Confronting the Paradox of Enrichment to the Metacommunity Perspective

Resource enrichment can potentially destabilize predator-prey dynamics. This phenomenon historically referred as the "paradox of enrichment" has mostly been explored in spatially homogenous environments. However, many predator-prey communities exchange organisms within spatially heterogeneous networks called metacommunities. This heterogeneity can result from uneven distribution of resources among communities and thus can lead to the spreading of local enrichment within metacommunities. Here, we adapted the original Rosenzweig-MacArthur predator-prey model, built to study the paradox of enrichment, to investigate the effect of regional enrichment and of its spatial distribution on predator-prey dynamics in metacommunities. We found that the potential for destabilization was depending on the connectivity among communities and the spatial distribution of enrichment. In one hand, we found that at low dispersal regional enrichment led to the destabilization of predator-prey dynamics. This destabilizing effect was more pronounced when the enrichment was uneven among communities. In the other hand, we found that high dispersal could stabilize the predator-prey dynamics when the enrichment was spatially heterogeneous. Our results illustrate that the destabilizing effect of enrichment can be dampened when the spatial scale of resource enrichment is lower than that of organismss movements (heterogeneous enrichment). From a conservation perspective, our results illustrate that spatial heterogeneity could decrease the regional extinction risk of species involved in specialized trophic interactions. From the perspective of biological control, our results show that the heterogeneous distribution of pest resource could favor or dampen outbreaks of pests and of their natural enemies, depending on the spatial scale of heterogeneity.


A. DYNAMICS OF METACOMMUNITIES WHEN DISPERSAL TENDS TO INFINITY
We consider a predator-prey metacommunity occupying M patches where prey growth follows a logistic shape, with any prey-dependent functional response f and predator mortality g. Dispersal depends on density in the departure patch.
Our aim is to investigate the properties of the solutions when d N and d P →+∞. In order to facilitate the analysis, we assume d N = d P =d. For any d, we denote (N i d , P i d ) the associated solution of (A.1).
First, the equation A.1 of the prey for each patch i is equivalent to: When d→+∞, the left-hand side of (A.2) tends to 0 and thus N i The same conclusion is obtained for the equation of the predator. This means that the prey and the predator have the same densities in all patches when dispersal is global and infinite. Set N ∞ and P ∞ these densities.
Thus, taking the sums of equations (A.1) over i leads to: When d→+∞, N i d → N ∞ and P i d → P ∞ , and the system of equation (A.3) tends to: is the harmonic mean of the carrying capacities K i in the different patches i.
Third, we conclude on the stability of the equilibrium for our specific model. If the functional response is of type II (Holling 1959) obtained the Rosenzweig-MacArthur model. This model has one equilibrium where the prey and the predator can coexist. At this equilibrium, predator density is positive when . By studying the sign of the determinant and of the trace of the Jacobian matrice of (A.4), it is easy to show that this equilibrium is stable when K < K thr with

B.1 Stability thresholds
We consider a predator-prey metacommunity occupying M patches where the carrying capacity in all the patches is assumed to be K 0 , which is below the destabilizing threshold of an isolated patch K thr . We consider that this metacommunity experiences regional enrichment E and we studied the critical value of regional enrichment, E thr , leading to the destabilization of metacommunity equilibrium in the limiting cases where there is no dispersal and where dispersal tends to infinity.

(a) No dispersal
As we consider that dispersal from patch to patch is equal to zero, the metacommunity equilibrium is stable if the equilibriums of each of the M isolated communities are stable.
Isolated communities have one equilibrium where the prey and the predator can coexist with positive densities if K i > m (a(e− mt h )). This equilibrium is stable if and only if the carrying capacity K i in this patch is lower than K thr = (e+ mt h ) (at h (e− mt h )). If the carrying capacity of at least one community crosses the threshold K thr , then metacommunity equilibrium is unstable.
When enrichment distribution is homogeneous (α=0), the carrying capacity in each patch is equal to K 0 +E/M. Thus, the equilibrium of the metacommunity is destabilized when the regional enrichment E exceeds the critical value E thr 0 (α = 0) = (K thr − K 0 )M .
When the spatial heterogeneity in enrichment distribution is maximal (α=1), the regional enrichment E is concentrated in only one patch. When enrichment leads the carrying capacity in this patch to be higher than K thr , the equilibrium of the metacommunity is destabilized.
Thus, the equilibrium of the metacommunity is destabilized when enrichment is higher than the critical value E thr 0 (α = 1) = (K thr − K 0 ) .

(b) Infinite dispersal
We consider that dispersal from patch to patch tends to +∞. The dynamics of this homogenized metacommunity are described by the equations (A.4).
When enrichment distribution is homogeneous (α=0), the carrying capacity in each patch is increased by the same amount E/M. Hence the carrying capacity is the same in all patches and is equal to K 0 +E/M. As a consequence, the regional carrying capacity of the prey simplify as: Thus, the equilibrium of the metacommunity is destabilized when enrichment is higher than the critical value E thr ∞ (α = 0) = (K thr − K 0 )M .
When the heterogeneity of enrichment distribution is maximal (α=1), only one patch receives enrichment and its carrying capacity is increased to K 0 +E whereas the carrying capacity in the other patches is equal to K 0 , Hence, the regional carrying capacity of the prey is: If the regional enrichment E tends to +∞ in (B.1), then K tends to a maximal value K max : Recall that the equilibrium of the homogenized metacommunity is stable if K < K thr where K thr depends only on predator parameters. Hence, there are two cases: • In the first case, where K 0 × M (M −1) < K thr , the regional enrichment E does never destabilize the equilibrium of the homogenized metacommunity, even the regional enrichment reaches infinite values.
In the second case, where K 0 × M (M −1) > K thr , the regional enrichment E destabilizes the equilibrium of the homogenized metacommunity when it is above the threshold given by: We found that ∂E thr

B.2 Comparison of stability thresholds
Now, we want to compare these four thresholds in order to study the effect of enrichment distribution and of dispersal on the stability of metacommunity equilibrium.