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Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment. (English) Zbl 1113.34042

Consider random 2D Lotka-Volterra systems

$\stackrel{˙}{x}=x\left(a\left({\xi }_{t}\right)-b\left({\xi }_{t}\right)y\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=y\left(-c\left({\xi }_{t}\right)+d\left({\xi }_{t}\right)x\right)$

for the size $x$ of prey and $y$ of predators, where $a,b,c,d:E\to {ℝ}_{+}\setminus \left\{0\right\}$ and the Markov process $\xi$ has two states $E=\left\{1,2\right\}$ only. Suppose that $\xi$ jumps at times which have conditionally independent, exponentially distributed increments. It is proved that, under the influence of so-called telegraph noise $\xi$, all positive trajectories of such a system always go out from any compact set of ${ℝ}_{+}^{2}$ 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 ${ℝ}_{+}^{2}$ or converges to the rest point. The escape of the trajectories from any compact set means ecologically that the system is neither permanent nor dissipative.

The authors extend investigations of Slatkin (1978) in 1D to 2D, known from ecology. Some simulation results are presented as well. For the probabilistic proofs, they use well-known techniques of 0-1 laws.

##### MSC:
 34F05 ODE with randomness 37H10 Generation, random and stochastic difference and differential equations 60H10 Stochastic ordinary differential equations 92D25 Population dynamics (general) 92D40 Ecology