Ecological communities with Lotka-Volterra dynamics

Ecological communities in heterogeneous environments assemble through the combined effect of species interaction and migration. Understanding the effect of these processes on the community properties is central to ecology. Here we study these processes for a single community subject to migration from a pool of species, with population dynamics described by the generalized Lotka-Volterra equations. We derive exact results for the phase diagram describing the dynamical behaviors, and for the diversity and species abundance distributions. A phase transition is found from a phase where a unique globally attractive fixed point exists to a phase where multiple dynamical attractors exist, leading to history-dependent community properties. The model is shown to possess a symmetry that also establishes a connection with other well-known models.

Figure 1

Communities assembled in different locations from a regional species pool. The model focuses on one such community.

Figure 2

The model exhibits three distinct dynamical behaviors, depending on model parameters. In the unique fixed-point (UFP) phase, a unique, stable fixed point that is resistent to invasion exists for any system. In multiple attractor (MA) phase, multiple dynamical attractors can be found for any system. These may be stable fixed points or other attractors, and the community composition is history dependent. In the third phase abundances grow without bound; here the Lotka-Volterra equations likely break down. The phases are shown for asymmetric interactions ($\gamma =0$) and equal carrying capacities, all set to $K_i=1$.

Figure 3

Diversity and species abundance distribution. $\varphi$ is the fraction of persistent ($N_i>0$) species. Solid and dashed lines show analytical predictions, exact in the UFP phase, left of the vertical dotted lines in (a). Parameter values are $\gamma =0$ and ${\sigma }_K^2=0$, and in (b) $\mu =4$ and $\sigma =1.1$.

Figure 4

Moments of the species abundance distribution. Symbols and parameter values are the same as in Fig. 3.

Figure 5

(a) Phase diagram for ${\sigma }_K=0$ and different $\gamma$. The $\gamma =0$ lines correspond to Fig. 2 in the main text. Crosses mark the transition to diverging solutions, found numerically. (b) Numerical check of the phase boundary between UFP and MA phases, the fraction of systems for which there are multiple solutions, at $\mu =4$ and $\gamma =1$. At large $S$ this fraction jumps sharply at the phase transition. The dashed line marks the analytically calculated transition point, $\sigma =1/\sqrt{2}$.

Figure 6

Phase boundary between UFP and MA phases for different values of ${\sigma }_K$ at $\gamma =0$. Solid lines correspond to the $\gamma =0$ phase boundaries in Fig. 5.

Figure 7

Fraction of species that may invade. Data plotted with dashed lines show the results after multiple invasion attempts.