Dynamics of two competing species in the presence of Lévy noise sources

We consider a Lotka-Volterra system of two competing species subject to multiplicative $\alpha$-stable Lévy noise. The interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence both of a periodic driving term and an additive $\alpha$-stable Lévy noise. We study the species dynamics, which is characterized by two different regimes, exclusion of one species and coexistence of both. We find quasiperiodic oscillations and stochastic resonance phenomenon in the dynamics of the competing species, analyzing the role of the Lévy noise sources.

Figure 1

(Color online) Two-dimensional trajectories of free diffusion of a particle subjected to noise sources with (a) Gaussian $\left(\alpha =2,\beta =0\right)$; (b) Lévy $\left(\alpha =1.5,\beta =0\right)$; (c) Cauchy-Lorentz $\left(\alpha =1,\beta =0\right)$; (d) Lévy-Smirnov $\left(\alpha =0.5,\beta =1\right)$ distributions. The values of the other parameters are $\mu =0$, $\sigma =1$. Arbitrary units are used in both axes.

Figure 2

(Color online) Probability density function, $P\left(X\right)$, of stable Lévy distribution for fixed values of $\mu$ and $\sigma$, and different values of $\alpha$ and $\beta$. Namely, (a) $\sigma =1$, $\mu =0$, $\beta =0$, and $\alpha =0.5,1.0,1.5$; (b) $\sigma =1$, $\mu =0$, $\alpha =1$, and $\beta =0.2,0.6,1.0$

Figure 3

(Color online) Three configurations of the time dependent bistable potential $U\left(\gamma ,t\right)$ of Eq. (11) at times $t=0,0.5,1$. The values of the potential parameters are $h=8$, $\eta =0.5$, $A=2h$, ${\omega }_0=\pi$.

Figure 4

(Color online) Time evolution of the interaction parameter $\gamma \left(t\right)$ (blue line), by numerical integration of Eq. (10) with zero noise intensity $D_{\gamma }$. The gray line indicates the threshold value $\gamma =1$: coexistence of both species $\left(\gamma <1\right)$ and exclusion of one species $\left(\gamma >1\right)$.

Figure 5

(Color online) Time evolution of the parameter $\gamma \left(t\right)$, by numerical integration of Eq. (10) with noise intensity $D_{\gamma }=0.5$ and $\beta =0$, for five values of the index $\alpha$, namely, (a) $\alpha =1.9$, (b) $\alpha =1.8$, (c) $\alpha =1.7$, (d) $\alpha =1.6$ (blue lines), and $\alpha =2.0$ (red points in all panels).

Figure 6

(Color online) Time series of the parameter $\gamma \left(t\right)$ [from Eq. (10)], for suitable noise intensities $\left(D_{\gamma }=5,5.5,6\right)$ and different values of the Lévy index $\alpha$, namely, $\alpha =1.9,1.8,1.7,1.6$ (blue lines), compared with the Gaussian case $\left(\alpha =2.0\right)$ (red points). The asymmetry parameter is $\beta =0$.

Figure 7

(Color online) Time series of the parameter $\gamma \left(t\right)$ [from Eq. (10)], for noise intensity $D_{\gamma }=7.5$ and different values of the Lévy index $\alpha$, namely, $\alpha =1.9,1.8,1.7,1.6$ (blue lines), compared with the Gaussian case $\left(\alpha =2.0\right)$ (red points). The asymmetry parameter is $\beta =0$.

Figure 8

(Color online) Signal-to-noise ratio $R$ of $\gamma \left(t\right)$ as a function of the noise intensity $D_{\gamma }$, for different values of the index $\alpha$ of the stable Lévy distribution, namely, $\alpha =1.6,1.7,1.8,1.9,2.0$.

Figure 9

(Color online) Spectral power amplification $\eta$ of $\gamma \left(t\right)$ as a function of the noise intensity $D_{\gamma }$, for different values of the index $\alpha$ of the stable Lévy distribution, namely, $\alpha =1.6,1.7,1.8,1.9,2.0$.

Figure 10

(Color online) Panels (a), (b), (c): time series of the two species densities $x$ (red points) and $y$ (blue lines) obtained from Eqs. (8, 9) with $\alpha =1.9$, $\beta =0$, $D_{\gamma }=5$ and (a) $D={10}^{-13}$, (b) $D=5\times {10}^{-6}$, (c) $D=0.5$. Panel (d): SNR of ${\left(x-y\right)}^2$ as a function of the noise intensity $D$.

Figure 11

(Color online) Panels (a), (b), (c): time series of the two species densities $x$ (red points) and $y$ (blue lines) obtained from Eqs. (8, 9) with $\alpha =1.8$, $\beta =0$, $D_{\gamma }=5.3$ and (a) $D={10}^{-14}$, (b) $D=5\times {10}^{-5}$, (c) $D=1$. Panel (d): SNR of ${\left(x-y\right)}^2$ as a function of the noise intensity $D$.

Figure 12

(Color online) Panels (a), (b), (c): time series of the two species $x$ (red points) and $y$ (blue lines) obtained from Eqs. (8, 9) with $\alpha =1.7$, $\beta =0$, $D_{\gamma }=5.5$, and (a) $D={10}^{-10}$, (b) $D={10}^{-4}$, (c) $D=0.5$. Panel (d): SNR of ${\left(x-y\right)}^2$ as a function of the noise intensity $D$.

Figure 13

(Color online) Time series of the two species $x$ (red points) and $y$ (blue lines) obtained from Eqs. (8, 9) with $\alpha =1.6$, $\beta =0$, $D_{\gamma }=5.7$ and $D=5\times {10}^{-13}$, (b) $D={10}^{-5}$, (c) $D=0.5$. Panel (d): SNR of ${\left(x-y\right)}^2$ as a function of the noise intensity $D$.

Figure 14

(Color online) Signal-to-noise ratio $R$ of ${\left(x-y\right)}^2$ as a function of the noise intensity $D$ for different values of the index $\alpha$.