Role of noise in population dynamics cycles

Noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase-forgetting quasicycles by converting damped into sustained oscillations. This function of noise becomes evident when considering Langevin equations whose deterministic part yields only damped oscillations. We formulate here a consistent and systematic approach to population dynamics, leading to a Fokker-Planck equation and the associate Langevin equations in accordance with this conceptual framework, founded on stochastic lattice-gas models that describe spatially structured predator-prey systems. Langevin equations in the population densities and predator-prey pair density are derived in two stages. First, a birth-and-death stochastic process in the space of prey and predator numbers and predator-prey pair number is obtained by a contraction method that reduces the degrees of freedom. Second, a van Kampen expansion in the inverse of system size is then performed to get the Fokker-Planck equation. We also study the time correlation function, the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence.

Figure 1

Transitions of the birth-and-death stochastic process in the space of prey and predator numbers. The transition to the east represents a prey birth with rate $A_{+0}$, to the northwest a prey death and a simultaneous predator birth with rate $A_{-+}$, to the south a predator death with rate $A_{0-}$. The full circles represent absorbing states.

Figure 2

(Color online) Simulations of the birth-and-death stochastic process for the case $a=b=0.475$ and $c=0.05$ and $N=1000$ showing oscillations of the type phase-forgetting quasicycles. (a) Prey and predator densities as functions of time (with arbitrary origin). (b) Time correlation functions against time lag.

Figure 3

Real eigenvalues ($-{\gamma }_1$ and $-{\gamma }_2$), real $\left(-\alpha \right)$ and imaginary $\left(\omega \right)$ parts of the complex eigenvalue as determined by the simple mean-field approximation. The vertical dotted line at $c^{\ast }=2/7=0.2857$ indicates the transition from oscillatory to ordinary behavior.

Figure 4

Real eigenvalues ($-{\gamma }_1$, $-{\gamma }_2$, and $-{\gamma }_3$), real $\left(-\alpha \right)$ and imaginary $\left(\omega \right)$ parts of the complex eigenvalue as determined by the mean-field approximation for the case $a=b$. The vertical dotted line at $c^{\ast }=0.1143$ indicates the transition from oscillatory to ordinary behavior.