Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave-type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.     

Weakly nonlinear analysis of a hyperbolic model for animal group formation

We consider an one-dimensional nonlocal hyperbolic model for group formation with application to self-organizing collectives of animals in homogeneous environments. Previous studies have shown that this model displays at least four complex spatial and spatiotemporal group patterns. Here, we use weakly nonlinear analysis to better understand the mechanisms involved in the formation of two of these patterns, namely stationary pulses and traveling trains. We show that both patterns arise through subcritical bifurcations from spatially homogeneous steady states. We then use these results to investigate the effect of two social interactions (attraction and alignment) on the structure of stationary and moving animal groups. While attraction makes the groups more compact, alignment has a dual effect, depending on whether the groups are stationary or moving. More precisely, increasing alignment makes the stationary groups compact, and the moving groups more elongated. Also, the results show the existence of a threshold for the total group density, above which, coordinated behaviors described by stationary and moving groups persist for a long time.      

Formation and deformation of patterns through diffusion

Simulating various patterns exhibited on biological forms with mathematical models has become an important supplement to theoretical biology. Models based on a certain mechanism are intended to provide explanations to the formation of a basic pattern. However, in real phenomena, among a basic pattern there always exist some difference between any two individuals. Such differences are consequences of environmental factors posed during the developmental processes. These factors, such as temperature, affect the diffusion rates of corresponding morphogenes which, in turn, alter a basic pattern to certain extent. We provide, in this paper, a quantitative characterization of this effect for a class of reaction-diffusion models.Mathematically, we study the emergence of stationary patterns and their dependence on diffusion rates for this class of models (RD-equations) with no-flux boundary conditions. The results are generalized to systems with homogeneous Dirichlet boundary conditions when the kinetic terms are odd functions. Through an analysis of the phase dynamics, we show that the deformation of stationary patterns, as the diffusion rates change, is governed by the variation of certain plane curves in the phase space. A constructive proof is given which shows explicitly how to obtain such curves.Applications of this study are illustrated with three model examples. We use these models to explain the biological implications of the mathematical features we investigated. Results from computer simulations are presented and compared with physical patterns.     

Multiple resource limitation: nonequilibrium coexistence of species in a competition model using a synthesizing unit

During the last two decades, the simple view of resource limitation by a single resource has been changed due to the realization that co-limitation by multiple resources is often an important determinant of species growth. Hence, the multiple resource limitation hypothesis needs to be taken into account, when communities of species competing for resources are considered. We present a multiple species–multiple resource competition model which is based on the concept of synthesizing unit to formulate the growth rates of species competing for interactive essential resources. Using this model, we demonstrate that a more mechanistic explanation of interactive effects of co-limitation may lead to the known complex dynamics including nonequilibrium states as oscillations and chaos. We compare our findings with earlier investigations on biological mechanisms that can predict the outcome of multispecies competition. Moreover, we show that this model yields a periodic state where more species than limiting complementary resources can coexist (supersaturation) in a homogeneous environment. We identify two novel mechanisms, how such a state can emerge: a transcritical bifurcation of a limit cycle and a transition from a heteroclinic cycle. Furthermore, we demonstrate the robustness of the phenomenon of supersaturation when the environmental conditions are varied.     

The role of convection in the standard fluid-mechanical model for morphogenesis

The effect of convection on reaction-diffusion instabilities in a visco-elastic medium is studied by using the standard continuum theory of a fluid mixture. The medium is assumed to be in local mechanical equilibrium, and convection is generated by pressure forces which arise if the equilibrium density of the medium changes with its composition. A linear stability analysis shows that reaction-diffusion instabilities proceeding from homogeneous steady states at rest are unmodified by induced convection to first order in concentration changes. We suggest that a non-linear analysis would show convection produces no new instabilities, as a linear analysis of inhomogeneous non-convecting stationary states shows that reaction-diffusion growth rates are reduced by convection at long wavelengths and are otherwise unchanged. For applications in embryology, numerical estimates suggest that convection can be ignored in reaction-diffusion mechanisms for pattern formation, and this conclusion is supported by a dimensional analysis.On leave from Department of Physics, Monash University, Clayton, Victoria 3168, Australia     

Receptor-based models with hysteresis for pattern formation in hydra

In this paper, we propose a new receptor-based model for pattern formation and regulation in a fresh-water polyp, namely hydra. The model is defined in the form of a system of reaction-diffusion equations with zero-flux boundary conditions coupled with a system of ordinary differential equations. The production of diffusible biochemical molecules has a hysteretic dependence on the density of these molecules and is modeled by additional ordinary differential equations. We study the hysteresis-driven mechanism of pattern formation and we demonstrate the advantages and constraints of its ability to explain different aspects of pattern formation and regulation in hydra. The properties of the model demonstrate a range of stationary and oscillatory spatially heterogeneous patterns, arising from multiple spatially homogeneous steady states and switches in the production rates.     

Remarks on the number of persistent states to a fragmented predator–prey model system

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.     

Biological Control Through Intraguild Predation: Case Studies in Pest Control,Invasive Species and Range Expansion

Intraguild predation (IGP), the interaction between species that eat each other and compete for shared resources, is ubiquitous in nature. We document its occurrence across a wide range of taxonomic groups and ecosystems with particular reference to non-indigenous species and agricultural pests. The consequences of IGP are complex and difficult to interpret. The purpose of this paper is to provide a modelling framework for the analysis of IGP in a spatial context. We start by considering a spatially homogeneous system and find the conditions for predator and prey to exclude each other, to coexist and for alternative stable states. Management alternatives for the control of invasive or pest species through IGP are presented for the spatially homogeneous system. We extend the model to include movement of predator and prey. In this spatial context, it is possible to switch between alternative stable steady states through local perturbations that give rise to travelling waves of extinction or control. The direction of the travelling wave depends on the details of the nonlinear intraguild interactions, but can be calculated explicitly. This spatial phenomenon suggests means by which invasions succeed or fail, and yields new methods for spatial biological control. Freshwater case studies are used to illustrate the outcomes.     

The asymptotic behavior of solutions of the buffered bistable system

In this paper, we study a model for calcium buffering with bistable nonlinearity. We present some results on the stability of equilibrium states and show that there exists a threshold phenomenon in our model. In comparing with the model without buffers, we see that stationary buffers cannot destroy the asymptotic stability of the associated equilibrium states and the threshold phenomenon. Moreover, we also investigate the propagation property of solutions with initial data being a disturbance of one of the stable states which is confined to a half-line. We show that the more stable state will eventually dominate the whole dynamics and that the speed of this propagation (or invading process) is positive.     

Quasi-elastic light scattering from migrating chemotactic bands of Escherichia coli.

We report the observation of migrating chemotactic bands of Escherichia coli in a buffer solution. The temporal development of the bacterial density profile is observed by the scattered light intensity as the band migrates through a stationary laser beam. We have made a preliminary analysis of the observed band profile with help of the Keller-Segel theory. The model accounts for only some aspects of the observed time evolution of the density profile. The microscopic motility characteristics of the E. coli in the band are simultaneously studied by photon correlation. The measured correlation functions are analyzed to obtain the spatial dependence of the half-width within the band. A simple analytical model is proposed to account for the contribution of the twiddle motion to the correlation function. By analyzing the correlation function as a superposition of straight-line and twiddle motions, we obtain a satisfactory agreement between the theory and the measured angular dependence of the line shape. As a consequence we are able to extract a parameter beta, which measures the average fraction of twiddling bacteria in the center of the band at a given time.     

Distribution of phylogenetic diversity under random extinction

Phylogenetic diversity is a measure for describing how much of an evolutionary tree is spanned by a subset of species. If one applies this to the unknown subset of current species that will still be present at some future time, then this ‘future phylogenetic diversity’ provides a measure of the impact of various extinction scenarios in biodiversity conservation. In this paper, we study the distribution of future phylogenetic diversity under a simple model of extinction (a generalized ‘field of bullets’ model). We show that the distribution of future phylogenetic diversity converges to a normal distribution as the number of species grows, under mild conditions, which are necessary. We also describe an algorithm to compute the distribution efficiently, provided the edge lengths are integral, and briefly outline the significance of our findings for biodiversity conservation.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Plasticity of death rates in stationary phase in Saccharomyces cerevisiae

For the species that have been most carefully studied, mortality rises with age and then plateaus or declines at advanced ages, except for yeast. Remarkably, mortality for yeast can rise, fall and rise again. In the present study we investigated (i) if this complicated shape could be modulated by environmental conditions by measuring mortality with different food media and temperature; (ii) if it is triggered by biological heterogeneity by measuring mortality in stationary phase in populations fractionated into subpopulations of young, virgin cells, and replicatively older, non-virgin cells. We also discussed the results of a staining method to measure viability instead of measuring the number of cells able to exit stationary phase and form a colony. We showed that different shapes of age-specific death rates were observed and that their appearance depended on the environmental conditions. Furthermore, biological heterogeneity explained the shapes of mortality with homogeneous populations of young, virgin cells exhibiting a simple shape of mortality in conditions under which more heterogeneous populations of older cells or unfractionated populations displayed complicated death rates. Finally, the staining method suggested that cells lost the capacity to exit stationary phase and to divide long before they died in stationary phase. These results explain a phenomenon that was puzzling because it appeared to reflect a radical departure from mortality patterns observed for other species.     

Two stationary nonhomogeneous Markov models of nucleotide sequence evolution

The general Markov model (GMM) of nucleotide substitution does not assume the evolutionary process to be stationary, reversible, or homogeneous. The GMM can be simplified by assuming the evolutionary process to be stationary. A stationary GMM is appropriate for analyses of phylogenetic data sets that are compositionally homogeneous; a data set is considered to be compositionally homogeneous if a statistical test does not detect significant differences in the marginal distributions of the sequences. Though the general time-reversible (GTR) model assumes stationarity, it also assumes reversibility and homogeneity. We propose two new stationary and nonhomogeneous models--one constrains the GMM to be reversible, whereas the other does not. The two models, coupled with the GTR model, comprise a set of nested models that can be used to test the assumptions of reversibility and homogeneity for stationary processes. The two models are extended to incorporate invariable sites and used to analyze a seven-taxon hominoid data set that displays compositional homogeneity. We show that within the class of stationary models, a nonhomogeneous model fits the hominoid data better than the GTR model. We note that if one considers a wider set of models that are not constrained to be stationary, then an even better fit can be obtained for the hominoid data. However, the methods for reducing model complexity from an extremely large set of nonstationary models are yet to be developed.     

Two-Species Migration and Clustering in Two-Dimensional Domains

We extend two-species models of individual aggregation or clustering to two-dimensional spatial domains, allowing for more realistic movement of the populations compared with one spatial dimension. We assume that the domain is bounded and that there is no flux into or out of the domain. The motion of the species is along fitness gradients which allow the species to seek out a resource. In the case of competition, species which exploit the resource alone will disperse while avoiding one another. In the case where one of the species is a predator or generalist predator which exploits the other species, that species will tend to move toward the prey species, while the prey will tend to avoid the predator. We focus on three primary types of interspecies interactions: competition, generalist predator–prey, and predator–prey. We discuss the existence and stability of uniform steady states. While transient behaviors including clustering and colony formation occur, our stability results and numerical evidence lead us to believe that the long-time behavior of these models is dominated by spatially homogeneous steady states when the spatial domain is convex. Motivated by this, we investigate heterogeneous resources and hazards and demonstrate how the advective dispersal of species in these environments leads to asymptotic steady states that retain spatial aggregation or clustering in regions of resource abundance and away from hazards or regions or resource scarcity.     

Finite time blow-up in some models of chemotaxis

We consider a class of models of chemotactic bacterial populations, introduced by Keller-Segel. For those models, we investigate the possibility of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-function. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relative strength of three mechanisms in competition: (i) the chemotactic attraction of bacteria towards regions of high concentration in substrate (ii) the rate of consumption of the substrate by the bacteria and (iii) (possibly) the diffusion of bacteria. The solutions we construct are radially symmetric, and therefore have no relation with the classical traveling wave solutions. Our scaling can be justified by a dimensional analysis. We give some evidence of numerical stability.     

Eigensolutions of nonlinear wave equations in one dimension

A class of nonlinear equations describing the steady propagation of a disturbance on the infinite interval in one dimensional space are shown under certain conditions to admit solution with a unique velocity of propagation. The class of equations describe both initial and final homogeneous steady states which are asymptotically stable with respect to uniform perturbations, in contrast to the Fisher equation, which does not.     

Towards a quantitative understanding of horizontal gene transfer: a kinetic model

In this work, we present a simple kinetic model of horizontal gene transfer. It describes the processes of gene duplication, mutation, gene transfer and the regulation of the total size of the genome for genetically homogeneous prokaryotic species or strains. The emerging nonlinear system of first-order differential equations can be linearized at the stationary point. For selected models, we give an analytical solution for the number of foreign and native genes within a species. We identify a regime characterized by a fast gene transfer rate and species with a mixed genome, a slow gene transfer regime with pure organisms, and a crossover region. The data are compared to experiments, and the biological implications of our model are discussed.