The impact of diffusion and stirring on the dynamics of interacting populations

We investigate the combined effects of diffusion and stirring on the dynamics of interacting populations which have spatial structure. Specifically we consider the marine phytoplankton and zooplankton populations, and model them as an excitable medium. The results are applicable to other biological and chemical systems. Under certain conditions the combination of diffusion and stirring is found to enhance the excitability, and hence population growth of the system. Diffusion is found to play an important role: too much and initial perturbations are smoothed away, too little and insufficient mixing takes place before the reaction is over. A key time-scale is the mix-down time, the time it takes for the spatial scale of a population to be reduced to that of a diffusively controlled filament. If the mix-down time is short compared to the reaction time-scale, then excitation of the system is suppressed. For intermediate values of the mix-down time the peak population can attain values many times that of a population without spatial structure. We highlight the importance of the spatial scale of the initial disturbance to the system.     

Coalescence of interacting cell populations

We analyse the coalescence of invasive cell populations by studying both the temporal and steady behaviour of a system of coupled reaction-diffusion equations. This problem is relevant to recent experimental observations of the dynamics of opposingly directed invasion waves of cells. Two cell types, u and v, are considered with the cell motility governed by linear or nonlinear diffusion. The cells proliferate logistically so that the long-term total cell density, u+v approaches a carrying capacity. The steady-state solutions for u and v are denoted u(s) and v(s). The steady solutions are spatially invariant and satisfy u(s)+v(s)=1. However, this expression is underdetermined so the relative proportion of each cell type u(s) and v(s) cannot be determined a priori. Various properties of this model are studied, such as how the relative proportion of u(s) and v(s) depends on the relative motility and relative proliferation rates. The model is analysed using a combination of numerical simulations and a comparison principle. This investigation unearths some novel outcomes regarding the role of overcrowding and cell death in this type of cell migration assay. These observations have relevance to experimental design and interpretation regarding the identification and parameterisation of mechanisms involved in cell invasion.     

Modelling the movement of interacting cell populations

Mathematical modelling of cell movement has traditionally focussed on a single population of cells, often moving in response to various chemical and environmental cues. In this paper, we consider models for movement in two or more interacting cell populations. We begin by discussing intuitive ideas underlying the extension of models for a single-cell population to two interacting populations. We then consider more formal model development using transition probability methods, and we discuss how the same equations can be obtained as the limiting form of a velocity-jump process. We illustrate the models we have developed via two examples. The first of these is a generic model for competing cell populations, and the second concerns aggregation in cell populations moving in response to chemical gradients.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

Persistence in models of three interacting predator-prey populations

This paper considers a class of deterministic models of three interacting populations with a view towards determining when all of the populations persist. In analytical terms persistence means that liminf_t→∞x(t)> 0 for each population x(t); in geometric terms, that each trajectory of the modeling system of differential equations is eventually bounded away from the coordinate planes. The class of systems considered allows three level food webs, two competing predators feeding on a single prey, or a single predator feeding on two competing prey populations. As a corollary to the last case it is shown that the addition of a predator can lead to persistence of a three population system where, without a predator, the two competing populations on the lower trophic level would have only one survivor. The basic models are of Kolmogorov type, and the results improve several previous theorems on persistence.     

Adaptation of interacting populations to the environmental temperature regime

The biological process of the passing of individuals from the active to the passive state when unfavourable environmental conditions occur has been formalized. This leads to a universal linear superstructing to nonlinear models of ecosystems. At certain rates of this jump, a stabilization of the oscillation instability is possible, and an increase in population vitality can occur. The adaptation processes were examined, and the areas of evolutionarily stable parameters for interacting population (competitors and antagonists) on regular temperature changes were found. A possibility of cyclic changes of parameters in the "prey-predator" system was shown.     

Control of neural information transmission by synaptic dynamics.

The computational processing of a neural system is strongly influenced by the dynamical characteristics of the information transmission between neurons. In this work, the control of neural information transmission by synaptic dynamics is investigated by means of a master-equation-based stochastic model of pre-synaptic release of neurotransmitter-containing vesicles. The model incorporates facilitation of vesicle fusion with the pre-synaptic membrane due to intracellular calcium ions and depletion of readily releasable vesicles. The message to be transmitted is coded by the pre-synaptic firing sequence, and the received signal corresponds to the post-synaptic membrane potential response. At the sending end, the stochastic character of the vesicle release contributes to the entropy of the probability distribution of the number of vesicles released and represents noise with respect to information transmission. At the receiving end, the generation of post-synaptic membrane potentials is influenced by the temporal behaviour of ionic currents and membrane charging and is determined by means of a low-dimensional model. The rate and temporal types of neural coding are compatible with limiting cases of the synaptic information transmission as a function of initial vesicle release probability and pre-synaptic firing rate. The effects of the nonlinear dependencies of the vesicle release probability on intracellular calcium concentration and number of available vesicles are analysed. The model is compared with phenomenological and reduced models, a principal advantage being the capability of also determining fluctuations of dynamic variables Copyright 2002 Academic Press.     

Analysis of two interacting bacterial populations with opposite substrate preferences

Interactions between two bacterial species which exhibit opposite substrate preferences in response to the same environmental conditions were investigated. Klebsiella oxytoca preferentially utilizes glucose over citrate as a substrate while Pseudomonas aeruginosa utilizes citrate over glucose. Both species were grown on single as well as mixed substrates, namely, D-glucose and citrate, with various ratios of inocula and substrates. Competition is observed between the two species with citrate alone while amensalism is observed with glucose alone. In mixed cultures-mixed substrates experiments, amensalism of K. oxytoca on P. aeruginosa prevailed until the exhaustion of glucose.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.     

Spatial dynamics of Norwegian tetraonid populations

Different species in a given site or population of a given species in different sites may fluctuate in synchrony if they are affected similarly by factors such as spatially autocorrelated climate, predation, or by dispersal between populations of one species. We used county wise time series of hunting bag records of four Norwegian tetraonid species covering 24 years to examine patterns of interspecific and intraspecific synchrony. We estimated synchrony at three spatial scales; national, regional (consisting of counties with similar climate), and county level. Ecologically related species with overlapping distributions exhibited strong synchrony across Norway, but there was much variation between the different regions and counties. Regions with a long coastline to both the North Sea and the Norwegian Ocean exhibited an overall stronger synchrony than those consisting of more continental areas. Intraspecific synchrony was generally low across all counties, but stronger synchrony between counties within regions defined by climatic conditions. Synchrony was negatively related to distance between populations in three of four species. Only the synchrony in willow ptarmigan showed a clear negative relationship with distance, while the other species had both strong positive and negative correlations at short distances. Strong interspecific synchrony between some species pairs within regions and weak intraspecific synchrony across counties within regions suggest a stronger synchronizing effect from environmental factors such as weather or predation and less effect from dispersal. Our results suggest that the complete tetraonid community is structured by environmental factors affecting the different species similarly and causes widespread interspecific synchrony. Local factors affecting the population dynamics nevertheless frequently forces neighbouring populations out of phase.     

The regulation of microtubule dynamics in Saccharomyces cerevisiae by three interacting plus-end tracking proteins

Microtubule plus-end tracking proteins (+TIPs) are a diverse group of molecules that regulate microtubule dynamics and interactions of microtubules with other cellular structures. Many +TIPs have affinity for each other but the functional significance of these associations is unclear. Here we investigate the physical and functional interactions among three +TIPs in S. cerevisiae, Stu2, Bik1, and Bim1. Two-hybrid, coimmunoprecipitation, and in vitro binding assays demonstrate that they associate in all pairwise combinations, although the interaction between Stu2 and Bim1 may be indirect. Three-hybrid assays indicate that these proteins compete for binding to each other. Thus, Stu2, Bik1, and Bim1 interact physically but do not appear to be arranged in a single unique complex. We examined the functional interactions among pairs of proteins by comparing cytoplasmic and spindle microtubule dynamics in cells lacking either one or both proteins. On cytoplasmic microtubules, Stu2 and Bim1 act cooperatively to regulate dynamics in G1 but not in preanaphase, whereas Bik1 acts independently from Stu2 and Bim1. On kinetochore microtubules, Bik1 and Bim1 are redundant for regulating dynamics, whereas Stu2 acts independently from Bik1 and Bim1. These results indicate that interactions among +TIPS can play important roles in the regulation of microtubule dynamics.     

Molecular dynamics decomposition of temperature-dependent elastic neutron scattering by a protein solution

Molecular dynamics simulations are performed of bovine pancreatic trypsin inhibitor in a cryosolution over a range of temperatures from 80 to 300 K and the origins identified of elastic dynamic neutron scattering from the solution. The elastic scattering and mean-square displacement calculated from the molecular dynamics trajectories are in reasonable agreement with experiments on a larger protein in the same solvent. The solvent and protein contributions to the scattering from the simulation model are determined. At lower temperatures (< approximately 200 K) or on shorter timescales ( approximately 10 ps) the scattering contributions are proportional to the isotopic nuclear scattering cross-sections of each component. However, for T > 200 K marked deviations from these cross-sections are seen due to differences in the dynamics of the components of the solution. Rapid activation of solvent diffusion leads to the variation with temperature of the total elastic intensity being determined largely by that of the solvent. At higher temperatures (>240 K) and longer times ( approximately 100 ps) the protein makes the only significant contribution to the scattering, the solvent scattering having moved out of the accessible time-space window. Decomposition of the protein mean-square displacement shows that the observed dynamical transition in the solution at 200-220 K involves activation of both internal motions and external whole-molecule rotational and translational diffusion. The proportion that the external dynamics contributes to the protein mean-square displacement increases to approximately 30 and 60% at 300 K on the 10- and 100-ps timescales, respectively.     

Bone morphogenetic proteins regulate neural tube closure by interacting with the apicobasal polarity pathway

During neural tube closure, specialized regions called hinge points (HPs) display dynamic and polarized cell behaviors necessary for converting the neural plate into a neural tube. The molecular bases of such cell behaviors (e.g. apical constriction, basal nuclear migration) are poorly understood. We have identified a two-dimensional canonical BMP activity gradient in the chick neural plate that results in low and temporally pulsed BMP activity at the ventral midline/median hinge point (MHP). Using in vivo manipulations, high-resolution imaging and biochemical analyses, we show that BMP attenuation is necessary and sufficient for MHP formation. Conversely, BMP overexpression abolishes MHP formation and prevents neural tube closure. We provide evidence that BMP modulation directs neural tube closure via the regulation of apicobasal polarity. First, BMP blockade produces partially polarized neural cells, which retain contact with the apical and basal surfaces but where basolateral proteins (LGL) become apically localized and apical junctional proteins (PAR3, ZO1) become targeted to endosomes. Second, direct LGL misexpression induces ectopic HPs identical to those produced by noggin or dominant-negative BMPR1A. Third, BMP-dependent biochemical interactions occur between the PAR3-PAR6-aPKC polarity complex and phosphorylated SMAD5 at apical junctions. Finally, partially polarized cells normally occur at the MHP, their frequencies inversely correlated with the BMP activity gradient in the neural plate. We propose that spatiotemporal modulation of the two-dimensional BMP gradient transiently alters cell polarity in targeted neuronal cells. This ensures that the neural plate is flexible enough to be focally bent and shaped into a neural tube, while retaining overall epithelial integrity.     

The neural substrate of spectral preference in Drosophila

Drosophila vision is mediated by inputs from three types of photoreceptor neurons; R1-R6 mediate achromatic motion detection, while R7 and R8 constitute two chromatic channels. Neural circuits for processing chromatic information are not known. Here, we identified the first-order interneurons downstream of the chromatic channels. Serial EM revealed that small-field projection neurons Tm5 and Tm9 receive direct synaptic input from R7 and R8, respectively, and indirect input from R1-R6, qualifying them to function as color-opponent neurons. Wide-field Dm8 amacrine neurons receive input from 13-16 UV-sensing R7s and provide output to projection neurons. Using a combinatorial expression system to manipulate activity in different neuron subtypes, we determined that Dm8 neurons are necessary and sufficient for flies to exhibit phototaxis toward ultraviolet instead of green light. We propose that Dm8 sacrifices spatial resolution for sensitivity by relaying signals from multiple R7s to projection neurons, which then provide output to higher visual centers.     

Stability in a class of discrete time models of interacting populations

Summary Effective Lyapunov and Lyapunov-like functions for a class of discrete time models of interacting populations are presented. These functions are constructed on the biologically meaningful principle that a viable population must absorb energy from external sources when its density is low and it must dissipate energy to the environment when its density is high. These functions can be used to establish that a discrete time model is globally stable or that its solutions are ultimately confined to an acceptable region of the state space. The latter is especially interesting when the model has chaotic solutions. These methods are applied to a single species model and a model of competition between two species.     

Study of a tri-trophic prey-dependent food chain model of interacting populations

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf–Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.     

Optimal open-loop desynchronization of neural oscillator populations

Deep brain stimulation (DBS) is an increasingly used medical treatment for various neurological disorders. While its mechanisms are not fully understood, experimental evidence suggests that through application of periodic electrical stimulation DBS may act to desynchronize pathologically synchronized populations of neurons resulting desirable changes to a larger brain circuit. However, the underlying mathematical mechanisms by which periodic stimulation can engender desynchronization in a coupled population of neurons is not well understood. In this work, a reduced phase-amplitude reduction framework is used to characterize the desynchronizing influence of periodic stimulation on a population of coupled oscillators. Subsequently, optimal control theory allows for the design of periodic, open-loop stimuli with the capacity to destabilize completely synchronized solutions while simultaneously stabilizing rotating block solutions. This framework exploits system nonlinearities in order to strategically modify unstable Floquet exponents. In the limit of weak neural coupling, it is shown that this method only requires information about the phase response curves of the individual neurons. The effects of noise and heterogeneity are also considered and numerical results are presented. This framework could ultimately be used to inform the design of more efficient deep brain stimulation waveforms for the treatment of neurological disease.