Interaction strength revisited—clarifying the role of energy flux for food web stability

Abstract

Interaction strength (IS) has been theoretically shown to play a major role in governing the stability and dynamics of food webs. Nonetheless, its definition has been varied and problematic, including a range of recent definitions based on biological rates associated with model parameters (e.g., attack rate). Results from food web theory have been used to argue that IS metrics based on energy flux ought to have a clear relationship with stability. Here, we use simple models to elucidate the actual relationship between local stability and a number of common IS metrics (total flux and per capita fluxes) as well as a more recently suggested metric. We find that the classical IS metrics map to stability in a more complex way than suggested by existing food web theory and that the new IS metric has a much clearer, and biologically interpretable, relationship with local stability. The total energy flux metric falls off existing theoretical predictions when the total resource productivity available to the consumer is reduced despite increased consumer attack rates. The density of a consumer can hence decrease when its attack rate increases. This effect, called the paradox of attack rate, is similar to the well-known hydra effect and can even cascade up a food chain to exclude a predator when consumer attack rate is increased.

Appendix

Fig. 6

Results from the Lotka-Volterra consumer-resource model when varying consumer attack rate. a Consumer-resource phase space with zero net growth isoclines for different consumer attack rates (a _cr  = 10 in black, a _cr   = 2 in dark gray, a _cr  = 1.4 in gray, a _cr  = 1 in light gray). Dots represent the equilibrium points. The dotted line is an example of a predator isocline, relevant for the food chain scenario, and since the phase space here is only represented by the consumer and resource density axis, it appears as a horizontal line at the consumer densities the predator requires to persist. b Consumer biomass (full line), resource biomass (dotted), consumer/resource biomass ratio (dashed), as well as λ _max (dot dashed) when varying a _cr . c Total energy flux (full line), per consumer energy flux (dashed), and per resource energy flux (dotted) when varying a _cr . Corresponds to RM (Fig. 2) in the main text

Fig. 7

Bifurcation over the intrinsic resource growth (r) for different values of consumer attack rate (a _cr ). a For the Lotka-Volterra model when a _cr  = 0.55 (gray line) and when a _cr  = 0.7 (black line), dotted lines are used when λ _max is real and full lines are used when λ _max is complex. b For the Rosenzweig-MacArthur model when a _cr  = 1.3 (corresponding to a real λ _max for r = 1, full black line), for a _cr  = 1. 4 (corresponding to a complex and stable λ _max, dot dashed line) and for a _cr  = 2 (corresponding to a complex and unstable λ _max, dashed line)

Fig. 8

Consumer-resource system where consumer mortality (m _c ) is varied, for the Lotka-Volterra (left panels) and the Rosenzweig-MacArthur (right panels) model. a–b Consumer biomass (full line), resource biomass (dotted), consumer/resource biomass ratio (dashed), and λ _max (dot dashed) (LV: a _cr  = 0.52, RM: a _cr  = 1.6). c-d Total energy flux (full line), per resource energy flux (dotted), and per consumer energy flux (dashed)

Fig. 9

Food chain where consumer mortality (m _c ) is varied for the Rosenzweig-MacArthur model. a Consumer (full line), resource (dotted), and predator biomass (dashed), as well as λ _max (dot dashed) (a _cr  = 1.6). b Total energy flux (full line), per resource energy flux (dotted), and per consumer energy flux (dashed). The parameter range when the predator is present is shown

Fig. 10

Relationships between stability (λ _max) and the total energy flux (between consumer and resource), when varying consumer attack rate (a _cr ) in a food chain. For the Lotka-Volterra model in a and the Rosenzweig-MacArthur model in b. Dashed lines are used for when λ _max is real and full lines for when λ _max complex

Fig. 11

Biomasses, fluxes, and stability measures for the Lotka-Volterra food chain case when the predator is present and consumer attack rate (a _cr ) is varied. a Resource biomass (dotted line), consumer biomass (full), predator biomass (dashed), and λ_max (dot dashed). b Total flux between consumer and resource (full line), flux per resource (dotted), and flux per consumer (dashed)