Recolonisation by diffusion can generate increasing rates of spread

Diffusion is one of the most frequently used assumptions to explain dispersal. Diffusion models and in particular reaction-diffusion equations usually lead to solutions moving at constant speeds, too slow compared to observations. As early as 1899, Reid had found that the rate of spread of tree species migrating to northern environments at the beginning of the Holocene was too fast to be explained by diffusive dispersal. Rapid spreading is generally explained using long distance dispersal events, modelled through integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, i.e. decaying slower than any exponential. We show here that classical reaction-diffusion models of the Fisher-Kolmogorov-Petrovsky-Piskunov type can produce patterns of colonisation very similar to those of IDEs, if the initial population is EU at the beginning of the considered colonisation event. Many similarities between reaction-diffusion models with EU initial data and IDEs with EU kernels are found; in particular comparable accelerating rates of spread and flattening of the solutions. There was previously no systematic mathematical theory for such reaction-diffusion models with EU initial data. Yet, EU initial data can easily be understood as consequences of colonisation-retraction events and lead to fast spreading and accelerating rates of spread without the long distance hypothesis.     

Modelling Population Dynamics in Realistic Landscapes with Linear Elements: A Mechanistic-Statistical Reaction-Diffusion Approach

We propose and develop a general approach based on reaction-diffusion equations for modelling a species dynamics in a realistic two-dimensional (2D) landscape crossed by linear one-dimensional (1D) corridors, such as roads, hedgerows or rivers. Our approach is based on a hybrid “2D/1D model”, i.e, a system of 2D and 1D reaction-diffusion equations with homogeneous coefficients, in which each equation describes the population dynamics in a given 2D or 1D element of the landscape. Using the example of the range expansion of the tiger mosquito Aedes albopictus in France and its main highways as 1D corridors, we show that the model can be fitted to realistic observation data. We develop a mechanistic-statistical approach, based on the coupling between a model of population dynamics and a probabilistic model of the observation process. This allows us to bridge the gap between the data (3 levels of infestation, at the scale of a French department) and the output of the model (population densities at each point of the landscape), and to estimate the model parameter values using a maximum-likelihood approach. Using classical model comparison criteria, we obtain a better fit and a better predictive power with the 2D/1D model than with a standard homogeneous reaction-diffusion model. This shows the potential importance of taking into account the effect of the corridors (highways in the present case) on species dynamics. With regard to the particular case of A. albopictus, the conclusion that highways played an important role in species range expansion in mainland France is consistent with recent findings from the literature.     

Diffusion, Cross-diffusion and Competitive Interaction

The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing’s instability of the corresponding reaction-diffusion system.H. Ninomiya was Partially supported by Grant-in-Aid for Young Scientists (No. (B)15740076), Japan Society for the Promotion of Science.     

A multi-structured epidemic problem with direct and indirect transmission in heterogeneous environments

In this work, we analyse a deterministic epidemic mathematical model motivated by the propagation of a hantavirus (Puumala hantavirus) within a bank vole population (Clethrionomys glareolus). The host population is split into juvenile and adult individuals. A heterogeneous spatial chronological age and infection age structure is considered, and also indirect transmission via the environment. Maturation rates for juvenile individuals are adult density-dependent. For the reaction-diffusion systems with age structures derived, we give global existence, uniqueness and global boundedness results. A model with transmission to humans is also studied here.     

Spatial patterns and travelling waves in population genetics.

We consider a reaction-diffusion equation to model a multi-allelic, single locus problem. The population can migrate in a homogeneous region and the diffusion rates depend upon the genotype. It is shown that if there is an equilibrium point with all alleles present and if this polymorphism is stable for the classical reaction system then it is also stable for the reaction-diffusion equation. Also a simplified model is used to investigate which allele will spread in the two-allele case. Alleles which are associated with large fitness and small dispersion do best.     

Target reproduction numbers for reaction-diffusion population models

A very important population threshold quantity is the target reproduction number, which is a measure of control effort required for a target prevention, intervention or control. This concept, as a generalization of type reproduction number, was first introduced in Shuai et al. (J Math Biol 67:1067–1082, 2013) for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. The current paper is devoted to the study of all target reproduction numbers for reaction-diffusion population models with compartmental structure. It turns out that the target reproduction number can be regarded as the basic reproduction number of a modified system, where the state of newborn individuals is limited to the target control set and the offspring from the non-target set is regarded as a part of the transition. In other words, the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. We also characterize the target reproduction number so that it can be easily computed numerically for reaction-diffusion models. At the end, we demonstrate our theoretical observations using two examples.

     

Does reaction-diffusion support the duality of fragmentation effect?

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal – or similarly harvesting – term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e., the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.     

The characteristics of epidemics and invasions with thresholds

In this paper we report the development of a highly efficient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensiohal reaction-diffusion models whose dynamics include a threshold or Allee effect. We prove that this methodology produces the correct results for single-component models which are generalizations of the Fisher model, and then demonstrate by numerical experimentation that analogous methods work for a wide class of epidemic and invasion models including the S-I and S-E-I epidemic models and the Rosenzweig-McArthur predator-prey model. As examplary application of this approach we consider the atto-fox effect in the classic reaction-diffusion model of rabies in the European fox population and show that the appropriate threshold for this model is within an order of magnitude of the peak disease incidence and thus has potentially significant effects on epidemic properties. We then make a careful re-parameterisation of the model and show that the velocities calculated with realistic thresholds differ surprisingly little from those calculated from threshold-free models. We conclude that an appropriately thresholded reaction-diffusion model provides a robust representation of the initial epidemic wave and thus provides a sound basis on which to begin a properly mechanistic modelling enterprise aimed at understanding the long-term persistence of the disease.     

Persistence,extinction, and critical patch number for island populations

Sufficient conditions are derived for persistence and extinction of a population inhabiting several islands. Discrete reaction-diffusion population models are analyzed which describe growth and diffusion of a population on a group of islands or a patch environment. A critical patch number is defined as the number of islands below which the population goes extinct on that group of islands. It is shown that population persistence on one island leads to population persistence for the entire archipelago. Both single-species and multi-species models are discussed.     

On some theory of monostable and bistable pure birth-jump integro-differential equations

We study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics, fire propagation, and cancer cell dynamics. We contrast the dynamics of this equation with those of the classical reaction-diffusion equation, where the reaction term models either logistic growth or a strong Allee effect. Recent evidence of an Allee effect has been found in plant dynamics during the germination process (due to seed predation) but not in the generation of seeds. This motivates where the Allee effect is included in our model. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species, which are analogous to those of the classical reaction-diffusion equation. A key finding is that in some cases a population which is initially below the Allee threshold in some area, even if small, will actually survive. This is in contrast to solutions of the classical reaction-diffusion with the same initial data. Another difference of note is the lack of regularization and an infinite number of discontinuous equilibrium solutions to the birth-jump model.     

The mechanism of formation of Liesegang structures around Dictyostelium discoideum

A theoretical study of the phenomenon of Liesegang structure formation induced by the Dictyostelium discoideum population in a medium containing folic acid was carried out. Using a "reaction-diffusion" model proposed in this work, it was shown that the formation of Liesegang structures around the Dictyostelium discoideum population depends on two competing processes: (a) inactivation of folic acid by vegetative amoebae and (b) the chemical reaction of folic acid with the products of amoeba metabolism, which results in the formation of insoluble sediment. The dependence of the model solutions on the geometric and functional parameters was studied. The results are in good agreement with experimental data.     

Effect of environmental fluctuations on invasion fronts

We determine the density profile and velocity of invasion fronts in one-dimensional infinite habitats in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction-diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain analytical expressions for the front profile and velocity by constructing a variational principle. The effect of the noise differs, depending on whether it affects the density-independent growth rate, the intraspecific competition term or the Allee threshold. Fluctuations in the density-independent growth rate increase the invasion velocity and the population density of the invaded area. Fluctuations in the competition term also change the population density of the invaded area, but modify the invasion velocity only for certain initial conditions. Fluctuations in the Allee threshold can induce pulled or pushed invasion fronts as well as invasion failure. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction-diffusion equations with multiplicative noise.     

Taxis-driven pattern formation in a predator-prey model with group defense

We consider a reaction-diffusion(-taxis) predator-prey system with group defense in the prey. Taxis-driven instability can occur if the group defense influences the taxis rate (Wang et al., 2017). We elaborate that this mechanism is indeed possible but biologically unlikely to be responsible for pattern formation in such a system. Conversely, we show that patterns in excitable media such as spatiotemporal Sierpinski gasket patterns occur in the reaction-diffusion model as well as in the reaction-diffusion-taxis model. If group defense leads to a dome-shaped functional response, these patterns can have a rescue effect on the predator population in an invasion scenario. Preytaxis with prey repulsion at high prey densities can intensify this mechanism leading to taxis-induced persistence. In particular, taxis can increase parameter regimes of successful invasions and decrease minimum introduction areas necessary for a successful invasion. Last, we consider the mean period of the irregular oscillations. As a result of the underlying mechanism of the patterns, this period is two orders of magnitude smaller than the period in the nonspatial system. Counter-intuitively, faster-moving predators lead to lower oscillation periods and eventually to extinction of the predator population. The study does not only provide valuable insights on theoretical spatially explicit predator-prey models with group defense but also comparisons of ecological data with model simulations.     

具常投放率的反应扩散系统的渐近性质

本文研究一类具常投放率的人口动力学中反应扩散系统的Neumann初边值问题,应用比较函数讨论其解的渐近性态,给出稳态解的存在条件．     

Rearranging agricultural landscapes towards habitat quality optimisation: In silico application to pest regulation

Modern agriculture suffers from its dependence on chemical inputs and subsequent impacts on health and environment. Alternatively, protecting crops against pests can be achieved through the reinforcement of regulation ecological services. Our work propounds a data-driven methodological framework to derive relevant agricultural landscape rearrangements enhancing populations of beneficial organisms regulating pests.Building on spatialised entomological and geographic data, we developed a parsimonious reaction–diffusion model describing the population dynamics of beneficial organisms. Parameter estimation was carried out in a Bayesian framework accounting for uncertainty in the measurement.Thousands of agricultural landscapes were generated under agronomic specifications dealt with as constraint satisfaction problems. Population dynamics was simulated on each landscape with the fitted reaction-diffusion model mentioned above, and two metrics of abundances allowed the assessment of the regulation performance of the landscape spatial arrangements. One metric is a mean field performance criterion assessing the regulation performance from the landscape composition only, the other is a spatial performance metric assessing the performance resulting from the whole landscape spatial configuration. The former is computed with a non-spatialised form of the population dynamics model, the latter results from the reaction-diffusion model of the population dynamics. Comparing these metrics enabled to quantify the impact of spatial arrangements, hence allowing arrangements proposals.This framework was applied to the case study of a ground beetle species involved in the biological regulation of weeds. The arrangement proposals abides by the productive agronomic constraint that is the landscape composition, while they allow for significant habitat quality enhancement (or deterioration) for the beneficial organism (or a pest). Minor adaptations of our integrated data-driven approach would suit numerous situations ranging from the provision of enhanced ecosystem services to land management for conservation.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

The spatio-temporal dynamics of neutral genetic diversity

The notions of pulled and pushed solutions of reaction-dispersal equations introduced by Garnier et al. (2012) and Roques et al. (2012) are based on a decomposition of the solutions into several components. In the framework of population dynamics, this decomposition is related to the spatio-temporal evolution of the genetic structure of a population. The pulled solutions describe a rapid erosion of neutral genetic diversity, while the pushed solutions are associated with a maintenance of diversity. This paper is a survey of the most recent applications of these notions to several standard models of population dynamics, including reaction-diffusion equations and systems and integro-differential equations. We describe several counterintuitive results, where unfavorable factors for the persistence and spreading of a population tend to promote diversity in this population. In particular, we show that the Allee effect, the existence of a competitor species, as well as the presence of climate constraints are factors which can promote diversity during a colonization. We also show that long distance dispersal events lead to a higher diversity, whereas the existence of a nonreproductive juvenile stage does not affect the neutral diversity in a range-expanding population.     

Cellular automata for contact ecoepidemic processes in predator–prey systems

Spatial ecoepidemic models, in which diseases affect interacting populations, are often explored through reaction-diffusion equations. However, cellular automata (CA) are a widely recognized tool for modelling spatial pattern formation that are broadly analagous to reaction diffusion equations, but provide greater flexibility in defining population dynamics. In this work we present a CA defined to mimic the prey–predators interactions while a pathogen is affecting, in turn, one population. We explore system equilibria, given different initial conditions and local interaction neighborhoods. Furthermore, in the various ecoepidemic systems considered we report the formation of waves and spirals: a key summary of how diseases may spread among individuals. Some inferences on the predators and infection eradication strategies are presented and supported by simulations results.