Effects of demographic stochasticity on biological community assembly on evolutionary time scales

We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the noise consist of species strongly coupled with each other and have stronger linear stability around the fixed-point populations than the corresponding noiseless model. The dynamics on evolutionary time scales for the predator-prey model are also altered by the noise. Approximate $1/f$ fluctuations are observed with noise, while $1/f^2$ fluctuations are found for the model without demographic noise.

Figure 1

(Color online) Typical time series of the exponential Shannon-Wiener diversity and the total population size plotted every 8000 generations for model A with $\mu =0.001$, $L=13$, and (a) $\kappa =1$ and (b) $\kappa =0$. The upper and lower curves in the figures show total population size and diversity, respectively.

Figure 2

(Color online) Time series of (a) exponential Shannon-Wiener diversity and (b) total population size for model B with $\mu =0.0005$ at several noise levels. The data are plotted every 8192 generations. In either figure, the curves correspond to $\kappa =0$, 0.1, 0.5, and 1.0 from top to bottom, respectively.

Figure 3

(Color online) (a) Species abundance distribution (SAD), (b) pdfs of the eigenvalues of the linear stability matrix, ${\lambda }_{\mathbf{S}}$, (c) frequency of the eigenvalues of the covariance matrix, ${\lambda }_{\mathbf{G}}$, for model B with $\kappa =1$ and $\kappa =0$. The data are normalized in the same way as the SADs. For $\kappa =0$, the data are obtained for three time intervals. The SADs show the number of species whose populations are within each bin region, for each community. The data were sampled every $1\times {10}^6$ generations and averaged over 18 independent runs. The fitting functions Eq. (8) are shown in (a) as guides to the eye. The fitting parameters are $\beta =2.14$ and $\gamma =0.0245$ for $\kappa =0$, and $\beta =3.4$ and $\gamma =0.0054$ for $\kappa =1$. Analogous data for model A are shown in Fig. 12 (Appendix ).

Figure 4

(Color online) Probability density function of (a) birth cost, $b_I$, (b) coupling constant to resource, ${\eta }_I$, and (c) interaction matrix elements, $\left|M_{IJ}\right|$, for fixed-point communities of model B. The data are obtained for $\kappa =1$ and $\kappa =0$.

Figure 5

(Color online) PSDs of (a) exponential Shannon-Wiener diversities and (b) total population sizes for model A with $\mu =0.001$ at several noise levels. In both figures, lines corresponding to $1/f$ are shown as guides to the eye. Data are averaged over six independent runs, and their statistical errors are also shown.

Figure 6

(Color online) (a) The probability density functions of the logarithmic derivative of the diversity, $dS\left(t\right)/dt$, for model A with $\mu =0.001$ at several noise levels. The data were averaged over 16 generations in each run and then averaged over six independent runs. (b) Log-log plot of the probability density functions of the duration of QSSs. The QSSs are estimated as the periods between times when $\left|dS\left(t\right)/dt\right|$ exceeds a cutoff (here, 0.02). The logarithmic derivative, $dS\left(t\right)/dt$, was averaged over 16 generations as in (a). The line corresponding to a $t^{-2}$ power law is shown as a guide to the eye.

Figure 7

(Color online) Species-lifetime distributions plotted on log-log scale for model A with $\mu =0.001$ and $L=13$ at several noise levels. The line corresponding to $t^{-2}$ is shown as a guide to the eye.

Figure 8

(Color online) PSDs of (a) exponential Shannon-Wiener diversities and (b) total population sizes for model B with $\mu =0.0005$ at several noise levels. The data are averaged over six independent runs and their (small) statistical errors are also shown. The straight lines in each figure represent $1/f^{\alpha }$ power laws with exponents $\alpha =1$ and 2 as guides to the eye. The shoulder in the population-size PSD at high frequencies is due to self-excited population oscillations.

Figure 9

(Color online) (a) Probability density functions of the logarithmic derivative of the diversity $\left[S\left(t+16\right)-S\left(t\right)\right]/16$ for model B with $\mu =0.0005$ at several noise levels. (b) Probability density functions of $\left\{\left[S\left(t+16\right)-S\left(t\right)\right]/16\right\}\times \overline{D}$, where $\overline{D}$ is the average diversity. The data are averaged over six independent runs and their statistical errors are also shown.

Figure 10

(Color online) Probability density functions of QSS duration for model B with several values of $\kappa$. The QSSs are estimated as the periods between times when $\left|dS\left(t\right)/dt\right|\times \overline{D}$ exceeds a cutoff (here, 0.1). The line corresponding to $1/t$ is shown as a guide to the eye.

Figure 11

(Color online) Species-lifetime distributions plotted on log-log scale for model B with $\mu =0.0005$ at several noise levels. The line corresponding to $t^{-2}$ is shown as a guide to the eye.

Figure 12

(Color online) (a) Species abundance distribution (SAD), (b) probability distribution functions (pdf) of the eigenvalues of the linear stability matrix, ${\lambda }_{\mathbf{S}}$, (c) pdf of the eigenvalues of the covariance matrix, ${\lambda }_{\mathbf{G}}$, for model A with several $\kappa$. The data are sampled every $1\times {10}^6$ generations. The SADs show the number of species whose populations are within each bin region, for each community. The fitting function for the SAD $\left(\kappa =1\right)$ is Eq. (8) with $\beta =4.5$ and $\gamma =0.006$. Compare with Fig. 3 for model B.

Figure 13

(Color online) Time series of (a) exponential Shannon-Wiener diversity and (b) total population size for the modified model B with $\mu =0.001$ and $L=20$ at $\kappa =0$ and 1. The data are plotted every 16384 generations. Compare to Fig. 2.

Figure 14

(Color online) PSDs of (a) exponential Shannon-Wiener diversities and (b) total population sizes for the modified model B. In both figures, lines corresponding to $1/f$ and $1/f^2$ are shown as guides to the eye. Data are averaged over six independent runs, and their statistical errors are also shown. Compare to Fig. 8.

Figure 15

(Color online) (a) Probability density function of $\left[S\left(t+16\right)-S\left(t\right)\right]/16\times \overline{D}$, where $\overline{D}$ is the average diversity, for the modified model B with $\kappa =0$ and 1. A Gaussian function is also shown as a guide to the eye. Compare to Fig. 9b. (b) The QSS duration distribution for $\kappa =1$. The cutoff to detect QSS is 0.012. Compare to Fig. 10. (c) Species lifetime distribution for $\kappa =0$ and 1. A power law $t^{-2}$ is also shown as a guide to the eye. Compare to Fig. 11.

Figure 16

(Color online) (a) Species abundance distribution for the modified model B with $\kappa =0$ and 1. (b) The distribution of the eigenvalues of the linear stability matrix, ${\lambda }_{\mathbf{S}}$ for the modified model B with $\kappa =1$ and 0. (c) The distribution of the square roots of the eigenvalues of the covariance matrix, ${\lambda }_{\mathbf{G}}$. The data are shown in the same way as the SADs, (binning in ${\text{log}}_2$ scale). Compare to Fig. 3.

Figure 17

(Color online) (a) The distribution of ${\eta }_I$. Species with large ${\eta }_I$ are more favored in the limit without demographic noise. (b) The distribution of the off-diagonal elements of the interaction matrix, $\left|M_{IJ}\right|$. Compare to Fig. 4.