Colored-noise-induced discontinuous transitions in symbiotic ecosystems

A symbiotic ecosystem is studied by means of the Lotka-Volterra stochastic model, using the generalized Verhulst self-regulation. The effect of fluctuating environment on the carrying capacity of a population is taken into account as dichotomous noise. The study is a follow-up of our investigation of symbiotic ecosystems subjected to three-level (trichotomous) noise [R. Mankin, A. Ainsaar, A. Haljas, and E. Reiter, Phys. Rev. E 65, 051108 (2002)]. Relying on the mean-field theory, an exact self-consistency equation for stationary states is derived. In some cases the mean field exhibits hysteresis as a function of noise parameters. It is established that random interactions with the environment can cause discontinuous transitions. The dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams. Predictions from the mean-field theory are compared with the results of numerical simulations. Our results provide a possible scenario for catastrophic shifts of population sizes observed in nature.

Figure 1

Solutions of the self-consistency equation (20) at different parameters $\beta$. The amplitude parameter $a^2=0.995$ and the straight line is determined by $\varrho =2,\kappa =0.8$. The mean-field solutions are given by the intersection points of different curves with a straight line. Two typical cases are shown: there is just one stable solution (full circles on curves $\beta =1.8$ and $\beta =3$), or two stable solutions and one unstable solution (empty circle on curve $\beta =2$).

Figure 2

The critical noise amplitude parameter $a_{0c}^2$ vs the system parameter $\beta$. In the case of large values of $\beta$ the critical noise amplitude saturates up to the value $a_{0c}^2\approx 0.9248$.

Figure 3

Stationary mean field $m$ vs coupling strength $J$ at different correlation times ${\tau }_c$. The system parameter $\beta =2$ and the noise amplitude $a_0=0.99$. In the case of ${\tau }_c=0.5$ the system shows hysteresis. Solid and dashed lines are stable and unstable solutions of the self-consistency equation (20), respectively. If the value of the mean field lies on the upper branch close to the point $F$ with $J=J_2$, a slight growth of $J$ induces a catastrophic transition of the system to another stable state with the value of the mean field corresponding to the point G.

Figure 4

A plot of the phase diagram in the $J–{\tau }_c$ plane at $a_0^2=0.980,\beta =2$. The shaded region in the figure corresponds to the coexistence region of two phases. The critical correlation time ${\tau }_c^{*}=0.655$. The borders of the coexistence region $J_1\left({\tau }_c\right)$ and $J_2\left({\tau }_c\right)$ are computed from Eq. (23). The dotted line $J_3$ corresponding to Eq. (24) separates the domains of the phase space, where the shapes of the probability density $P(x,r)$ are qualitatively different.

Figure 5

The critical correlation time ${\tau }_c^{*}$ vs the noise amplitude $a_0$ for some values of the system parameter $\beta$. The filled dots on the $a_0^2$ axis correspond to values of the critical noise amplitude $a_{0c}\left(\beta \right):a_{0c}^2\left(1.1\right)=0.2828,a_{0c}^2\left(1.5\right)=0.6208,a_{0c}^2\left(2\right)=0.7148$, and $a_{0c}^2\left(3\right)=0.7686$. At the point $a_0=1$ the critical correlation time approaches infinity.

Figure 6

Plot of the mean value of the population density $⟨X⟩$ as a function of the noise correlation time ${\tau }_c$ at $a_0^2=0.939,\beta =2$, and $J=6$. The solid and dashed lines correspond to the stable and unstable solutions of Eq. (20), respectively. Hysteresis of the mean value $⟨X⟩$ appears. The squares are obtained by means of computer simulations of the system (12) with $N=500$.