Abstract

We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka--Volterra system for an exponential random time after which it switches to a different deterministic Lotka--Volterra system. This switching procedure is then repeated. The resulting process is a piecewise deterministic Markov process (PDMP). In the case when the equilibrium points of the two deterministic Lotka--Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between 0 and $\infty$. This proves a conjecture of Takeuchi, Du, Hieu, and Sato [J. Math. Anal. Appl., 323 (2006), pp. 938--957]. The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume $(Y_t, I_t)$ is a PDMP that evolves according to $\frac{dY_t}{dt}=A_{I_t} Y_t$, where $A_0,A_1$ are $2\times2$ matrices and $I_t$ is a Markov chain on $\{0,1\}$ with transition rates $k_0,k_1>0$. If the matrices $A_0$ and $A_1$ are not proportional and are of the form $ A_i := (\begin{smallmatrix} \alpha_i & \beta_i \\ \gamma_i & -\alpha_i \end{smallmatrix}) $ with $\alpha_i^2 + \beta_i \gamma_i < 0$, then there exists $\lambda >0$ such that $\lim_{t \to \infty} \frac{\log \| Y_t \|}{t} = \lambda.$

Keywords

  1. piecewise deterministic Markov processes
  2. random switching
  3. population dynamics
  4. Lyapunov exponents
  5. Lotka--Volterra
  6. telegraph noise

MSC codes

  1. 60J99
  2. 34F05
  3. 37H15
  4. 37A50
  5. 92D25

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3625 - 3640
ISSN (online): 1095-7154

History

Submitted: 22 June 2018
Accepted: 8 July 2019
Published online: 5 September 2019

Keywords

  1. piecewise deterministic Markov processes
  2. random switching
  3. population dynamics
  4. Lyapunov exponents
  5. Lotka--Volterra
  6. telegraph noise

MSC codes

  1. 60J99
  2. 34F05
  3. 37H15
  4. 37A50
  5. 92D25

Authors

Affiliations

Funding Information

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung https://doi.org/10.13039/501100001711 : 2000020-149871/1

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