Emergence of diversity in a model ecosystem

The biological requirements for an ecosystem to develop and maintain species diversity are in general unknown. Here we consider a model ecosystem of sessile and mutually excluding organisms competing for space [Mathiesen et al. Phys. Rev. Lett. 107, 188101 (2011)]. Competition is controlled by an interaction network with fixed links chosen by a Bernoulli process. New species are introduced in the system at a predefined rate. In the limit of small introduction rates, the system becomes bistable and can undergo a phase transition from a state of low diversity to high diversity. We suggest that isolated patches of metapopulations formed by the collapse of cyclic relations are essential for the transition to the state of high diversity.

Figure 1

(a) Figure of a system of size $L=1200$ with $\gamma =0.05$ and $\alpha =0.01$. (b) Distributions for the total species sizes (squares) and patch sizes (circles) for different $L$ (left panel, $\gamma =0.05$) and different $\gamma$ (right panel, $L=200$) in the steady state $\alpha =0.01$ for $L=1200$, and $\alpha =0.0025$ for $L=200$. The data for $L=200$ are averaged over $4\times {10}^8$ time steps. The vertical axis shows the probability scaled by the system size $L^2$ of finding a species or patch of given size.

Figure 2

(a) The time-averaged diversity $D$ as a function of $\gamma$ for various $\alpha$. A sharp transition is seen in the limit of $\alpha \to 0$. (This corresponds to quasistatic simulation, explained later in the text.) The connecting lines are for guiding the eye. (b) Probability density function of the diversity measured over time for a system of size $L=200$. The distributions are shown for a few values of $\gamma$ below and above the critical value ${\gamma }_c$. We see that for $\gamma \ge 0.06$ the distributions are bimodular. The density is shown along a logarithmic axis.

Figure 3

(a) Time evolution of the patchiness (line) and diversity (line with shaded area) for $\alpha =0.0125$ and $\gamma =0.07$ for a system of size $L=200$. For illustration, typical snapshots from another simulation data set with the same parameters are shown for low- and high-diversity states. (b) Relation between the patchiness and diversity for $\alpha =0.0125$ and $\gamma =0.07$, averaged over 100 transitions between low and high diversity. The transition from low to high (filled circles) diversity is defined as a development where diversity increases from 3 to above 40. The reverse transition is marked by filled squares, and defined as a development where the diversity decreases from 40 to below 3. The patchiness is averaged for each value of diversity $D$, and the error bars show the standard deviations. As marked by the open circles, the patchiness is independent of the diversity as long as the system is staying in the high-diversity state.

Figure 4

Stochastic patch creation in a system when a new species “light blue (middle darkness)” can invade yellow (lightest), but at the same time light blue (middle darkness) can be invaded by one of the species (“Brown”, darkest). The figure illustrates that patches can be created when species form this purely hierarchical relationship in a noisy system.

Figure 5

Investigation of breakdown of cycles: Average number of patches that are left when cycles of length $3,4,5$, and 6 collapse due to fluctuations associated with the stochastic update of the system. The dashed line corresponds to the scaling $P\propto L^{0.75}$. The inset shows the average lifetime of the cycles of length $2,...,6$ as a function of system size.

Figure 6

Quasistatic simulation of a system of size $L=200$. (a) Diversity of a steady state where short cycles of lengths below $2,3,4,...$ are prevented from forming in the system. Removing cycles of length 2 is the same as allowing all cycles $\ge 3$. The connecting lines are for guiding the eye. (b),(c) Probability of having an active cycle of a certain length at a time $\tau$ after the introduction of a new species for, respectively, (b) $\gamma =0.05$ and (c) $\gamma =0.025$. Notice that the probabilities do not add up to 1, which reflects the fact that the majority of new species does not activate a cyclic relation.