Asymptotic Behavior of a Stochastic Two-Species Competition Model under the Effect of Disease

This paper is concerned with a stochastic two-species competition model under the effect of disease. It is assumed that one of the competing populations is vulnerable to an infections disease. By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model.Then, the asymptotic pathwise behavior of the model is given via the exponential martingale inequality and Borel-Cantelli lemma. Next, we find a new method to prove the boundedness of the pth moment of the global positive solution. Then, sufficient conditions for extinction and persistence in mean are obtained. Furthermore, by constructing a suitable Lyapunov function, we investigate the asymptotic behavior of the stochasticmodel around the interior equilibrium of the deterministic model. At last, some numerical simulations are introduced to justify the analytical results. The results in this paper extend the previous related results.


Introduction
Populations that compete for common resources are well known among ecologists.They are classically modeled by observing their interactions that hinder the growth of both populations and are thus described by negative bilinear terms in all the relevant differential equations.The classic twospecies Lotka-Volterra competition model takes the form There is an extensive literature concerned with model (1) and related deterministic models (see [1][2][3] and the references therein) and we here do not mention them in detail.As mentioned in [4], another major problem in today's modern society is the spread of infectious diseases.A detailed account of modeling and the study of epidemic diseases can be found in the literature [5,6].The population biology of infectious diseases has also been presented in [7].In [8], the authors studied the dynamics of two competing species when one of them is subject to a disease.In [4], the authors assumed that () and () are competing for the same resource and assumed that the disease spreads only in one of the competing species, denoted by ().They specified the healthy individuals (), the healthy individuals  1 (), and the infected individuals of the latter species denoted by  2 ().Moreover, they studied the following two-competing-species model under the effect of infectious disease d () =  () [ −  () −  1 () −  2 ()] d, d 1 () =  1 () [ − ê () −  ( 1 () +  2 ()) −  2 ()] d, d 2 () =  2 () [ 1 () −  () −  ( 1 () +  2 ())] d (2) with initial value (0) =  0 ,  1 (0) =  10 , and  2 (0) =  20 .The parameters in (2) are defined as follows:  and  are the intrinsic growth rates of the populations () and  1 (), respectively. is the loss rate in population () due to the competitor  1 () and  is the loss rate in population () due to the competitor  2 ().ê is the loss rate in population  1 () due to the competitor () and  is the loss rate in population  2 () due to the competitor ().,  are intraspecific coefficients of (),  1 (), and  2 (). is the transmission rate of the infection.Denote R 3 + = {(, .) ∈ R 3 :  > 0,  > 0,  > 0}.From [4], we know that all solutions of model (2) will lie in the region as  → ∞, for all positive initial values ( 0 ,  10 ,  20 ) ∈ R 3 + .Moreover, the interior equilibrium Ê = (x, ŷ1 , ŷ2 ) of model ( 2) is feasible when  > , ( + ) > ê,  > ( 1 +  3 ), and ( − )( +  − ê) >  2 .Here where  1 = / 2 ,  2 = ( +  − ê)/ 2 ,  3 = ( − )/ 2 ,  4 = ( − )( +  − ê)/ 2 − /.However, the population dynamics in the real world are often disturbed by some uncertain factors while the stochastic population model is more in line with actual situation.During the past decades, a great deal of attention has been paid to the study of stochastic biological model (see [9][10][11][12][13][14][15][16][17][18][19][20]).In [12], the authors discussed a two-species stochastic nonautonomous Lotka-Volterra competition model.Some sufficient conditions on the boundedness, extinction, nonpersistence in the mean, and weak persistence of solutions are established.In [13], the authors investigated the optimal harvesting problem for a stochastic delay competitive Lotka-Volterra model with L é vy jumps.In [14], the authors studied the permanence and asymptotical behavior of stochastic prey-predator model with Markovian switching.In [15], the authors investigated the stochastic competitive models in a polluted environment.In [16], the authors explored an impulsive stochastic infected predator-prey system with L é vy jumps and delays.In [20], the authors considered a stochastic susceptible-infective epidemic model in a polluted environment, which incorporates both environmental fluctuations and pollution.
Parameter perturbation induced by white noise is an important and common form to describe the effect of stochasticity.In this paper, we perturb the intrinsic growth rates  and  in model (2) with white noise; that is, where  1 (),  2 () are mutually independent Brownian motions and  2 1 ,  2 2 denote the intensities of the white noise.Then corresponding to the deterministic model ( 2), the stochastic model takes the following form with initial value (0) =  0 ,  1 (0) =  10 , and  2 (0) =  20 .Here  = { 1 (),  2 (),  ≥ 0} represents the two-dimensional standard Brownian motion defined on a compete probability space (Ω, F, P) with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all -null sets).All meanings of the parameters are exact to or similar to those for (2).
The remaining part of this paper is organized as follows.The proof of the existence and the uniqueness for the global positive solution of model (6) for any positive initial value is given in Section 2. An important asymptotic property of the model is obtained by using the exponential martingale inequality and Borel-Cantelli lemma in Section 3. In Section 4, the stochastically ultimate boundedness of the positive solution is examined.In Section 5, sufficient conditions for extinction and persistence in mean of model (6) are established.In Section 6, by constructing a suitable Lyapunov function, we investigate the asymptotic behaviors of the stochastic model (6) around the interior equilibrium of the deterministic model.Numerical simulations under certain parameters are presented to illustrate our main results in Section 7. Finally, a few comments will conclude the paper.

Existence and Uniqueness of Positive Solution
Since (),  1 (), and  2 () in model ( 6) are the size of the populations at time , we are interested only in the positive solutions of model (6).However, the coefficients of (6) do not satisfy the linear growth condition; the classical theory of stochastic differential equations is not applicable directly.Next, by using comparison theorem of stochastic differential equations, we show that model (6) has unique positive global solution with positive initial value.

Asymptotic Property
In this section, by using the exponential martingale inequality and Borel-Cantelli lemma, we investigate an important asymptotic property of positive solutions of model (6)

Stochastically Ultimate Boundedness
In this section, we continue to examine the stochastically ultimate boundedness which means that the solution is ultimately bounded with the large probability.Firstly, its definition will be given.
en, for any  ≥ 1, (44 at is, the solutions of model ( ) are uniformly bounded in the th moment.
Proof.For population , applying Itô's formula to   leads to Taking expectation on both sides of the above inequality, we can derive which implies the differentiability of Then, from Lemma 5 and the comparison theorem, it follows that For population , denote () =  1 () +  2 () and  0 =  10 +  20 .Applying Itô's formula to   leads to Taking expectation on both sides of the above inequality, we can derive which implies the differentiability of Using the H ö lder inequality, we have Then, from Lemma 5 and the comparison theorem, it follows that Note that  ≥ 1.
Since  1 () and  2 () are positive for all  ≥ 0, it follows that, for  = 1, 2, The proof is therefore complete.

Extinction and Persistence
In this section, we will investigate the extinction and persistence of the population.In order to obtain our main results, several lemmas will be given.For the sake of convenience and simplicity, we first introduce the following notation: Lemma 8 (see [26]).
(i) If Δ 1 < 0 and Δ 2 ≤ 0, then lim →∞ () = 0 a.s.Furthermore, if Δ 3 ≥ 0, then which, together with the conditions of Theorem 10, yields Then by applying the strong law of large numbers and Remark 3, we have lim sup which, together with the conditions of Theorem 10, yields Let Ω 3 = { ∈ Ω : lim →∞ (, ) = 0}; then (85) implies P(Ω 3 ) = 1.Hence, for any  ∈ Ω for any  ≥  1 .Note that Then by (II) in Lemma 8 and the arbitrariness of , one can observe that lim inf Therefore, we can derive that lim (ii) A similar discussion to that in the above for (i), we also have the desired assertion (ii).This completes the proof.
If we do not consider the effect of disease, then model ( 6) can be degraded into the following model with initial value (0) =  0 and  1 (0) =  10 .Model (102) is the same as the stochastic competitive population model ( 0 ) discussed in [15].
From Theorems 9 and 10, we can get the following corollaries with the proof being omitted.(i) If  < 0.5 2 1 , then the population  will go to extinction almost surely; (ii) If  < 0.5 2  2 , then the population  1 will go to extinction almost surely.
(i) If Δ 1 < 0, then the population  will go to extinction almost surely; if Δ 4 < 0, then the population  1 will go to extinction almost surely.
Remark .Corollaries 11 and 12 are consistent with Theorems 9 and 10 in [15].Moreover, if one considers the effects of the disease, from Theorem 10 we know that the conditions for population extinction and persistence will be more complicated.Therefore, our work can be seen as the extension of [15].

Numerical Simulations
In this section, we make numerical simulations to illustrate our results by using the Milstein method (see, e.g., [27]).The numerical simulations of population dynamics are carried out for the academic tests with the arbitrary values of the vital rates and other parameters which do not correspond to some specific biological populations and exhibit only the theoretical properties of numerical solutions of the considered model.In the figures, the red lines, blue lines, and green lines represent the trajectories of populations (),  1 (), and  2 (), respectively.Here we give numerical simulations of model ( 6) with the same initial values  0 = 1,  10 = 0.9, and  20 = 0.1.
As can be seen from Figures 1 and 2, regardless of the size of the interspecific competition rate  and the rate of infection , as long as  < 0.5  the healthy individuals (), the healthy individuals  1 (), and the infected individuals  2 () will go to extinction.
From Figures 3 and 4, we can see that although the environmental noise is relatively small, population extinction can also occur when the competition coefficients satisfy certain conditions.It can be seen from Figure 2 that the population  is dominant in competition, while the population  is at a disadvantage, and the individuals with infectious diseases in population  will go to extinction.Moreover, populations  and  2 will go to extinction and population  1 will be weakly persistent in mean almost surely.However, from Figure 3, we can see that the population  is dominant in competition, while the population  is at a disadvantage, and the individuals with infectious diseases in population  will go to extinction.That is, populations  1 and  2 will go to extinction and population  will be weakly persistent in mean almost surely.

Conclusion and Discussion
In this paper, we consider a stochastic two-species competition model under the effect of disease.By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model.Then, an important asymptotic property of model is given via the exponential martingale inequality and Borel-Cantelli lemma.Next, we prove the boundedness of the th moment of the global positive solution.Then, sufficient conditions for extinction and persistence in mean are obtained.Furthermore, by constructing suitable Lyapunov function, we investigate the asymptotic behavior of the stochastic system around the interior equilibrium of the deterministic model.
From Remark 13, we know that our work can be seen as the extension of [15].Some interesting problems deserve further consideration.One may incorporate the Markovian switching into the stochastic population model (6).Since model ( 6) may be perturbed by the telegraph noise which can make the model switch from one environmental regime to another, one may also introduce the jumps into the stochastic model (6).Since population systems may suffer severe environmental perturbations, such as tsunami, volcanoes, avian influenza, SARS, floods, hurricanes, earthquakes, and toxic pollutants, we leave this for future consideration.

Theorem 7 .
Solutions of model ( ) are stochastically ultimate bounded.