Stability of consumer–resource interactions in periodic environments

Abstract

Periodic fluctuations in abiotic conditions are ubiquitous across a range of temporal scales and regulate the structure and function of ecosystems through dynamic biotic responses that are adapted to these external forces. Research has suggested that certain environmental signatures may play a crucial role in the maintenance of biodiversity and the stability of food webs, while others argue that coupled oscillators ought to promote chaos. As such, numerous uncertainties remain regarding the intersection of temporal environmental patterns and biological responses, and we lack a general understanding of the implications for food web stability. Alarmingly, global change is altering the nature of both environmental rhythms and biological rates. Here, we develop a general theory for how continuous periodic variation in productivity, across temporal scales, influences the stability of consumer–resource interactions: a fundamental building block of food webs. Our results suggest that consumer–resource dynamics under environmental forcing are highly complex and depend on asymmetries in both the speed of forcing relative to underlying dynamics and in local stability properties. These asymmetries allow for environmentally driven stabilization under fast forcing, relative to underlying dynamics, as well as extremely complex and unstable dynamics at slower periodicities. Our results also suggest that changes in naturally occurring periodicities from climate change may lead to precipitous shifts in dynamics and stability.

Phase synchronization between culture and climate forcing

“…Note, that in our case, where BBCs emerge from the multiplicative stochastic innovation forcing. This is different from recently proposed models [32,40,41], which generate these cycles as deterministic biotic/demographic limit cycles-even without external periodic forcing.…”

“…In worst cases, this can lead to migration [61], political tensions and possibly also wars, which in turn can further limit food supplies [62,63]. -Non-autonomous forcing [32] introduced in the relaxation term of the resource equation to mimic climate variability (e.g. Milanković cycles) can synchronize both population density and to a lesser degree cultural development (figure 5).…”

“…where η denotes the rate of resource replenishment back to an unperturbed environmental carrying capacity K 0 , which in later sensitivity experiments will be chosen as a non-autonomous forcing K 0 ( t ) = K 0 * [ 1 + κ normalcos ⁡ ( normalΩ t ) ] , to mimic the effect of temporally varying climatic conditions [32], associated with, for example, Milanković cycles [1]. The analytical solution to this equation for time-varying ρ can be formally expressed as…”

Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

“…Theorem 1 ([30]). Under condition (14), the first population displaces the second one within a community of two competitors.…”

“…Taking into account seasonality can qualitatively change the conclusions regarding population dynamics, since parameter periodicity can, in particular, produce quasi-periodic and chaotic solution behavior [10,11]. Interestingly, seasonality can have a stabilizing as well as destabilizing effect on dynamics [12][13][14].…”