Extinction in the Lotka-Volterra model

Birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey interaction. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.

Figure 1

Orbits of constant $G=\left(0.01,0.1,0.4,1,1.7,2.7,4.2\right)$ in units of $\sqrt{\sigma \mu }$. The evolution proceeds clockwise around the mean-field fixed point of $N_1=N_2=100$.

Figure 2

Typical run of the stochastic simulation of the model (1) for $N=100$ and $ϵ=1$.

Figure 3

Extinction probability in time $t$ from ${10}^5$ simulation trials ($N=100$, $ϵ=1$). Time is in units of $1/\sqrt{\sigma \mu }$.

Figure 4

Logarithm of the survival probability at long times for the data presented in Fig. 3.

Figure 5

Time constant ${\tau }_l$ of exponential decay, Eq. (5), versus $N$ for $ϵ=1$.

Figure 6

Extinction probabilities for $ϵ=2$ and $ϵ=1/2$; $N=100$.

Figure 7

Plot of $E_0$ vs $\text{ln}ϵ$; $N=100$.

Figure 8

Logarithm of the extinction probability at short times for the data of Fig. 3.

Figure 9

Orbits of constant $G$ in the new coordinates

Figure 10

$X\left(G\right)$ for $ϵ=1$. At $G\to \infty$ the function converges to $X_0=2.39$

Figure 11

Numerically calculated $\sqrt{T\left(X\right)D\left(X\right)}$ for $ϵ=1$. The function diverges at $X=X_0=2.39$

Figure 12

$E_0$ numerically calculated via matrix diagonalization vs number of rows in matrix

Figure 13

Function $X_0^{-1}\left(ϵ\right)$.

Figure 14

Function $E_0\left(ϵ\right)$; dots—results of stochastic simulation, full line—operator $\widehat{H}$ diagonalization.

Figure 15

Extinction probability of the two models: crosses $N=100$, $ϵ=5$; triangles $N=400$, $ϵ=10$. In both cases ${\tau }_l\approx 4.0$, while ${\tau }_s\approx 8.8$.

Figure 16

Numerically calculated $D\left(G\right)$ for $ϵ=1$ fit with analytically predicted $D\left(G\right)$.

Figure 17

Numerically calculated $T\left(G\right)$ for $ϵ=1$ fit with analytically predicted $T\left(G\right)$.