Stochastic Delay Lotka-Volterra Model
© Lian Baosheng et al. 2011
Received: 15 October 2010
Accepted: 20 January 2011
Published: 7 February 2011
This paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process, namely the delay Lotka-Volterra model. The stochastic version of this process appears to have some intriguing properties such as pathwise estimation and asymptotic moment estimation. Indeed, their solutions will be stochastically ultimately bounded.
As is well known, Lotka-Volterra Model is nonlinear and tractable models of predator-prey system. The predator-prey system is also studied in many papers. In the last few years, Mao et al. change the deterministic model in this field into the stochastic delay model. and give it more important properties [1–8].
Fluctuations play an important role for the self-organization of nonlinear systems; we will study their influence on a simple nonlinear model of interacting populations, that is, the Lotka-Volterra model. A simple analysis shows the result that the system allows extreme behaviour, leading to the extinction of both of their species or to the extinction of the predator and explosion of the prey. For example, in Mao et al. [1–8], we can see that once the population dynamics are corporate into the deterministic subclasses of the delay Lotka-Voterra model, the stochastic model will bear more attractive properties: the solutions will be be stochastically ultimately bounded, and their pathwise estimation and asymptotic moment estimation will be well done.
It is clear that the upper inequalities are the key conditions in the stochastic R&D model in economic growth model.
We use the ordinary result of the polynomial functions.
2. Positive and Global Solutions
Let be a complete probability space with filtration satisfying the usual conditions, that is, it is increasing and right continuous while contains all -null sets . Moreover, let be an -dimensional Brownian motion defined on the filtered space and . Finally, denote the trace norm of a matrix by (where denotes the transpose of a vector or matrix ) and its operator norm by . Moreover, let and denote by the family of continuous functions from to .
The coefficients of (1.1) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of (1.1) may explode at a finite time.
let us emphasize the important feature of this theorem. It is well known that a deterministic equation may explode to infinity at a finite time for some system parameters and . However, the explosion will no longer happen as long as conditions (1.2) and (1.3) hold. In other words, this result reveals the important property that conditions (1.2) and (1.3) suppress the explosion for the equation. The following theorem shows that this solution is positive and global.
3. Stochastically Ultimate Boundedness
Theorem 2.1 shows that under simple hypothesis conditions (1.2), (1.3), and (2.1), the solutions of (1.1) will remain in the positive cone . This nice positive property provides us with a great opportunity to construct other types of Lyapunov functions to discuss how the solutions vary in in more detail.
As mentioned in Section 2, the nonexplosion property in a population dynamical system is often not good enough but the property of ultimate boundedness is more desired. Let us now give the definition of stochastically ultimate boundedness.
4. Asymptotic Pathwise Estimation
5. Further Topic
The explanations in population dynamic of the conditions (1.2), (1.3), and (2.1) for (1.1) are worth pointing out. Each species has a special ability to inhibit the fast growth; the relationship of the species is the role of either species competition ( ), or a low level of cooperation ( , but they are small enough).
This paper is supported by the National Natural Science Foundation of China (10901126), research direction: Theory and Applications of Stochastic Differential Equations.
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