of (5) and it has an ergodic property.Results on dynamic in a patchy environment have largely been restricted to extinction analysis which means that the population system will survive or die out in the future.In Section 5, we give sufficient conditions for extinction.In Sections 6 and 7, we make numerical simulation to confirm the effect of white noise intensity and the diffusion coefficient on the species and give a conclusion.Finally, for the completeness of the paper, we give an Appendix containing some results which will be used in other sections.

Positive and Global Solutions
As the solution of SDE (5) has biological significance, it should be nonnegative.Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy a linear growth condition and a local Lipschitz condition (cf.Mao [13]).However, the coefficients of SDE (5) do not satisfy a linear growth condition, though they are locally Lipschitz continuous.In this section, we will use a method similar to Mao et al. [5,Theorem 2.1] to prove that the solution of SDE ( 5) is nonnegative and global.
Theorem 3. Let Assumption 1 hold.For any given initial value (0) ∈   + , there is a unique positive solution () of system (5), and the solution will remain in   + with probability 1.
Proof.Define a  2 -function  :   + →  + by The nonnegativity of this function can be observed from  − 1 − log  ≥ 0 on  > 0 with equality holding if and only if  = 1.For  ∈   + , applying Itô's formula, we have where and by Assumption 1, we know that there exists a positive constant number  satisfying and  is independent of   and .By a proof similar to Mao et al. [5,Theorem 2.1], we obtain the desired assertion.

Stochastic Permanence and Persistence in Mean
In this section, we will investigate the persistence under two different meanings: stochastic permanence and persistence in mean.
The proof of Theorem 7 is a simple application of the Chebyshev inequality and Lemma 6.
Since the solution of SDE ( 5) is positive, by the classical comparison theorem of stochastic differential equations [14], we can obtain the lemma.

Persistence in Mean.
In this section, we will investigate persistence in mean.First we introduce one definition.
Definition 11.SDE ( 5) is said to be persistent in mean, if there exist positive constants   ,   ( = 1, 2, . . ., ) such that the solution () of SDE ( 5) has the following property: From the result in [12], we know that lim inf Using the above conclusions, we get the following lemmas.
Lemma 12. Suppose that Assumptions 1 and 2 are satisfied.Then the solution () of SDE (5) with any initial value (0) ∈   + has the following property: An application of the Burkholder-Davis-Gundy inequality (see [12,14]) and the Hölder inequality (see [12]) yields where σ = max 1≤≤ {  }.This together with (39) yields lim sup We observe from (41) that there is a positive constant  * such that Let  > 0 be arbitrary.Then, by the well-known Chebyshev inequality, we have Applying the Borel-Cantelli lemma (see [12]), for almost all  ∈ Ω, we obtain that sup holds for all but finitely many .Hence, we have a  0 () such that (44) holds whenever  ≥  0 , for almost all  ∈ Ω.
Proof.Assume that  :   + →  + is defined as in (37).From the inequality (28) of Lemma 9 and (36) of Lemma 13, one can derive that lim By virtue of the Itô's formula and the Cauchy inequality, we have Abstract and Applied Analysis where () is a martingale defined by with (0) = 0.The quadratic variation of this martingale is By the strong law of large numbers for martingales (see [11]), we have lim It finally follows from (49) by dividing by  on both sides and then letting  → ∞; that is, which implies that lim sup On the other hand, from Lemma 12, we know that lim inf Let Thus the required assertion follows.

Stationary Distribution
In this section, we investigate that there is a stationary distribution for SDE (5) instead of asymptotically stable equilibria.In order to ensure that system (1) has a globally stable positive equilibrium point  * = ( * 1 ,  * 2 , . . .,  *  ), we need to introduce the following lemmas.(c) all of the principal minors of  are positive; that is, Proof.We can obtain by Lemma 15 that if   > ∑  =1   ( = 1, 2, . . ., ), then all of the principal minors of   are positive, and from (a) and (c) of Lemma 16, we know that   is a nonsingular M-matrix.
Since all of the principal minors of  and   are the same, so  is also a nonsingular -matrix.
From Lemma 17, we know that if  is a nonsingular -matrix, then the real part of each eigenvalue of  is positive based on (a) and (d) of Lemma 16, and we can also deduce that the maximum of the real part eigenvalues of negative matrix  is less than 0. This together with Theorem 3 of [2]  ) .
(61) By Itô's formula, we have From ( 60), we know that Substituting ( 63) into (62) one sees that Using the inequality where By Assumption 1, we know that the quadratic coefficients are less than zero.The following proof of ergodicity is similar to Theorem 3.2 in [10].Note that lies entirely in   + .We can take  to be a neighborhood of the ellipsoid with  ⊂   =   + , so for  ∈   \ ,  ≤ − ( is a positive constant), which implies that the condition (B2) in Assumption A.1 (see the Appendix) is satisfied.By Remark A.3 and Lemma A.4 and using the similar method as [10], we can prove that (A1) is also satisfied (see page 349 of [15]).Therefore, the stochastic system (5) has a stable stationary distribution (⋅) and it is ergodic.

Extinction
We know that, if Assumption 1 holds, the solution of ODE (1) converges to a positive equilibrium point or is unbounded, so the population will not become extinct, and by Theorem 10, we note that if the condition   >  2   /2 ( = 1, 2, . . ., ) is also satisfied, that is, the white noise intensity is smaller, then the species will be stochastically permanent and persistent in mean.We will show in this section that if the noise is sufficiently large, the solution to the associated SDE (5) will become extinct with probability 1.
Proof.Define  :   + →  + as in (37).Using Itô's formula, one can derive that Figure 1: The pictures on the left are the solutions of stochastic system (73) and the corresponding undisturbed system, and the blue lines and the black lines represent them, respectively.The middle of the subgraphs is the histogram of stochastic system (73) and the subgraphs on the right are normal quantile-quantile plots of the values of the paths  1 () and  2 ().The stochastic system is stochastically permanent and has a stationary distribution. 1 = 0.1,  2 = 0.09.
Following the scaling method of (48) and applying the Cauchy inequality and Assumption 1, we find Integrating both sides of the above inequality (69) from 0 to  gives log  ( ()) ≤ log  ( (0 where () is a martingale defined in the proof of Theorem 14.By the strong law of large numbers for martingales (see [11]), we have lim It finally follows from (70) by dividing by  on both sides and then letting  → ∞; that is, lim sup Thus the required assertion follows.
Case 1.The effect of different white noise intensity on the population.
In Figure 1, we choose  1 = 0.1,  2 = 0.09.Obviously Assumption 2 holds and the SDE (73) is stochastically permanent and persistent in mean.We compute  2 = (1/2)  1).The left pictures in Figure 1 show that the stochastic system imitate the deterministic system.The right subgraphs are the normal quantile-quantile plots of the values of the paths  1 () and  2 (), and they are similar to the straight lines.This means that the distribution is approximately standard normal distribution.The scatter plot of  1 () and  2 () is Figure 3(a); we find that almost all   Figure 4: The pictures on the left are the solutions of stochastic system (73) and the corresponding undisturbed system, and the blue lines and the black lines represent them, respectively.The right subgraphs are the histogram of stochastic system (73). 1 = 0.2,  2 = 0.8. 12 = 0.6,  21 = 0.5.population distribution lies in a small neighborhood, which can be imagined as a circular or elliptic region centered at ( * 1 ,  * 2 ).Hence, although there is no equilibrium of the stochastic system (73) as the deterministic system, it is stochastically permanent, persistent in mean and has the ergodic property by Theorems 10, 14, and 18.
In Figure 2, we choose  1 = 0.2,  2 = 0.3.The populations of  1 and  2 suffer relatively large white noise.By comparing Figure 1, we can see that in Figure 2  its scatter distributes in a larger area (see the scatter picture in Figure 3 Comparing with small white noise as in Figures 1 and 2, we choose  1 = 0.9,  2 = 1.0 in Figure 7.Both  1 and  2 suffer large white noise.We find that   < (1/2)    Figure 7: The subgraphs are defined as in Figure 4;  1 = 0.9,  2 = 1.The populations of  1 and  2 will become extinct.
Case 2. The effect of different diffusion coefficient on the population.
In Figure 4, we select  1 = 0.2,  2 = 0.8.The conditions of Theorems 10, 14, and 18 are satisfied. 2 suffers relatively large white noise.From the left pictures of  1 () in Figures 2 and  4, we see that the fluctuations of the two curves are different and the reason is that larger white noise of  2 impacts  1 in Figure 4.In other words, due to the presence of diffusion, the relatively big white noise intensity in the individual patches will be evenly distributed to the other patches, which reduces the risk of extinction of the population.Therefore, system (73) is stochastically permanent and has a stationary distribution.
In Figure 5, we choose  12 = 0.01,  21 = 0,  1 = 0.2, and  2 = 0.8.Figures 4 and 5 have the same white noise intensity but have different diffusion coefficients.Because there is no diffusion effects, we can see that  2 will die out from Figure 5 and the scatter plot Figure 6(b), that is to say, the isolated patches may become extinct if the white noise is large.

Conclusion
In this paper, we study the stochastic logistic single-species model with nonlinear directed diffusion among  patches.
First, we divide the white noise intensity into small, medium, and large three cases, and through numerical simulation, we can more clearly understand the important role played by the white noise in biological populations.From these figures, we find that when the white noise is small, system (73) imitates its deterministic system and it is stochastically permanent and persistent in mean and has a stationary distribution (see Figures 1 and 3(a)).When the white noise is relatively large in some groups, it will bring relatively large deviation (see Figures 2, 3(b), 4, and 6(a)) but will not bring the species extinction due to the presence of diffusion.But, when the noise is sufficiently large in all the groups (see Figure 7), the species will become extinct even if diffusion exists.We also study the effect of different diffusion coefficient on the species and we find that isolated plaque affected by big white noise may become extinct if the diffusion coefficient is very small or equals zero (see Figures 5 and  6(b)).
In the real world, the large white noise may be bad weather, serious epidemic, which can be considered as the decisive factor responsible for the extinction of populations.Diffusion phenomena, however, play a crucial role in the development of biological populations, and human activities without control will affect the biological diffusion process which is likely to cause fatal consequences.Therefore, our research and analysis on population have great practical significance.

Figure 2 :
Figure2: The subgraphs are defined as in Figure1. 1 = 0.2,  2 = 0.3.The stochastic system is stochastically permanent and persistent in mean and has a stationary distribution.