Dynamics of Nonautonomous Stochastic Gilpin-Ayala Competition Model with Jumps
The nonautonomous stochastic Gilpin-Ayala competition model driven by Lévy noise is considered. First, it is shown that this model has a global positive solution. Then, we discuss the asymptotic behavior of the solution including moment and pathwise estimation. Finally, sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution are established.
Population dynamics that describes and understands changes in populations over time is an important subfield of ecology. One of the famous models for population dynamics is the traditional Lotka-Volterra competition system which has received a lot of attention and has been studied extensively because of its theoretical and practical significance. It was suggested by Lotka  and Volterra  in the 1920s and was described by where represents the population size of species at time , is the rate of growth at time , represents the effect of interspecies or intraspecies at time , and is the carrying capacity of the th species in the absence of other species at time . There is an extensive literature concerned with the dynamics of (1); see, for example, Gopalsamy , Kuang , Liao and Tang , and Xiao and Li . In particular, the books by Gopalsamy  and Kuang  are good references in this area. However, the Lotka-Volterra model has often been severely criticized. One of the criticisms is that in such a model the rate of change in the density of species is a linear function of densities of the interacting species. Hence, in order to yield significantly more accurate and realistic results, in 1973, Gilpin and Ayala  proposed a more complicated model as follows: where are the parameters to modify the classical Lotka-Volterra model. Since then, some related research has appeared in the literature . And He and his coauthors in  considered Gilpin-Ayala competition system with impulses; they investigated some properties, such as permanence, extinction, and global attraction, of the determined Gilpin-Ayala competition system. They mainly considered the impulses effects on the long behavior of the system.
On the other hand, population dynamics in the real world is inevitably affected by environment noise which is an important component in an ecosystem. So it is necessary and important to consider the corresponding stochastic population model. For the stochastic population models, we refer the reader to [10–13] mainly dealing with Lotka-Volterra model and [14–16] mainly dealing with stochastic Gilpin-Ayala models. Furthermore, the population may suffer from sudden environmental shocks, for example, earthquakes, hurricanes, epidemics, and so forth. However, the above mentioned stochastic models cannot explain such phenomena. To explain these phenomena, we introduce a jump process into the stochastic population dynamics. As far as we know, there are few papers considering the stochastic population models with jumps [10, 17, 18]. However, still no scholars investigated the nonautonomous stochastic Gilpin-Ayala competitive model with Poisson jumps or Lévy jumps. Zhang and Wang  discussed the asymptotic behavior of stochastic Gilpin-Ayala mutualism model with jumps: where the intrinsic exponential growth rate and the effects of interspecific (for ) and intraspecific (for ) interaction are constants. In this model, some asymptotic property was obtained, especially, and the explicit solution was obtained in one-dimension case. However, in  it assumed that all parameters are constants in time; they are naturally subject to fluctuation in time; if a model which takes into account such fluctuations is desired, it must be non-autonomous.
Hence, motivated by the above discussion, in this paper, we consider the nonautonomous stochastic Gilpin-Ayala competition models perturbed by Lévy noise; that is, where is a -dimensional Brownian motion defined on the filtered probability space. , is a real-valued Poisson counting measure with characteristic measure on a measurable subset of with , . is the intrinsic exponential growth rate for the th species at time . represent the effects of interspecific and intraspecific interaction at time ; and are nonnegative constants. Functions , , and are continuous, bounded, and nonnegative on ; are measurable functions.
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions; that is, it is increasing and right continuous while contains all -null sets, and let , and denotes the Euclidian norm in .
If is a continuous and bounded function on , then define For any constant sequence , define
The rest of the paper is arranged as follows. In Section 2, we prove that the solution of system (4) is global and positive and it does not explode at a finite time. In Section 3, we analyze the asymptotic moment behavior of the solution. The other different properties of the solution, asymptotic pathwise estimation, extinction, nonpersistence, and weak persistence, are obtained in Sections 4 and 5, respectively. In Section 6, we construct a numerical example with some figures to illustrate our main results. At last, we close the paper with conclusions in Section 7.
2. Global Positive Solutions
The solution of (4) denotes the population sizes of th species at time ; it is natural to require it to be positive. 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 the linear growth condition and local Lipschitz condition (see, e.g., ). However, the coefficients of system (4) are locally Lipschitz continuous but do not satisfy the linear growth condition, so the solution of system (4) may explode at a finite time. Under some simple hypothesis, the following theorem shows that this solution is positive and global.
In what follows we always denote by a generic constant whose values may vary for its different appearances. To show the main result let us recall the following facts.
Consider a -dimensional jump-diffusion process satisfying with initial condition , where and are measurable functions and is a -matrix-valued function. Here is -valued Brownian motion, and is a compensated Poisson random measure on which is independent of .
The following conclusion is given by [20, Lemma 2].
Lemma 1. Let , which is a zero matrix, and , , and satisfy locally Lipschitz condition. Set If then, for , where denotes the solution of (7) starting from at time .
Before stating the main results in this paper, we give the following assumptions for the jump-diffusion coefficient.
(H1) Assume that, for any , , , and , and for each there exists such that whenever with .
(H2) Assume further that for there exist constants and such that, for , , where as .
Theorem 2. Let the assumptions (H1) and (H2) hold. Then, for any given initial value , (4) has a unique global solution for any a.s.
Proof. Since, by (14), the coefficients of drift and diffusion are locally Lipschitz continuous, hence, for any given initial value , there is a unique local solution on , where is the explosion time. By (4) we can get that This, together with , yields that for any . On the other hand, due to (13), for (4) the condition (10) holds. Thus according to Lemma 1, we immediately get that, for any , a.s. Next, to show that this solution is global, we need to show that a.s. Let be sufficiently large for every component of such that . For each integer , define the stopping time where throughout this paper we set (as usual denotes the empty set). Clearly, is increasing as . Set , whence a.s. If we can show that a.s., then a.s., and a.s., for all . In other words, to complete the proof all we need to show is that a.s. To show this statement, we define nonnegative -function where . Let and be arbitrary. For any , we apply Itô’s formula to and obtain where, for , It is not difficult to estimate On the other hand, using (15) and knowing that for any , , we deduce Thus, for , Therefore Integrating the last inequality from to and then taking expectations yield For each , define Recalling the inequality (22), it is easy to see that Then we obtain that, for some constant , From this inequality, we immediately get . Since is arbitrary, we must have which completes the proof.
3. Asymptotic Moment Estimation
In the previous section, we show that the solution of (4) is positive and will not explode in any finite time. This nice positive property allows us to further discuss asymptotic moment estimation for the solution of (4).
Theorem 3. Let conditions of Theorem 2 hold. Then, for any initial data and , there exists a constant , such that
Proof. For any , let as in Theorem 2. For any , define a stopping time From the argument of Theorem 2, we have and a.s. as . Using Itô’s formula to implies where Obviously, for , there exists a positive constant such that , and we have Integrating 0 into and taking expectation give Hence, letting , we get and the desired conclusion follows.
If we note the inequality we claim that, for any , th moment of the solution of system (4) is also asymptotically bounded, which is stated in the following corollary.
In the following theorem, we prove that the average in time of the moment of the solution to (4) is bounded.
Theorem 5. Under the conditions of Theorem 2, for any and any initial value , consider the following.(1)If , then (2)If , then
Proof. For , using the same Lyapunov function and by (19) and (20), we get
According to (15) of the assumption (H2), we conclude that the term
has an upper positive bound (dependent on ). Therefore for Case ,
are two local martingales vanishing at . Taking expectations on both sides of (44) gives
Hence, dividing by on both sides and from inequality (38), Case follows immediately.
Case : if , we can also get that and taking expectations on both sides yields Hence, the assertion of Case follows as before.
4. Pathwise Estimation
In the previous section we have discussed how the solution of system (4) varies in in moment. In this section we examine pathwise properties of the solution of system (4); that is, we consider some limit inequalities for growth rates of the population size of species, which shows how the solution of system (4) varies in . First, we need the following exponential martingale inequality with jumps [21, Theorem ] to prove the main result in this part.
Lemma 6. Assume that and are both predictable -adapted processes such that for any Then for any constants where
Theorem 7. Let the conditions of Theorem 2 hold. Assume further that there exists a constant such that If or , then there exists , independent of initial value , such that the solution , , of (4) has the property
Proof. Let be the Lyapunov function; for convenience, we denote and . Applying Itô’s formula to yields
Integrating 0 into yields where are two local martingales. By virtue of Lemma 6, for any we have Choose , , and , where , , in the above equation. Since , using the Borel-Cantelli lemma yields that there exists an with such that for any an integer can be found such that for and . Thus (55) results in for and . For any and , compute that When or , we obtain Noting the inequality , , we have In the light of Taylor’s series expansion, for , , , and , where lies between 0 and . Thus For and , we have a decomposition for as For and , we have . Hence On the other hand, recalling the fundamental inequality and observing for and , we have Consequently, for and Hence, substituting (60), (62), and (69) into (59), we arrive at Moreover, by the condition (52), we can deduce that for and Thus, combining (61), (62), and (69) with (71), when or , there exists a constant such that Hence, for any , , and , In particular, for and , which implies that The desired assertion then follows by letting and .
Corollary 8. Under conditions of Theorem 7, one has
5. Extinction and Nonpersistence
One of the important problems in population biology is whether all species of one system will become extinct over time. There are many factors including natural and man-made factors which may cause the extinction of some species, such as low birth rate, high death rate, decreasing habitats, and aggravating living environment. Sometimes, small even large population of some species may be destroyed by some extraordinary perturbation. To proceed, we need some appropriate definitions of extinction and persistence. Hallam and Ma proposed the concepts of weak persistence , weak persistence in the mean , and strong persistence in the mean  for some deterministic models. Liu and Wang [25, 26] applied thes concepts to stochastic logistic models. In this section, we show, that if the noise is sufficiently large, the solution of the system (4) will become extinct with probability one, nonpersistent in time average, or weakly persistent.
Definition 9. Stochastic population dynamics (4) is said to be extinct with probability 1 if, for every initial datum , the solution , , has the property
Definition 10. The population is said to be nonpersistent in time average with probability 1 if a.s.
Definition 11. The population is said to be weakly persistent with probability 1 if a.s.
Theorem 12. Let conditions of Theorem 2 hold. If for some , , and hold, then th population will become extinct with probability 1.
Proof. Applying Itô’s formula to and integrating 0 into yield
are the local martingales with the quadratic variations
Since , by the strong law of large numbers for martingales, we have that a.s.
On the other hand, for every integer , by using the exponential martingale inequality, we have It then follows from the Borel-Cantelli lemma that for almost all , there is a random integer such that, for , Substituting this inequality into (79) gives Consequently for , , Taking superior limit on both sides and using (78), one can see that Therefore a.s.; that is, th population will become extinct with probability 1.
6. Examples and Simulations
In this section we provide a numerical example to substantiate the analytical findings for the stochastic model system reported in the previous sections.
Example 1. Consider the two-species nonautonomous stochastic Gilpin-Ayala system with jumps where and represent the population sizes of species.