In ecology, prey switching refers to a predator's adaptive change of habitat or diet in response to prey abundance. In this paper, we study piecewise-smooth models of predator-prey interactions with a linear trade-off in a predator's prey preference. We consider optimally foraging predators and derive a model for a 1 predator-2 prey interaction with a tilted switching manifold between the two sides of discontinuous vector fields. We show that the 1 predator-2 prey system undergoes a novel adding-sliding-like (center to two-part periodic orbit; "C2PO") bifurcation in which the prey ratio transitions from constant to time-dependent. Further away from the bifurcation point, the period of the oscillating prey ratio period doubles, suggesting a possible cascade to chaos. We compare our model predictions with data and demonstrate that we successfully capture the periodicity in the ratio between the predator's preferred and alternative prey types in data on freshwater plankton. Our study suggests that it is useful to investigate prey ratio as a possible indicator of how population dynamics can be influenced by ecosystem diversity. 1 We have chosen to follow this generally accepted convention instead of planktic, which is the correct adjective.A standard Lotka-Volterra model for one predator and two prey that allows diversity in the prey community predicts extinction of the prey type that has a smaller capacity to survive [22]. Therefore, such a framework is inappropriate for investigations of ciliate-phytoplankton dynamics, because several predators and prey coexist, and there is a known preference towards one of the phytoplankton prey [32]. One way to resolve this discrepancy is to examine adaptive predator behavior in response to changes in prey densities. There have been many investigations of prey switching [33], in which predators express preference for more abundant prey. For example, prey switching has been demonstrated to promote coexistence of competing prey species [42] and to decrease prey competition due to a shared predator [1]. As a result, prey switching has been suggested as a candidate mechanism for coexistence in communities with diverse prey [2].In smooth differential-equation models for population dynamics, prey switching can be represented using a Holling type-III functional response [17], in which a sigmoid function gives the increased degree of predation with increasing densities of a principal prey. In a 1 predator-1 prey system, a Holling type-III functional response corresponds to a situation in which predation is low at low prey densities but saturates quickly at a high value when prey is abundant. Such a functional response was observed in a system of protist predators and their yeast prey [15]. A low abundance of yeast resulted in increased sedimentation of yeast at the bottom of a glass tube and the walls, so prey were safe from predation, which resulted in a decrease of predator concentration when the prey concentration was below a threshold. Predator preference towards more abundant prey ...

Experimental studies show that human pain sensitivity varies across the 24-hour day, with the lowest sensitivity usually occurring during the afternoon. Patients suffering from neuropathic pain, or nerve damage, experience an inversion in the daily modulation of pain sensitivity, with the highest sensitivity usually occurring during the early afternoon. Processing of painful stimulation occurs in the dorsal horn (DH), an area of the spinal cord that receives input from peripheral tissues via several types of primary afferent nerve fibers. The DH circuit is composed of different populations of neurons, including excitatory and inhibitory interneurons, and projection neurons, which constitute the majority of the output from the DH to the brain. In this work, we develop a mathematical model of the dorsal horn neural circuit to investigate mechanisms for the daily modulation of pain sensitivity. The model describes average firing rates of excitatory and inhibitory interneuron populations and projection neurons, whose activity is directly correlated with experienced pain. Response in afferent fibers to peripheral stimulation is simulated by a Poisson process generating nerve fiber spike trains at variable firing rates. Model parameters for fiber response to stimulation and the excitability properties of neuronal populations are constrained by experimental results found in the literature, leading to qualitative agreement between modeled responses to pain and experimental observations. We validate our model by reproducing the wind-up of pain response to repeated stimulation. We apply the model to investigate daily modulatory effects on pain inhibition, in which response to painful stimuli is reduced by subsequent non-painful stimuli. Finally, we use the model to propose a mechanism for the observed inversion of the daily rhythmicity of pain sensation under neuropathic pain conditions. Underlying mechanisms for the shift in rhythmicity have not been identified experimentally, but our model results predict that experimentally-observed dysregulation of inhibition within the DH neural circuit may be responsible. The model provides an accessible, biophysical framework that will be valuable for experimental and clinical investigations of diverse physiological processes modulating pain processing in humans.

This proceedings paper is the first in a series of three papers developing mathematical models for the complex relationship between pain and the sleep-wake cycle. Here, we briefly review what is known about the relationship between pain and the sleep-wake cycle in humans and laboratory rodents in an effort to identify constraints for the models. While it is well accepted that sleep behavior is regulated by a daily (circadian) timekeeping system and homeostatic sleep drive, the joint modulation of these two primary biological processes on pain sensitivity has not been considered. Under experimental conditions, pain sensitivity varies across the 24 h day, with highest sensitivity occurring during the evening in humans. Pain sensitivity is also modulated by sleep behavior, with pain sensitivity increasing in response to the build up of homeostatic sleep pressure following sleep deprivation or sleep disruption. To explore the interaction between these two biological pro- The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/098269 doi: bioRxiv preprint first posted online Jan. 9, 2017; 2 Hagenauer, Crodelle, Piltz, Toporikova, Ferguson and Booth cesses using modeling, we first compare the magnitude of their effects across a variety of experimental pain studies in humans. To do this comparison, we normalize the results from experimental pain studies relative to the range of physiologicallymeaningful stimulation levels. Following this normalization, we find that the estimated impact of the daily rhythm and of sleep deprivation on experimental pain measurements is surprisingly consistent across different pain modalities. We also review evidence documenting the impact of circadian rhythms and sleep deprivation on the neural circuitry in the spinal cord underlying pain sensation. The characterization of sleep-dependent and circadian influences on pain sensitivity in this review paper is used to develop and constrain the mathematical models introduced in the two companion articles.

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