Unification of Small and Large Time Scales for Biological Evolution: Deviations from Power Law

We develop a unified model that describes both “micro” and “macro” evolutions within a single theoretical framework. The ecosystem is described as a dynamic network; the population dynamics at each node of this network describes the “microevolution” over ecological time scales (i.e., birth, ageing, and natural death of individual organisms), while the appearance of new nodes, the slow changes of the links, and the disappearance of existing nodes accounts for the “macroevolution” over geological time scales (i.e., the origination, evolution, and extinction of species). In contrast to several earlier claims in the literature, we observe strong deviations from power law in the regime of long lifetimes.

Figure 1

The total number of species $N(t)$ is plotted against time; the corresponding parameter set is $N_{\mathrm{m}\mathrm{a}\mathrm{x}}=100$, $n_{\mathrm{m}\mathrm{a}\mathrm{x}}=1000$, $C=0.1$, $p_{sp}=0.001$, $p_{\mathrm{m}\mathrm{u}\mathrm{t}}=0.001$.

Figure 2

The total number of organisms $n(t)$ is plotted against time over a relatively short part of the evolution of the ecosystem. The parameter values are identical to those in Fig. 1.

Figure 3

Log-log plots of the distributions of the lifetimes of the species in an ecosystem with (a) $n_{\mathrm{m}\mathrm{a}\mathrm{x}}=100$, (b) $n_{\mathrm{m}\mathrm{a}\mathrm{x}}=1000$. In (b), the data beyond the lifetime of 10 are not shown to emphasize the initial power law regime (symbolized by the straight line with slope $-2$. The other common parameters for both the figures are $N_{\mathrm{m}\mathrm{a}\mathrm{x}}=50$, $p_{sp}=0.01$, $p_{\mathrm{m}\mathrm{u}\mathrm{t}}=0.001$. The symbols □, +, ×, and * in (a) correspond to maximum simulation times $5\times {10}^3$, $5\times {10}^4$, $5\times {10}^5$, and $5\times {10}^6$, respectively, while the symbols +, ×, and * in (b) correspond to the maximum simulation times ${10}^4$, $5\times {10}^4$, and $4\times {10}^5$, respectively. Each of the data points has been obtained by averaging over 18 to 176 runs, each starting from a new random initial state.

Figure 4

Semilog plot of the distributions of $X$ (+), $X_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (×), and $X_{\mathrm{r}\mathrm{e}\mathrm{p}}$ (*); the parameter values are the same as those in Fig. 1.

Figure 5

The distribution of $M$; the parameter values are the same as those in Fig. 1.