Cattaneo models for chemosensitive movement

We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models to classical models and we develop an effective numerical scheme.     

Derivation of hyperbolic models for chemosensitive movement

A Chapman-Enskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocity-jump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocity-jump process. On the other hand, this approach provides a unified framework which includes previous models obtained by ad hoc methods or methods of moments. Numerical simulations are also performed and are motivated by recent experiments with human endothelial cells on matrigel. Their movements lead to the formation of networks that are interpreted as the beginning of a vasculature. These structures cannot be explained by parabolic models but are recovered by numerical experiments on hyperbolic models. Our kinetic model suggests that some kind of local interactions might be enough to explain them.Acknowledgement The authors thank M. Mirshahi (INSERM E355 - Faculté de Médecine de Paris VI) for fruitful discussions and providing experimental data. Helpful discussions on numerical and modeling issues with A. Gamba and M. Lemou are gratefully acknowledged. This work was also partially supported by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282.     

Differential movement and movement bias models for marine protected areas

Marine protected areas (MPAs) are promoted as a tool to protect overfished stocks and increase fishery yields. Previous models suggested that adult mobility modified effects of MPAs by reducing densities of fish inside reserves, but increasing yields (i.e., increasing densities outside of MPAs). Empirical studies contradicted this prediction: as mobility increased, the relative density of fishes inside MPAs (relative to outside) increased or stayed constant. We hypothesized that this disparity between theoretical and empirical results was the result of differential movement of fish inside versus outside the MPA. We, therefore, developed a model with unequal and discontinuous diffusion, and analyzed its steady state and stability. We determined the abundance in the fishing grounds, the yield, the total abundance and the log ratio at steady-state and examined their response to adult mobility (while keeping the relative inequity in the diffusion constant). Abundance in the fishing grounds and yield increased, while total abundance and log-ratio decreased, as mobility increased. These results were all qualitatively consistent with the previous models assuming uniform diffusivity. Thus, the mismatch between empirical and theoretical results must result from other processes or other forms of differential movement. Therefore, we modified our original model by assuming that species located on the boundary of the MPA will preferentially move towards the MPA. This localized movement bias model gives rise to steady state profiles that can differ radically from the profiles in the unbiased model, especially when the bias is large. Moreover, for sufficiently large bias values, the monotonicity of the four measures with increased mobility is reversed, when compared with our original model. Thus, the movement bias model reconciles empirical data and theoretical results.     

Analytical and numerical tools for diffusion-based movement models

I present a general diffusion-based modeling framework for the analysis of animal movements in heterogeneous landscapes, including terms representing advection, mortality, and edge-mediated behavior. I use adjoint operator theory to develop mathematical machinery for the assessment of a number of biologically relevant quantities, such as occupancy times, hitting probabilities, quasi-stationary distributions, the backwards equation, and conditional probability densities. I derive finite-element approximations, which can be used to obtain numerical solutions in domains which do not allow for an analytical treatment. As an example, I model the movements of the butterfly Melitaea cinxia in an island consisting of a set of habitat patches and the intervening matrix habitat. I illustrate the behavior of the model and the mathematical theory by examining the effects of a hypothetical movement barrier and advection caused by prevailing wind conditions.     

Evaluating models for classifying movement of whale-watching vessels

A continued rise in global ocean vessel activity has led to growing concerns for the health of whales around the world. Of particular interest is the increase in recreation vessels, including those related to whale-watching activities. However, there is an absence of established procedures to identify vessels engaged in whale-watching, thus limiting the ability to quantify whale-watching impacts on whales. This study evaluates three computational classification models and their ability to utilize Automatic Identification System (AIS) data to describe wildlife-viewing vessel behaviour. These models include a density-based spatial clustering application with noise (DBSCAN), a hidden Markov model (HMM), and logistic regression (LR), all of which have been previously used to classify vessel behaviour in industries, such as fishing, shipping, and marine security. The results of each model's classification were validated against observed whale sighting data using statistical performance and accuracy metrics. The findings suggest that all three classification models sufficiently detect wildlife-viewing behaviour, but the HMM and LR had preferable performance metrics compared to DBSCAN. Further, although LR provides an informative glance at which AIS variables are most important to detecting wildlife-viewing events, the HMM has comparable performance metrics and requires less data processing. Therefore, this study recommends the use of HMM due to its computational efficiency and because it provides an accurate classification of wildlife-viewing behaviour for whale-watching vessels. The results of this study can be used to support policy decisions, monitor regulation compliance, and inform marine conservation initiatives.     

Compartmental models with restricted movement

A restriction is imposed on the number of particles that can possibly move at any time from a compartment, so that any other particles present in the compartment must wait until such particles have moved out. The equations for such a system are formulated and the solution is given for a single compartment system; increased variability of the compartmental particle count is one effect of this restriction.     

Extended foot-ankle musculoskeletal models for application in movement analysis

Multibody simulations of human motion require representative models of the anatomical structures. A model that captures the complexity of the foot is still lacking. In the present work, two detailed 3D multibody foot-ankle models generated based on CT scans using a semi-automatic tool are described. The proposed models consists of five rigid segments (talus, calcaneus, midfoot, forefoot and toes), connected by five joints (ankle, subtalar, midtarsal, tarsometatarsal and metatarsophalangeal), one with 15DOF and the other with 8DOF. The calculated kinematics of both models were evaluated using gait trials and compared against literature, both presenting realistic results. An inverse dynamic analysis was performed for the 8DOF model, again presenting feasible dynamic results.     

State-space models of individual animal movement

Detailed observation of the movement of individual animals offers the potential to understand spatial population processes as the ultimate consequence of individual behaviour, physiological constraints and fine-scale environmental influences. However, movement data from individuals are intrinsically stochastic and often subject to severe observation error. Linking such complex data to dynamical models of movement is a major challenge for animal ecology. Here, we review a statistical approach, state-space modelling, which involves changing how we analyse movement data and draw inferences about the behaviours that shape it. The statistical robustness and predictive ability of state-space models make them the most promising avenue towards a new type of movement ecology that fuses insights from the study of animal behaviour, biogeography and spatial population dynamics.     

Considerations on models of movement detection

A general characterization of multi-input movement detection models is given in terms of the Volterra series formalism. When nonlinearities of order higher than the second are negligible, an n-input system can be decomposed into a set of 2-input systems, summing linearly. For a (symmetrical) 2-input system which has significant nonlinearities only up to the second order, the correlation model is its most general expression, if the infinite time average of the output is taken. Specific observations from optomotor experiments (e.g. phase invariance and contrast frequency dependence) can be interpreted in a general way in terms of properties of the Volterra representation.     

Free boundary models for mosquito range movement driven by climate warming

Journal of Mathematical Biology - As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and...     

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Incorporating movement into models of grey seal population dynamics

1. One of the most difficult problems in developing spatially explicit models of population dynamics is the validation and parameterization of the movement process. We show how movement models derived from capture-recapture analysis can be improved by incorporating them into a spatially explicit metapopulation model that is fitted to a time series of abundance data. 2. We applied multisite capture-recapture analysis techniques to photo-identification data collected from female grey seals at the four main breeding colonies in the North Sea between 1999 and 2001. The best-fitting movement models were then incorporated into state-space metapopulation models that explicitly accounted for demographic and observational stochasticity. 3. These metapopulation models were fitted to a 20-year time series of pup production data for each colony using a Bayesian approach. The best-fitting model, based on the Akaike Information Criterion (AIC), had only a single movement parameter, whose confidence interval was 82% less than that obtained from the capture-recapture study, but there was some support for a model that included an effect of distance between colonies. 4. The state-space modelling provided improved estimates of other demographic parameters. 5. The incorporation of movement, and the way in which it was modelled, affected both local and regional dynamics. These differences were most evident as colonies approached their carrying capacities, suggesting that our ability to discriminate between models should improve as the length of the grey seal time series increases.     

Bayesian models of eye movement selection with retinotopic maps

Among the various possible criteria guiding eye movement selection, we investigate the role of position uncertainty in the peripheral visual field. In particular, we suggest that, in everyday life situations of object tracking, eye movement selection probably includes a principle of reduction of uncertainty. To evaluate this hypothesis, we confront the movement predictions of computational models with human results from a psychophysical task. This task is a freely moving eye version of the multiple object tracking task, where the eye movements may be used to compensate for low peripheral resolution. We design several Bayesian models of eye movement selection with increasing complexity, whose layered structures are inspired by the neurobiology of the brain areas implied in this process. Finally, we compare the relative performances of these models with regard to the prediction of the recorded human movements, and show the advantage of taking explicitly into account uncertainty for the prediction of eye movements.     

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.     

Multiscale models for movement in oriented environments and their application to hilltopping in butterflies

Hilltopping butterflies direct their movement in response to topography, facilitating mating encounters via accumulation at summits. In this paper, we take hilltopping as a case study to explore the impact of complex orienteering cues on population dynamics. The modelling employs a standard multiscale framework, in which an individual’s movement path is described as a stochastic ‘velocity-jump’ process and scaling applied to generate a macroscopic model capable of simulating large populations in landscapes. In this manner, the terms and parameters of the macroscopic model directly relate to statistical inputs of the individual-level model (mean speeds, turning rates and turning distributions). Applied to hilltopping in butterflies, we demonstrate how hilltopping acts to aggregate populations at summits, optimising mating for low-density species. However, for abundant populations, hilltopping is not only less effective but also possibly disadvantageous, with hilltopping males recording a lower mating rate than their non-hilltopping competitors.     

Characteristic spatial and temporal scales unify models of animal movement

Animal movements have been modeled with diffusion at large scales and with more detailed movement models at smaller scales. We argue that the biologically relevant behavior of a wide class of movement models can be efficiently summarized with two parameters: the characteristic temporal and spatial scales of movement. We define these scales so that they describe movement behavior both at short scales (through the velocity autocorrelation function) and at long scales (through the diffusion coefficient). We derive these scales for two types of commonly used movement models: the discrete-step correlated random walk, with either constant or random step intervals, and the continuous-time correlated velocity model. For a given set of characteristic scales, the models produce very similar trajectories and encounter rates between moving searchers and stationary targets. Thus, we argue that characteristic scales provide a unifying currency that can be used to parameterize a wide range of ecological phenomena related to movement.