Fluctuations and correlations in lattice models for predator-prey interaction

Including spatial structure and stochastic noise invalidates the classical Lotka-Volterra picture of stable regular population cycles emerging in models for predator-prey interactions. Growth-limiting terms for the prey induce a continuous extinction threshold for the predator population whose critical properties are in the directed percolation universality class. We discuss the robustness of this scenario by considering an ecologically inspired stochastic lattice predator-prey model variant where the predation process includes next-nearest-neighbor interactions. We find that the corresponding stochastic model reproduces the above scenario in dimensions $1<d⩽4$, in contrast with the mean-field theory, which predicts a first-order phase transition. However, the mean-field features are recovered upon allowing for nearest-neighbor particle exchange processes, provided these are sufficiently fast.

Figure 1

Flows in the phase plane from integrating Eqs. (1, 2) for $\mu =\sigma =\eta =1$, with $\delta =4$ (a) and $\delta =9$ (b). (a): $(0,1)$ is the only stable fixed point (node). (b): There is an additional stable (node) active fixed point $(a_1^{*},b_1^{*})=(1∕3,1∕3)$; while $(0,0)$ and $(a_2^{*},b_2^{*})=(1∕6,2∕3)$ are unstable. The slopes of the separatrices at $(a_2^{*},b_2^{*})$ are $\approx 1.126$ (dashed line) and $\approx -1.568$ (see text).

Figure 2

(Color online) Average stationary prey density $b^{*}$ vs $\delta$ for $\eta =\sigma =2\mu =2$. (a): DP-like transitions on ${256}^2$, ${50}^3$, and ${20}^4$ lattices when $\mathcal{D}=0$. (b): Effect of the stirring on a ${512}^2$ lattice. For $\mathcal{D}=0$, there is a continuous transition (center curve, black), while for $\mathcal{D}=10$ (sufficient stirring) two stable branches emerge, and there is a first-order transition; the top (red) branch corresponds to predator extinction, and the bottom (blue) branch is associated with a coexistence phase.

Figure 3

(Color online) Average stationary density of predators in the absence of stirring on a $512\times 512$ lattice with $\eta =\sigma =2\mu =2$ and $\mathcal{D}=0$: Existence of a DP-like phase transition at ${\delta }_c\approx 11.72$ with exponent $\beta \approx 0.584$ (see inset).

Figure 4

Phase portrait of the generalized “NNN” lattice predator-prey model (on a $256\times 256$ lattice) with rates $\eta =\mu =\sigma =1$, $\delta =10$ and different exchange rates. From (a) to (c): stirring rate $\mathcal{D}=0,2,5$. (See text).