Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment

• Published in: Journal of mathematical analysis and applications 2006-11, Vol.323 (2), p.938-957
• Main Authors: Takeuchi, Y., Du, N.H., Hieu, N.T., Sato, K.
• Format: Article
• Language: English
• Subjects: Biological and medical sciences; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Lotka–Volterra equation; Mathematical analysis; Mathematics; Mathematics in biology. Statistical analysis. Models. Metrology. Data processing in biology (general aspects); Ordinary differential equations; Predator–prey model; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic analysis; Telegraph noise
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.11.009

In this paper, we consider the evolution of a system composed of two predator–prey deterministic systems described by Lotka–Volterra equations in random environment. It is proved that under the influence of telegraph noise, all positive trajectories of such a system always go out from any compact set of int R + 2 with probability one if two rest points of the two systems do not coincide. In case where they have the rest point in common, the trajectory either leaves from any compact set of int R + 2 or converges to the rest point. The escape of the trajectories from any compact set means that the system is neither permanent nor dissipative.