Ecological communities from random generalized Lotka-Volterra dynamics with nonlinear feedback

We investigate the outcome of generalized Lotka-Volterra dynamics of ecological communities with random interaction coefficients and nonlinear feedback. We show in simulations that the saturation of nonlinear feedback stabilizes the dynamics. This is confirmed in an analytical generating-functional approach to generalized Lotka-Volterra equations with piecewise linear saturating response. For such systems we are able to derive self-consistent relations governing the stable fixed-point phase and to carry out a linear stability analysis to predict the onset of unstable behavior. We investigate in detail the combined effects of the mean, variance, and covariance of the random interaction coefficients, and the saturation value of the nonlinear response. We find that stability and diversity increases with the introduction of nonlinear feedback, where decreasing the saturation value has a similar effect to decreasing the covariance. We also find cooperation to no longer have a detrimental effect on stability with nonlinear feedback, and the order parameters mean abundance and diversity to be less dependent on the symmetry of interactions with stronger saturation.

Figure 1

Phase diagram obtained from simulations of the generalized Lotka-Volterra system with $N=200$. Panel (a) shows the case $a\to \infty$ (i.e., linear feedback), panels (b) and (c) are for nonlinear feedback, with $a=2$ in (b), and $a=0.5$ in (c). Simulations are for $\mu =0$. The colors indicate the dominant outcome in each part of parameter space, with red (medium gray) representing a unique stable fixed point, green (light gray) multiple fixed points, blue (dark gray) indicating parameters for which the dynamics do not converge, and white indicating unbounded growth of species abundances. The solid line in panel (a) describes the onset of instability as derived from theory (see Refs. [24, 25, 43]).

Figure 2

Species abundance distribution for the case $a=0.5$ and $\mu =0$ for different values of the heterogeneity parameter $\sigma$. The upper row [panels (a–d)] is for $\gamma =-1$, the middle row [(e–h)] for $\gamma =0$, and the lower row [panels (i–l)] for $\gamma =1$. On the left [panels (a, e, i)] we show $z_2$ (upper line) and $z_1$ (lower line), remaining panels show the species abundance distributions, for $\sigma ={10}^{-1}$ [panels (b, f, j)], $\sigma ={10}^{-0.5}$ [panels (c, g, k)], and $\sigma =1$ [panels (d, h, l)]. Solid lines are theoretical predictions for the species abundance distribution, shaded histograms are from simulations.

Figure 3

Comparison of theoretical predictions (lines) for the characteristic order parameters against simulations (markers). Data is for $\mu =0$, $\gamma =-1(•),-0.5\left(\blacksquare \right),0\left(★\right),0.5(\times )$, and $\gamma =1(+)$. This is for the model with piecewise linear feedback. Left-hand column [panels (a, c, e)] is for $a=2$, right-hand column [panels (b, d, f)] for $a=0.5$. The vertical dashed lines mark the onset of instability as predicted by the theory; analytical predictions can no longer be expected to match with simulations in the phase to the right of the dashed lines. The graphs show the fraction of species not saturated $\varphi$, the mean abundance $M^{*}$, and the diversity $M^2/q$ as functions of $\sigma$.

Figure 4

Onset of instability for the model with piecewise linear feedback. Left-hand column [panels (a, c)] is for $a=2$, right-hand column [panels (b, d)] for $a=0.5$. The vertical dashed lines mark the onset of instability as predicted by the theory. The graphs show $d$ (top) and $h$ (bottom). Data is for $\mu =0$, and $\gamma =-1(•),-0.5\left(\blacksquare \right),0\left(★\right),0.5(\times ),1(+)$.

Figure 5

Phase diagram obtained from simulations of the generalized Lotka-Volterra systems with piecewise linear feedback with $N=200$. As in Fig. 1 the colors (gray shading) indicate the dominant outcome in each part of parameter space [red (medium gray): unique stable fixed point; green (light gray): multiple fixed points; blue (dark gray): dynamics do not converge]. Solid black lines show the onset of instability as predicted from the generating-functional approach. Data is for $\mu =0$, panel (a) for $a=2$, panel (b) for $a=0.5$.

Figure 6

Onset of instability for nonlinear feedback. Left-hand column [panels (a, c)] is for $a=2$, right-hand column [panels (b, d)] for $a=0.5$. The vertical dashed lines mark the onset of instability predicted for the piecewise feedback function from theory. The graphs show $d$ (top) and $h$ (bottom). Data is for $\mu =0$, $\gamma =-1(•),-0.5\left(\blacksquare \right),0\left(★\right),0.5(\times ),1(+)$.

Figure 7

Comparison of theoretical predictions for $M^{*}$ and diversity for the piecewise feedback (lines) against simulations for nonlinear feedback (markers). Left-hand column [panels (a, c)] is for $a=2$, right-hand column [panels (b, d)] for $a=0.5$. Data is for $\mu =0$, $\gamma =-1(•),-0.5\left(\blacksquare \right),0\left(★\right),0.5(\times ),1(+)$.

Figure 8

Critical value of the heterogeneity parameter, ${\sigma }_c$, plotted as a color map in the $a-\gamma$ plane at fixed $\mu =0$. Higher values of ${\sigma }_c$ indicate higher stability. The dashed line indicates $a=1$ (saturation parameter equal to carrying capacity); see text for further discussion.

Figure 9

Critical ${\sigma }_c$ plotted as a color map in the $\mu -\gamma$ plane at fixed $a=2$ (left) and $a=0.5$ (right). Higher values of ${\sigma }_c$ indicate higher stability. The data along the dashed line indicates values of ${\sigma }_c$ for $\mu =0$, these are as previously given in the black lines of Fig. 5.

Figure 10

Location of the onset of instability ${\sigma }_c$, and fraction of unsaturated species, $\varphi$, for varying $\mu$ at $a=2$ (left) an $a=0.5$ (right). Results in the figure are from the theory; the different values for $\gamma$ are indicated by different symbols $[\gamma =-0.5(\blacksquare ),0(★),0.5(\times ),1(+\left)\right]$.