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.     

Blind spectral decomposition of single-cell fluorescence by parallel factor analysis

Simultaneous measurement of multiple signaling molecules is essential to investigate their relations and interactions in living cells. Although a wide variety of fluorescent probes are currently available, the number of probes that can be applied simultaneously is often limited by the overlaps among their fluorescence spectra. We developed the experimental system to measure and analyze many overlapping fluorescent components in single cells. It is based on the recording of two-dimensional single-cell fluorescence spectra and on the blind spectral decomposition of fluorescence data by method of parallel factor analysis. Because this method does not require any preknowledge about the shapes of individual component spectra, it can be applied to the specimens that contain fluorescent components with unknown spectra. By examining the performance using the mixture solutions of fluorescent indicators, it was confirmed that >10 largely overlapping spectral components could be easily separated. The effectiveness in the physiological experiments was proven in the applications to the temporal analysis of intracellular Ca(2+) concentration and pH, as well as the intrinsic fluorescent components, in single mouse oocytes.     

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.     

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.     

Pyk2 regulates cell-edge protrusion dynamics by interacting with Crk

Focal adhesion kinase (FAK) is well established as a regulator of cell migration, but whether and how the closely related proline-rich tyrosine kinase 2 (Pyk2) regulates fibroblast motility is still under debate. Using mouse embryonic fibroblasts (MEFs) from Pyk2^–/– mice, we show here, for the first time, that lack of Pyk2 significantly impairs both random and directed fibroblast motility. Pyk2^–/– MEFs show reduced cell-edge protrusion dynamics, which is dependent on both the kinase and protein–protein binding activities of Pyk2. Using bioinformatics analysis of in vitro high- throughput screens followed by text mining, we identified CrkI/II as novel substrates and interactors of Pyk2. Knockdown of CrkI/II shows altered dynamics of cell-edge protrusions, which is similar to the phenotype observed in Pyk2^–/– MEFs. Moreover, epistasis experiments suggest that Pyk2 regulates the dynamics of cell-edge protrusions via direct and indirect interactions with Crk that enable both activation and down-regulation of Crk-mediated cytoskeletal signaling. This complex mechanism may enable fine-tuning of cell-edge protrusion dynamics and consequent cell migration on the one hand together with tight regulation of cell motility, a process that should be strictly limited to specific time and context in normal cells, on the other hand.     

Spatially localized interactive neural populations—II: Stability and dynamics of excitatory sets

The stability characteristics and dynamical behavior of a system of mutually excitatory neurons in close spatial proximity are investigated with a mathematical model. The model predicts the existence of uniform, intermediate levels of activity other than those of no activity and maximal activity. The model also, yeilds a good explanation of data obtained from periglomerular neurons in the olfactory bulb of the cat. This paper is based on a dissertation submitted by the author to Rensselaer Polytechnic Institute in partial fulfillment of the requirements for the degree of Doctor of Philosophyy. This work was supported in part by a Fellowship sponsored by the IBM Corporation and in part by the National Science Foundation under Grant MCS75-08328.     

Fixed point sensitivity analysis of interacting structured populations

Sensitivity analysis of structured populations is a useful tool in population ecology. Historically, methodological development of sensitivity analysis has focused on the sensitivity of eigenvalues in linear matrix models, and on single populations. More recently there have been extensions to the sensitivity of nonlinear models, and to communities of interacting populations. Here we derive a fully general mathematical expression for the sensitivity of equilibrium abundances in communities of interacting structured populations. Our method yields the response of an arbitrary function of the stage class abundances to perturbations of any model parameters. As a demonstration, we apply this sensitivity analysis to a two-species model of ontogenetic niche shift where each species has two stage classes, juveniles and adults. In the context of this model, we demonstrate that our theory is quite robust to violating two of its technical assumptions: the assumption that the community is at a point equilibrium and the assumption of infinitesimally small parameter perturbations. Our results on the sensitivity of a community are also interpreted in a niche theoretical context: we determine how the niche of a structured population is composed of the niches of the individual states, and how the sensitivity of the community depends on niche segregation.     

On some Markov models of certain interacting populations

We give a stochastic foundation to the Volterra prey-predator population in the following case. We take Volterra's predator equations and let a free host birth and death process support the evolution of the predator population. The purpose of this article is to present a rigorous population sample path construction of this interacted predator process and study the properties of this interacted process. The constructions yields a strong Markov process. The existence of steady-state distribution for the interacted predator process means the existence of equilibrium population level. We find a necessary and sufficient condition for the existence of a steady-state distribution. Next we see that if the host process possesses a steady-state distribution, so does the interacted predator process and this distribution satisfies a difference equation. For special choices of the auto death and interaction parametersa andb of the predator, whenever the host process visits the particular statea ^*=a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.     

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.     

Population dynamics of size-structured populations

[First paragraph]Many population cohorts are structured by different attributes: space and age are the most common of these. For many applications this structure is just as important as the temporal evolution with discrete or continuous time. For these applications modellers have to cope with this greatly increased complexity and cannot afford to neglect it. Different modellers proceed in a variety of ways to suit the circumstances and tools available. Nigel Barlow introduced me (in 1988) to the idea of using a limited number of spatial or age compartments, each of which was structurally uniform with well-defined interactive links between compartments. His view, if I interpret it correctly, was that this enabled us to avoid the technically difficult partial differential equation analysis. Of course, there are a wide variety of opinions about this, but I have recently used this approach in an analysis of a plankton– nutrient model, where Nigel was clearly right. We would have never succeeded in using the lovely path- following (in parameter space) computer algorithms had we not adopted Nigel’s approach. This paper addresses a situation where compartments were not, in fact, sufficient: where the population is structured by size or DNA content. Data sets of plant-root-cells, plankton, muscle cells, and cancer cells, often show a steady (in time) size distribution that appears asymptotically and structurally stable. (“Steady-size” means it is constant in shape as time changes.) About the same time as Nigel and I began (in Palmerston North around 1988) looking at a spatially compartmentalised model for tuberculosis in possums, another biologist, Dr Paul Gandar, asked whether we could explain this with a relatively simple model which, when validated and embedded in the cell physiological context, could have the capability of making robust predictions.     

Location coding by opponent neural populations in the auditory cortex

Although the auditory cortex plays a necessary role in sound localization, physiological investigations in the cortex reveal inhomogeneous sampling of auditory space that is difficult to reconcile with localization behavior under the assumption of local spatial coding. Most neurons respond maximally to sounds located far to the left or right side, with few neurons tuned to the frontal midline. Paradoxically, psychophysical studies show optimal spatial acuity across the frontal midline. In this paper, we revisit the problem of inhomogeneous spatial sampling in three fields of cat auditory cortex. In each field, we confirm that neural responses tend to be greatest for lateral positions, but show the greatest modulation for near-midline source locations. Moreover, identification of source locations based on cortical responses shows sharp discrimination of left from right but relatively inaccurate discrimination of locations within each half of space. Motivated by these findings, we explore an opponent-process theory in which sound-source locations are represented by differences in the activity of two broadly tuned channels formed by contra- and ipsilaterally preferring neurons. Finally, we demonstrate a simple model, based on spike-count differences across cortical populations, that provides bias-free, level-invariant localization—and thus also a solution to the “binding problem” of associating spatial information with other nonspatial attributes of sounds.     

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.     

Harvest and dynamics of duck populations

The role of harvest in the dynamics of waterfowl populations continues to be debated among scientists and managers. Our perception is that interested members of the public and some managers believe that harvest influences North American duck populations based on calls for more conservative harvest regulations. A recent review of harvest and population dynamics of North American mallard (Anas platyrhynchos) populations (Pöysä et al. 2004) reached similar conclusions. Because of the importance of this issue, we reviewed the evidence for an impact of harvest on duck populations. Our understanding of the effects of harvest is limited because harvest effects are typically confounded with those of population density; regulations are typically most liberal when populations are greatest. This problem also exists in the current Adaptive Harvest Management Program (Conn and Kendall 2004). Consequently, even where harvest appears additive to other mortality, this may be an artifact of ignoring effects of population density. Overall, we found no compelling evidence for strong additive effects of harvest on survival in duck populations that could not be explained by other factors. © 2012 The Wildlife Society.