Coexistence and Survival in Conservative Lotka-Volterra Networks

Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network’s interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.

Figure 1

Two interaction topologies specifying the conservative LV systems. (a) The general cyclic four species systems (4SS). (b) The general cyclic five species system (5SS) as a natural extension of the rock-papers-scissors configuration (RPS) [44].

Figure 2

Coexistence and survival in the general cyclic 4SS are controlled by the Pfaffian of the interaction matrix. (a) For $\mathrm{Pf}(G_4)=0$, one obtains coexistence of all species on periodic orbits. (b) Deterministic survival diagram: for $\mathrm{Pf}(G_4)<0$, species $A$, $B$, and $D$ survive in a stable RPS configuration, whereas $A$, $C$, and $D$ survive for $\mathrm{Pf}(G_4)>0$.

Figure 3

Stability of the cyclic 5SS. (a) For the interaction scheme [left inset of (b)], one obtains coexistence of all species for the critical rate $k_{AB}=5$. (b) Stability of the stochastic system, reflected by the extinction time $T_{\mathrm{ext}}$, peaks at the critical rate, which becomes more pronounced as $N\to \infty$. We find a scaling law for $T_{\mathrm{ext}}$ in the distance to the critical rate (right inset). Initial conditions were chosen as $\mathbf{x}(0)=1/5\times \mathbf{1}$. Larger line gap corresponds to smaller $N$.