Tangled nature model of evolutionary dynamics reconsidered: Structural and dynamical effects of trait inheritance

Based on the stochastic dynamics of interacting agents which reproduce, mutate, and die, the tangled nature model (TNM) describes key emergent features of biological and cultural ecosystems' evolution. While trait inheritance is not included in many applications, i.e., the interactions of an agent and those of its mutated offspring are taken to be uncorrelated, in the family of TNMs introduced in this work correlations of varying strength are parametrized by a positive integer $K$. We first show that the interactions generated by our rule are nearly independent of $K$. Consequently, the structural and dynamical effects of trait inheritance can be studied independently of effects related to the form of the interactions. We then show that changing $K$ strengthens the core structure of the ecology, leads to population abundance distributions better approximated by log-normal probability densities, and increases the probability that a species extant at time $t_w$ also survives at $t>t_w$. Finally, survival probabilities of species are shown to decay as powers of the ratio $t/t_w$, a so-called pure aging behavior usually seen in glassy systems of physical origin. We find a quantitative dynamical effect of trait inheritance, namely, that increasing the value of $K$ numerically decreases the decay exponent of the species survival probability.

Figure 1

The PDF of the interaction strengths for two different values of $K$. The theoretical PDF of the product of two standard Gaussian deviates with zero mean is also plotted but is indistinguishable from the $K=1$ curve at the resolution level of the plot.

Figure 2

The normalized correlation function for species pair interactions is plotted on a logarithmic scale as a function of the number $m$ of point mutations which one of the two species undergoes.

Figure 3

A three-dimensional rendering of a single ecology after 10 000 generations. Left, $K=1$; right, $K=5$. Species are represented by spheres of volume proportional to their population. The mutualistic interactions connecting core species are represented by lines with the same color as the species they affect and with a thickness proportional to their strength. Interactions linking cloud species are omitted for clarity.

Figure 4

Each line represents the population of a different species in the frozen ecology obtained by instantaneously setting the mutation rate to zero after $1\times {10}^4$ generations. The left and right panels show uncorrelated and correlated interactions, respectively.

Figure 5

The population abundance distributions vs the logarithm of the species population for $K=1$ (left) and $K=5$ (right), respectively. Lines are Gaussian fits. Notice the clear separation on the horizontal scale of the two maxima of the distribution. Insets zoom in on core species data at large populations.

Figure 6

Species persistence vs time. A new cohort of extant species is registered for each $t=2^k, k=1,2,3,\cdots$, and the corresponding fraction surviving at later times is plotted on a logarithmic scale. Not all cohorts are shown for graphical reasons. $K=1$ data are shown by squares, and $K=5$ data by circles. Lines are only guides to the eye.

Figure 7

The same data as in Fig. 6, now plotted as a function of $t/t_w$. $K=1$ data are shown by squares, and $K=5$ data by circles. The lines are least squares fits to power laws $y=a{(t/t_w)}^b$, where $b=-0.283\left(14\right)$ and $b=-0.117\left(6\right)$ for $K=1$ and $K=5$, respectively. All available cohorts have been used in the fits.