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

Consider random 2D Lotka-Volterra systems
\[ \dot{x} = x (a(\xi_t)-b(\xi_t)y) , \quad \dot{y} = y (-c(\xi_t)+d(\xi_t)x) \]
for the size \(x\) of prey and \(y\) of predators, where \(a,b,c,d : E \to \mathbb R_+ \setminus \{0\}\) and the Markov process \(\xi\) has two states \(E=\{1,2\}\) 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 \(\mathbb R_{+}^{2}\) 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 \(\mathbb R_{+}^{2}\) 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.


34F05 Ordinary differential equations and systems with randomness
37H10 Generation, random and stochastic difference and differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
92D40 Ecology
Full Text: DOI


[1] Ballys, M.; Le Dung; Jones, D.A.; Smith, H.L., Effects of random mortality on microbial growth and competition in a flow reactor, SIAM J. appl. math., 57, 2, 573-596, (1998), 374-402 · Zbl 1051.92039
[2] Chesson, P.L.; Warner, R.R., Environmental variability promotes coexistence in lottery competitive systems, Amer. natur., 117, 923-943, (1981)
[3] Du, N.H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behavior of lotka – volterra competition systems: nonautonomous bistable case and the effect of telegraph noise, J. comput. appl. math., 170, 399-422, (2004) · Zbl 1089.34047
[4] Du, N.H.; Kon, R.; Sato, K.; Takeuchi, Y., Evolution of periodic population systems under random environment, Tohoku math. J., 57, 447-468, (2005) · Zbl 1117.34052
[5] Farkas, M., Periodic motions, (1994), Springer-Verlag New York · Zbl 0805.34037
[6] Gihman, I.I.; Skorohod, A.V., The theory of stochastic processes, (1979), Springer-Verlag Berlin · Zbl 0404.60061
[7] Hanski, I.; Turchin, P.; Korpimäki, E.; Henttonen, H., Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos, Nature, 364, 232-235, (1994)
[8] Hofbauer, J.; Sigmund, K., Evolutionary game and population dynamics, (1998), Cambridge Univ. Press Cambridge · Zbl 0914.90287
[9] Gilpin, M.E., Predator – prey communities, (1975), Princeton Univ. Press
[10] Levin, A., Dispersion and population interactions, Amer. nature, 108, 207-228, (1974)
[11] Loreau, M., Coexistence of temporally segregated competitors in a cyclic environment, Theoret. population biol., 36, 181-201, (1989) · Zbl 0683.92019
[12] Mao, X.; Sabais, S.; Renshaw, E., Asymptotic behavior of stochastic lotka – volterra model, J. math. anal., 287, 141-156, (2003)
[13] Namba, T.; Takahashi, S., Competitive coexistence in a seasonally fluctuating environment: II. multiple stable states and invasion success, Theoret. population biol., 44, 374-402, (1993) · Zbl 0791.92024
[14] Randall, J.S., A stochastic predator – prey model, Irish math. soc. bull., 48, 57-63, (2002) · Zbl 1267.60084
[15] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256, (1978)
[16] Takeuchi, Y., Global dynamical properties of lotka – volterra systems, (1996), World Scientific · Zbl 0844.34006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.