A model of microbial contamination of a water reservoirEngel, Robert; Normand, Mark; Horowitz, Joseph; Peleg, Micha
doi: 10.1006/bulm.2001.0238pmid: 11732173
A three year record of daily fecal coliform counts in a Massachusetts water reservoir has the appearance of an irregular time series punctuated by outbursts of varying duration. The pattern is described in terms of a probabilistic model where the fluctuations in the ‘regular’ and ‘explosive’ regimes are governed by two sets of probabilities. It has been assumed that the random oscillations has a lognormal distribution, and that once an explosion threshold has been exceeded the increments or decrements in the population size have fixed probability distributions. The threshold for triggering an outburst was estimated by examining the randomness of the autocorrelation function of the record after it is filtered to eliminate peaks of progressively increasing magnitude. Once the threshold has been identified, the mean and standard deviation of the underlying lognormal distribution could be estimated directly from remains found in the record after all the peaks were removed. The probabilities of an increment and decrement during the outbursts and their relative magnitudes could also be estimated using simple formulas. These estimated parameter values were then used to generate realistic records with known threshold levels, which were subsequently used to assess the procedure’s feasibility and sensitivity.
One-vesicle hypothesis for neurotransmitter release: A possible molecular mechanismYusim, K.; Parnas, H.; Segel, L.
doi: 10.1006/bulm.2001.0247pmid: 11732174
Neurotransmitter-containing vesicles are clustered in release sites. Although a given site can contain tens of vesicles, there is evidence that under a wide range of conditions, following an action potential, rarely is more than one vesicle released from each site. Such findings led to the one vesicle hypothesis, for which this paper suggests a molecular mechanism. The release of a vesicle from a site provides a transient high concentration of transmitter in that site. It is proposed here that the local high transmitter concentration interrupts further vesicle releases from the same release site. The suggested mechanism for this ‘release interruption’ is based on a theory of release control by the authors wherein inhibitory transmitter autoreceptors play a central role. (That transmitter binding to these autoreceptors can inhibit release on a fast time scale has recently been shown experimentally.) A detailed kinetic scheme is presented for the proposed mechanism. Stochastic simulations of this scheme demonstrate how the mechanism accounts for the one vesicle hypothesis. In agreement with recent experiments, the simulations also show that changes in conditions that affect the release process can cause frequent release of more than one vesicle per site.
Model of droplet dynamics in the Argentine ant Linepithema humile (Mayr)Theraulaz, Guy; Bonabeau, Eric; Sauwens, Christian; Deneubourg, Jean-Louis; Lioni, Arnaud; Libert, François; Passera, Luc; Solé, Ricard
doi: 10.1006/bulm.2001.0260pmid: 11732177
The formation of droplets of ants Linepithema humile (Mayr) is observed under certain experimental conditions: a fluctuating aggregate forms at the end of a rod and a droplet containing up to 40 ants eventually falls down. When the flux of incoming ants is sufficient, this process can continue for several hours, leading to the formation and fall of tens of droplets. Previous work indicates that the time series of drop-to-drop intervals may result from a nonlinear low-dimensional dynamics, and the interdrop increments exhibit long-range anticorrelations. A model of aggregation and droplet formation, based on experimental observations, is introduced and shown to reproduce these properties.
Critical conditions for phytoplankton bloomsEbert, Ute; Arrayás, Manuel; Temme, Nico; Sommeijer, Ben; Huisman, Jef
doi: 10.1006/bulm.2001.0261pmid: 11732178
We motivate and analyse a reaction—advection—diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton—water—light-system characterized by the same dimensionless parameters will show the same dynamics.
Two-category model of task allocation with application to ant societiesBrandts, Wendy; Longtin, André; Trainor, Lynn
doi: 10.1006/bulm.2001.0262pmid: 11732179
In many network models of interacting units such as cells or insects, the coupling coefficients between units are independent of the state of the units. Here we analyze the temporal behavior of units that can switch between two ‘category’ states according to rules that involve category-dependent coupling coefficients. The behaviors of the category populations resulting from the asynchronous random updating of units are first classified according to the signs of the coupling coefficients using numerical simulations. They range from isolated fixed points to lines of fixed points and stochastic attractors. These behaviors are then explained analytically using iterated function systems and birth-death jump processes. The main inspiration for our work comes from studies of non-hierarchical task allocation in, e.g., harvester ant colonies where temporal fluctuations in the numbers of ants engaged in various tasks occur as circumstances require and depend on interactions between ants. We identify interaction types that produce quick recovery from perturbations to an asymptotic behavior whose characteristics are function of the coupling coefficients between ants as well as between ants and their environment. We also compute analytically the probability density of the population numbers, and show that perturbations in our model decay twice as fast as in a model with random switching dynamics. A subset of the interaction types between ants yields intrinsic stochastic asymptotic behaviors which could account for some of the experimentally observed fluctuations. Such noisy trajectories are shown to be random walks with state-dependent biases in the ‘category population’ phase space. With an external stimulus, the parameters of the category-switching rules become time-dependent. Depending on the growth rate of the stimulus in comparison to its population-dependent decay rate, the dynamics may qualitatively differ from the case without stimulus. Our simple two-category model provides a framework for understanding the rich variety of behaviors in network dynamics with state-dependent coupling coefficients, and especially in task allocation processes with many tasks.
An algebraic-combinatorial model for the identification and mapping of biochemical pathwaysOliveira, Joseph; Bailey, Colin; Jones-Oliveira, Janet; Dixon, David
doi: 10.1006/bulm.2001.0263pmid: 11732180
We develop the mathematical machinery for the construction of an algebraic-combinatorial model using Petri nets to construct an oriented matroid representation of biochemical pathways. For demonstration purposes, we use a model metabolic pathway example from the literature to derive a general biochemical reaction network model. The biomolecular networks define a connectivity matrix that identifies a linear representation of a Petri net. The sub-circuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariants, that are called S-(state) and T-(transition) invariants. Each invariant has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. We prove that the family of signed sub-circuits has an implicit order that defines an oriented matroid. The oriented matroid is then used to identify the feasible sub-circuit pathways that span the biochemical network as the positive cycles in a hyper-digraph.