Weak Interactions and Instability Cascades

Food web theory states that a weak interactor which is positioned in the food web such that it tends to deflect, or mute, energy away from a potentially oscillating consumer-resource interaction often enhances community persistence and stability. Here we examine how adding other weak interactions (predation/harvesting) on the stabilizing weak interactor alters the stability of food web using a set of well-established food web models/modules. We show that such “weak on weak” interaction chains drive an indirect dynamic cascade that can rapidly ignite a distant consumer-resource oscillator. Nonetheless, we also show that the “weak on weak” interactions are still more stable than the food web without them, and so weak interactions still generally act to stabilize food webs. Rather, these results are best interpreted to say that the degree of the stabilizing effect of a given important weak interaction can be severely compromised by other weak interactions (including weak harvesting).


Weak Interactions and Instability Cascades
. The local minima and maxima for consumer density C1 in the diamond food webs with different position of oscillator interaction(s) (see Fig. S3). In a, c and e, immediately after stabilizing agent invades the system by increasing its attack rate, the system becomes stable.  Figure S5. The 8 species food web configurations. In a, stabilizing agent, C2 is a stabilizing weak interactor in such a position in the food web that it tends to deflect, or mute, energy away from potentially oscillating strong consumer-resource interactions, denoted as oscillator. Dashed arrows represent weak interactions. In b, a predator, P1 on the stabilizing agent is added to the food web.   Figure S6. The local minima and maxima for consumer density C1 in the complex food web with 8 species (see Fig. S5). In a, immediately after stabilizing agent, C2 establishes the interaction with R1 by increasing its attack rate, the system becomes relatively stable. In b, the attack rate of stabilizing agent, C2 shown as asterisk in a, is used and attack rate of predator, P1 on the stabilizing agent is gradually increased.
Immediately after the predator starts to invade, the system becomes unstable.  harvesting is added to the system as shown in d. In d, vertical line represents the attack Max/Min

The food-web models analyzed in the main text
All models are derived from the well-known Rosenzweig-MacArthur food chain equations 20 , where R, is the resource density, C, is the consumer density, P, is the predator density, r is the intrinsic growth rate of the resource, K is the carrying capacity, ai is the attack rate of species i, hi is the handling time of species i, mi is the mortality rate of species i and e is the assimilation rate. The models are specified as follows:   Fig. 2b, aT was changed gradually.  Fig. 3b, aT1 was changed gradually.

a. A diamond food-web
(1) In the diamond food-web with consumer-resource oscillator case (Fig. S4a) Fig. S4c, aC2 was changed gradually.

The model for a complex food web with 8 species
We conducted the experiment using a complex food web model with 8 species (Fig.   S5a) with the same procedure as those shown in the text (Figs. 1, 3). The model is specified as follows:

The food web models with parameters based on metabolic allometry
We conducted the same experiment with those shown in Figs 1a where, R, is the resource density, C, is the consumer density, P, is the predator density,  Fig. S7c, yc2 was changed gradually.