Colored-noise-induced Hopf bifurcations in predator-prey communities

A broad class of $(N+1)$-species ratio-dependent predator-prey stochastic models, which consist of one predator population and $N$ prey populations, is considered. The effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise. In the framework of the mean-field theory, approximate self-consistency equations for prey-populations mean density and for predator-population density are derived (to the first order in the noise variance). In some cases, the mean field exhibits Hopf bifurcations as a function of noise correlation time. The corresponding transitions are found to be reentrant, e.g., the periodic orbit appears above a critical value of the noise correlation time, but disappears again at a higher value of the noise correlation time. The nonmonotonous dependence of the critical control parameter on the noise correlation time is found, and the conditions for the occurrence of Hopf bifurcations are presented. Our results provide a possible scenario for environmental-fluctuations-induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature.

Figure 1

The emergence of the Hopf bifurcation (and reentrant transition) to noise correlation time variations [$\Delta y≔y\left(t\right)-y_s$, $\Delta x≔x\left(t\right)-x_s$]. The orbits are computed by means of Eqs. (8, 19, A4), at the parameter values $c=b=\gamma =1$, $\beta =0.00078$, $d=0.0005$, $\alpha =1.696216$, $a^2=0.01$, and ${\sigma }^2=\chi =0$. (a) Stable focus at the noise switching rate $\nu =0.01$. Note that $\alpha <{\alpha }_{\mathit{cr}}\left(\nu \right)=1.697688$. (b) The stable limit cycle (the closed middle curve in the figure) at a moderate value of the switching rate $\nu =0.5$, ${\alpha }_{\mathit{cr}}\left(\nu \right)<\alpha <{\alpha }_{\mathit{cr}0}$. (c) The stable equlibrium point and orbit computed at the switching rate $\nu =10$. The initial conditions are: $x_0=0.391250$ and $y_0=0.219101$. The corresponding equilibrium point is determined by $x_s=0.391249$ and $y_s=0.219000$, while the condition $\alpha <{\alpha }_{\mathit{cr}}\left(\nu \right)=1.697609$ is fulfilled. For more details of panels (a), (b), and (c) see the text.

Figure 2

The dependence of the critical capturing rate ${\alpha }_{\mathit{cr}}$, $\Delta {\alpha }_{\mathit{cr}}≔{\alpha }_{\mathit{cr}}-{\alpha }_{\mathit{cr}0}$, on the noise switching rate $\nu$, obtained from Eqs. (13, 14). The noise variance $a^2=0.01$, the parameter $c=1$, and ${\alpha }_{\mathit{cr}0}$ is computed from Eq. (18). (a) The case of $\omega <2\eta$: $d=0.5$, $\beta =0.78$, $\omega =0.179487$, and $G\left(r\right)=0.56$. The critical capturing rate by absence of noise ${\alpha }_{\mathit{cr}0}=2.002230$. (b) The case of $\omega >2\eta$: $d=0.1$, $\beta =0.92$, $\omega =0.0891$, $G\left(r\right)=8.2$, and ${\alpha }_{\mathit{cr}0}=1.102150$. Note the nonmonotonous dependence of ${\alpha }_{\mathit{cr}}$ on $\nu$.

Figure 3

(a) Stable limit cycle around the equilibrium point, $\Delta x=x\left(t\right)-x_s$, $\Delta y=y\left(t\right)-y_s$, $x_s=0.282641$, $y_s=0.158279$, for the system (8) with Eqs. (19, A4) at $a^2=0.01$, $d=0.5$, $\beta =0.78$, $b=\gamma =c=1$, $\alpha =2.00120$, and $\nu =0.5$; $\Delta x≔x\left(t\right)-x_s$, $\Delta y≔y\left(t\right)-y_s$. The necessary conditions for the existence of a noise-induced limit cycle ${\alpha }_{\mathit{cr}}\left(\nu \right)<\alpha <{\alpha }_{\mathit{cr}0}$ are satisfied: ${\alpha }_{\mathit{cr}}\left(\nu \right)=2.00096$ and ${\alpha }_{\mathit{cr}0}=2.00223$. (b) Time evolution of mean prey density $x$ (upper curve) and predator density $y$ (lower curve) obtained by solving Eqs. (8, 19) for the limit cycle. The period of oscillations $T\approx 28.05$. Note that the predator-prey cycles exhibit a delay of $\Delta t=4.14$ between prey and predator maxima. The time is dimensionless as scaled by Eq. (7).

Figure 4

Typical dependence of $\Delta \alpha ={\alpha }_{\mathit{cr}0}-{\alpha }_{\mathit{cr}}\left({\nu }_m\right)$ on the exponent $c$ [See Eq. (2)] computed from Eqs. (13, 14, 18), while ${\alpha }_{\mathit{cr}}\left({\nu }_m\right)$ is the minimal value of critical capturing rate as $\nu$ is varied (cf. Fig. 2). Parameter values $a^2=0.01$, $d=0.5$, $\beta =0.78$, $\omega =0.179487$, and $G\left(r\right)=0.56$ are used.