Modelling the effect of temperature on the range expansion of species by reaction-diffusion equations

The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.     

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

Spatio-temporal patterns of gene flow and dispersal under temperature increase

Recent data show that the Earth climate is undergoing a change at a rate which is outstanding in geologic history. Temperature is one of the major driving forces of gene flow and dispersal. In this paper the spatial dynamics of genetic dispersal is studied under the auspices of temperature increase by means of a mathematical model. The main elements genetics, competition and dispersal are combined in a coherent approach by a system of coupled partial differential equations with non-linear reaction terms describing population dynamics, genetic exchange and competition. Temperature reaction norms are conferred by a two allele system. The non-linearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. Here we show how invasion processes in form of travelling waves are initiated by a temperature rise.     

Roles of dispersal, stochasticity, and nonlinear dynamics in the spatial structuring of seasonal natural enemy–victim populations

Natural enemy–victim systems may exhibit a range of dynamic space–time patterns. We used a theoretical framework to study spatiotemporal structuring in a transient natural enemy–victim system subject to differential rates of dispersal, stochastic forcing, and nonlinear dynamics. Highly mobile natural enemies that attacked less mobile victims were locally spatially segregated from each other when governed by approximate linear dynamics. In contrast, in nonlinear dynamical systems, such as cyclic populations, interacting species achieved local aggregation with each other regardless of dispersal rates, and aggregation was enhanced specifically when highly mobile enemies attacked less mobile victims. These patterns of spatial aggregation held under varying levels of stochastic forcing. This work thus shows a range of dynamic spatial patterns in interacting-species models, and how spatial aggregation between natural enemies and victims can be achieved in locally unstable populations that are linked through dispersal.     

带扩散的Barbour血吸虫病模型的定性分析(英文)

这篇文章主要考虑由常微分方程组和偏微分方程组构成的Barbour血吸虫病模型.偏微分系统是反映空间和时间分布的反应扩散系统.对模型的定性性质进行了分析.利用比较原理得出解的一致有上界性.同时利用能量方法证明出椭圆系统在扩散系数的一定范围内没有非常数的正稳态解.     

Dispersal, disease and life-history evolution

Discrete-time susceptible-infective-susceptible (S-I-S) disease transmission models can exhibit bistability (alternative stable equilibria) over a wide range of parameter values. We illustrate the richness generated by such 'simple' non-linear systems in the study of two patch epidemic models with disease-enhanced or disease-suppressed dispersal. Dispersal between patches can have a profound impact on local patch disease dynamics. In fact, dispersal between patches may give rise to bistability in parameter regimes without bistability in single patch models.     

Spatial dynamics in model plant communities: what do we really know?

A variety of models have shown that spatial dynamics and small-scale endogenous heterogeneity (e.g., forest gaps or local resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real systems. We synthesize results from three different kinds of models: interacting particle systems, moment equations for spatial point processes, and metapopulation or patch models. Studies using all three frameworks agree that spatial dynamics need not enhance coexistence nor slow down dynamics; their effects depend on the underlying competitive interactions in the community. When similar species would coexist in a nonspatial habitat, endogenous spatial structure inhibits coexistence and slows dynamics. When a dominant species disperses poorly and the weaker species has higher fecundity or better dispersal, competition-colonization trade-offs enhance coexistence. Even when species have equal dispersal and per-generation fecundity, spatial successional niches where the weaker and faster-growing species can rapidly exploit ephemeral local resources can enhance coexistence. When interspecific competition is strong, spatial dynamics reduce founder control at large scales and short dispersal becomes advantageous. We describe a series of empirical tests to detect and distinguish among the suggested scenarios.     

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.     

Modeling bird species distribution change in fire prone Mediterranean landscapes: incorporating species dispersal and landscape dynamics

Species distribution models (SDMs) have traditionally been founded on the assumption that species distributions are in equilibrium with environmental conditions and that these species–environment relationships can be used to estimate species responses to environmental changes. Insight into the validity of this assumption can be obtained from comparing the performance of correlative species distribution models with more complex hybrid approaches, i.e. correlative and process‐based models that explicitly include ecological processes, thereby accounting for mismatches between habitat suitability and species occupancy patterns. Here we compared the ability of correlative SDMs and hybrid models, which can accommodate non‐equilibrium situations arising from dispersal constraints, to reproduce the distribution dynamics of the ortolan bunting Emberiza hortulana in highly dynamic, early successional, fire driven Mediterranean landscapes. Whereas, habitat availability was derived from a correlative statistical SDM, occupancy was modeled using a hybrid approach combining a grid‐based, spatially‐explicit population model that explicitly included bird dispersal with the correlative model. We compared species occupancy patterns under the equilibrium assumption and different scenarios of species dispersal capabilities. To evaluate the predictive capability of the different models, we used independent species data collected in areas affected to different degree by fires. In accordance with the view that disturbance leads to a disparity between the suitable habitat and the occupancy patterns of the ortolan bunting, our results indicated that hybrid modeling approaches were superior to correlative models in predicting species spatial dynamics. Furthermore, hybrid models that incorporated short dispersal distances were more likely to reproduce the observed changes in ortolan bunting distribution patterns, suggesting that dispersal plays a key role in limiting the colonization of recently burnt areas. We conclude that SDMs used in a dynamic context can be significantly improved by using combined hybrid modeling approaches that explicitly account for interactions between key ecological constraints such as dispersal and habitat suitability that drive species response to environmental changes.     

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

On mathematical theory of selection: continuous time population dynamics

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.     

Shadows of predation: habitat-selecting consumers eclipse competition between coexisting prey

Ecologists have made substantial progress evaluating the influences of adaptive behaviors on population dynamics and communities. But no-one has examined the joint influences of stochastic variation, predators, and density-dependent habitat selection on our interpretations of species coexistence. I begin the search with simulation models of habitat isodars (lines along which the fitness of individuals is identical in two or more habitats) assuming ideal-free habitat selection by two prey species exploited by a habitat-selecting parasitoid predator. The models include both regulating and non-regulating stochasticity. The intriguing results include the following: (1) all three species often achieved a true ideal-free distribution; (2) predators reduced prey population sizes and increased the frequency of local habitat extinctions; (3) despite the predator's differential reduction of prey densities, there was no evidence of apparent competition; (4) all species exhibited pulses of dispersal associated with donor–receiver population dynamics; (5) isodars produced valid estimates of competition between prey only in constant environments lacking habitat-selecting predators; (6) habitat-selection by predators forced prey into their preferred habitats; (7) the resulting ghost of competition obscured the prey species' competitive interaction; (8) isodars correctly revealed density-dependent habitat selection by the predator. Overall, the results appeared to depend primarily on the predator's habitat choice, rather than on prey trade-offs between competitive ability and reduced value (handling time) to the predator. Habitat selection theory, and its revelation via isodars, can thus provide considerable insight into processes affecting real communities, and most especially if ecologists assess carefully the constraints for their analysis and interpretation. Nevertheless, isodars designed to measure competition are likely to be most reliable in donor-controlled or experimental systems where regulating stochasticity has relatively little influence on prey dynamics.     

Differential equations models for interacting wild and transgenic mosquito populations

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling-II-type functional mating rate that incorporates Allee effects, in order to account for mating difficulty when the size of the total mosquito populations is small. We investigate the existence and stability of both boundary and positive equilibria. We show that the Holling-II-type model is the more realistic and, by means of numerical simulations, that it exhibits richer dynamics.     

Differential equations models for interacting wild and transgenic mosquito populations

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling-II-type functional mating rate that incorporates Allee effects, in order to account for mating difficulty when the size of the total mosquito populations is small. We investigate the existence and stability of both boundary and positive equilibria. We show that the Holling-II-type model is the more realistic and, by means of numerical simulations, that it exhibits richer dynamics.     

Global dispersal reduces local diversity

Metapopulation models and stepping-stone models in genetics are based on very different underlying dispersal structures, yet it can be difficult to distinguish the behaviour of the two kinds of models. We demonstrate a striking qualitative difference in the equilibrium behaviour possible with these two kinds of dispersal. If, in a local patch, there are multiple stable equilibria (and consequently an unstable equilibrium), we demonstrate that, for the spatial system with a metapopulation structure, at equilibrium every patch has to be near one of the stable equilibria. This contrasts with the clinal structure possible with a stepping-stone or continuous space model; thus the result can be used to deduce qualitative information about the form of dispersal from observations of allele frequencies.     

Mathematical modelling of mosquito dispersal in a heterogeneous environment

Mosquito dispersal is a key behavioural factor that affects the persistence and resurgence of several vector-borne diseases. Spatial heterogeneity of mosquito resources, such as hosts and breeding sites, affects mosquito dispersal behaviour and consequently affects mosquito population structures, human exposure to vectors, and the ability to control disease transmission. In this paper, we develop and simulate a discrete-space continuous-time mathematical model to investigate the impact of dispersal and heterogeneous distribution of resources on the distribution and dynamics of mosquito populations. We build an ordinary differential equation model of the mosquito life cycle and replicate it across a hexagonal grid (multi-patch system) that represents two-dimensional space. We use the model to estimate mosquito dispersal distances and to evaluate the effect of spatial repellents as a vector control strategy. We find evidence of association between heterogeneity, dispersal, spatial distribution of resources, and mosquito population dynamics. Random distribution of repellents reduces the distance moved by mosquitoes, offering a promising strategy for disease control.