Landscape refuges delay resistance of the European corn borer to Bt-maize: a demo-genetic dynamic model

We constructed a reaction-diffusion model of the development of resistance to transgenic insecticidal Bt crops in pest populations. Kostitzin’s demo-genetic model describes local interactions between three competing pest genotypes with alleles conferring resistance or susceptibility to transgenic plants, the spatial spread of insects being modelled by diffusion. This new approach makes it possible to combine a spatial demographic model of population dynamics with classical genetic theory. We used this model to examine the effects of pest dispersal and of the size and shape of the refuge on the efficiency of the “high-dose/refuge” strategy, which was designed to prevent the development of resistance in populations of insect pests, such as the European corn borer, Ostrinia nubilalis Hübner (Lepidoptera, Crambidae). We found that, with realistic combinations of refuge size and pest dispersal, the development of resistance could be considerably delayed. With a small to medium-sized farming area, contiguous refuge plots are more efficient than a larger number of smaller refuge patches. We also show that the formal coupling of classical Fisher–Haldane–Wright population genetics equations with diffusion terms inaccurately describes the development of resistance in a spatially heterogeneous pest population, notably overestimating the speed with which Bt resistance is selected in populations of pests targeted by Bt crops.     

A spatial model of the development of pest resistance to a transgenic insecticidal crop: European corn borer on Bt maize

A mathematical model was constructed to describe the evolution of resistance to the Bacillus thuringiensis toxin (Bt) in an insect pest (European corn borer) population on a transgenic crop (Bt corn). The model comprises a set of partial differential equations of the reaction-diffusion type; local interactions of three competing pest genotypes formed by alleles of Bt resistance and susceptibility are described as in the Kostitzin model, and the spread of insects is modeled as diffusion. The model was used to evaluate the influence of pest characteristics on the efficacy of the high-dose/refuge strategy aiming to prevent or delay the spread of Bt resistance in pest populations. It was shown, by contrast, that a model based on Fisher-Haldane-Wright equations and formally incorporating a diffusion term cannot adequately describe the evolution of Bt resistance in a spatially inhomogeneous pest population. Further development of the proposed demo-genetic model is discussed.     

How patch configuration affects the impact of disturbances on metapopulation persistence

Disturbances affect metapopulations directly through reductions in population size and indirectly through habitat modification. We consider how metapopulation persistence is affected by different disturbance regimes and the way in which disturbances spread, when metapopulations are compact or elongated, using a stochastic spatially explicit model which includes metapopulation and habitat dynamics. We discover that the risk of population extinction is larger for spatially aggregated disturbances than for spatially random disturbances. By changing the spatial configuration of the patches in the system--leading to different proportions of edge and interior patches--we demonstrate that the probability of metapopulation extinction is smaller when the metapopulation is more compact. Both of these results become more pronounced when colonization connectivity decreases. Our results have important management implication as edge patches, which are invariably considered to be less important, may play an important role as disturbance refugia.     

Spatial model of development of resistance to transgenic insecticidal crop in a pest as applied to the European corn borer

A mathematical model of the development of resistance to transgenic Bt-crop in insect-pest populations as applied to the European corn borer has been constructed. The model represents a system of differential equations in special derivatives of the reaction-diffusion type. Local interactions of three competing pest genotypes formed by the allele of Bt-resistance and allele of susceptibility, are described by the same relations as in the Kostitsyn model, and the spread of insects is modelled as diffusion. The model was used to estimate the influence of the pest on the efficiency of the "high dose-refuge" strategy, which is intended for the suppression of development of Bt-resistance in pest populations. It was shown that formal integration of the Fisher-Haldane-Wright model and the diffusion term cannot adequately describe the development of Bt-resistance in a spatially heterogeneous pest population. A further development of the model is discussed.     

A Suite of Models to Support the Quantitative Assessment of Spread in Pest Risk Analysis

Pest Risk Analyses (PRAs) are conducted worldwide to decide whether and how exotic plant pests should be regulated to prevent invasion. There is an increasing demand for science-based risk mapping in PRA. Spread plays a key role in determining the potential distribution of pests, but there is no suitable spread modelling tool available for pest risk analysts. Existing models are species specific, biologically and technically complex, and data hungry. Here we present a set of four simple and generic spread models that can be parameterised with limited data. Simulations with these models generate maps of the potential expansion of an invasive species at continental scale. The models have one to three biological parameters. They differ in whether they treat spatial processes implicitly or explicitly, and in whether they consider pest density or pest presence/absence only. The four models represent four complementary perspectives on the process of invasion and, because they have different initial conditions, they can be considered as alternative scenarios. All models take into account habitat distribution and climate. We present an application of each of the four models to the western corn rootworm, Diabrotica virgifera virgifera, using historic data on its spread in Europe. Further tests as proof of concept were conducted with a broad range of taxa (insects, nematodes, plants, and plant pathogens). Pest risk analysts, the intended model users, found the model outputs to be generally credible and useful. The estimation of parameters from data requires insights into population dynamics theory, and this requires guidance. If used appropriately, these generic spread models provide a transparent and objective tool for evaluating the potential spread of pests in PRAs. Further work is needed to validate models, build familiarity in the user community and create a database of species parameters to help realize their potential in PRA practice.     

Bi-trophic mathematical model for pest adaptation to a resistant crop

A version of the Lotka-Volterra predator-prey model with logistic crop growth is modified to explore the rate of adaptation of a herbivore to a pest-resistant crop. This provides a phenotypic model for the evolution of resistance in a population comprising three different pest types each defined by differing parameter values for respiration rate and crop palatability. Expressions estimating the rates of increase of the fitter pest types are obtained as a function of the food qualities, and respiration and mortality rates. Potential strategies for delaying the rate of adaptation with regard to the expressions derived above, via the use of pest-susceptible refugia and natural enemies, are discussed. Although the model is formulated as one in which a single gene is the factor conferring resistance it can be interpreted and used independently of this.     

Population consequences of constitutive and inducible plant resistance: herbivore spatial spread

Abstract Little attention has been paid to the impact that constitutive and inducible plant resistance traits will have on herbivore spatial dynamics. We investigate mathematical models in which herbivore demographic rates and movement rates respond to host plant quality, which in turn is determined by constitutive and inducible resistance. Models with and without induced resistance yield the same analytic expression for the asymptotic speed at which a herbivore population will spread through an initially uninduced plant population, suggesting that induced resistance will have no effect on the rate of invasion of herbivores that respond to plant resistance on small spatial scales. In contrast, constitutive resistance will influence the speed of an invasion. If herbivore movement is quite sensitive to plant quality, an increase in constitutive resistance can actually accelerate the rate of herbivore spread even while it reduces the herbivore's intrinsic rate of increase. In other scenarios, the rate of invasion attains a maximum at intermediate levels of constitutive resistance. These results argue that our view of plant resistance should be broadened to include herbivore movement if we are to understand fully the implications of differences in resistance for the dynamics of herbivore populations in natural and managed settings.     

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.     

Simulating the impact of cross resistance between Bt toxins in transformed clover and apples in New Zealand

Simulation were conducted to guide development of resistance management strategies aimed at prolonging the usable life of B. thuringiensis (Bt) endotoxins in multiple cropping situations, where different crops expressing Bt endotoxins are host plants for a common pest. We used the New Zealand apple and clover model ecosystem to explore the relative impact on the rate of resistance development of varying levels of cross-resistance between different toxins expressed in these 2 potentially Bt-transformed crops. These 2 crops are hosts for a complex of leaf-rollers in New Zealand, including the light-brown apple moth, used here as the model pest. Cross-resistance was varied between 0.0 and 0.5 (zero to partial cross-resistance) to allow for the case in which selection by one plant has a potential effect on resistance to the toxin in another plant. The largest factor affecting the evolution of resistance was the total habitat area occupied by transgenic orchards. The proportion of the clover habitat that was transformed was also an important factor, even in the absence of cross-resistance. The effect of increasing the proportion of the second transformed crop (clover) acted on resistance evolution mainly by reducing the external refuge of susceptibility for the transgenic orchards. Hence, the ecological implications of reducing the available source of susceptible insects from clover, which can help to slow resistance development in the orchard ecosystem, had a more significant impact than the presence of cross-resistance. Partial cross-resistance between different toxins in the separate crops was overall of relatively minor importance. These simulations have implications for deployment decisions for individual transformed crops in multiple cropping systems, where there is the potential for the crops to serve as refugees for each other. These decisions may need to focus less on cross-resistance between toxins, than on economic trade-offs between the relative roles of individual crops as refugia maintaining susceptibility in the system as a whole.     

An interfacial approach to regional segregation of two competing species mediated by a predator

We consider the problem of coexistence of two competing species mediated by the presence of a predator. We employ a reaction-diffusion model equation with Lotka-Volterra interaction, and speculate that the possibility of coexistence is,enhanced by differences in the diffusion rates of the prey and their predator. In the limit where the diffusion rate of the prey tends to zero, a new equation is derived and the dynamics of spatial segregation is discussed by means of the interfacial dynamics approach. Also, we show that spatial segregation permits periodic and chaotic dynamics for certain parameter ranges.     

Front dynamics in fractional-order epidemic models

A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic's front and a power-law decay of the front's leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics.     

Invasion theory and biological control

Recent advances in the mathematical theory of invasion dynamics have much to offer to biological control. Here we synthesize several results concerning the spatiotemporal dynamics that occur when a biocontrol agent spreads into a population of an invading pest species. We outline conditions under which specialist and generalist predators can influence the density and rate of spatial spread of the pest, including the rather stringent conditions under which a specialist predator can successfully reverse a pest invasion. We next discuss the connections between long distance dispersal and invasive spread, emphasizing the different consequences of fast spreading pests and predators. Recent theory has considered the effects of population stage-structure on invasion dynamics, and we discuss how population demography affects the biological control of invading pests. Because low population densities generally characterize early stages of an invasion, we discuss the lessons invasion theory teaches concerning the detectability of invasions. Stochasticity and density-dependent dynamics are common features of many real invasions, influencing both the spatial character (e.g. patchiness) of pest invasions and the success of biocontrol agents. We conclude by outlining theoretical results delineating how stochastic effects and complex dynamics generated by density dependence can facilitate or impede biological pest control.     

Effect of the rate of phytoplankton growth on the spaced-time dynamics of plankton communities in an homogeneous environment

We use a conceptual mathematical reaction-diffusion model to investigate the mechanisms of spatial structure formation and complex temporal dynamics of plankton in a heterogeneous environment. We take into account basic trophic interactions, namely, "prey-predator" interactions between phytoplankton, zooplankton, and fish in upper layers of natural waters. We consider plankton as a passive contaminant in turbulent waters. We show that plankton structure formation can result from the difference in phytoplankton growth rate in neighboring habitats. Phytoplankton and zooplankton biomass is shown to undergo both regular and chaotic oscillations. The fish predation rate substantially affects the spatial and temporal dynamics of plankton in a heterogeneous environment.     

Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease

A greater understanding of the rate at which emerging disease advances spatially has both ecological and applied significance. Analyzing the spread of vector-borne disease can be relatively complex when the vector's acquisition of a pathogen and subsequent transmission to a host occur in different life stages. A contemporary example is Lyme disease. A long-lived tick vector acquires infection during the larval blood meal and transmits it as a nymph. We present a reaction-diffusion model for the ecological dynamics governing the velocity of the current epidemic's spread. We find that the equilibrium density of infectious tick nymphs (hence the risk of human disease) can depend on density-independent survival interacting with biotic effects on the tick's stage structure. The local risk of infection reaches a maximum at an intermediate level of adult tick mortality and at an intermediate rate of juvenile tick attacks on mammalian hosts. If the juvenile tick attack rate is low, an increase generates both a greater density of infectious nymphs and an increased spatial velocity. However, if the juvenile attack rate is relatively high, nymph density may decline while the epidemic's velocity still increases. Velocities of simulated two-dimensional epidemics correlate with the model pathogen's basic reproductive number (R0), but calculating R0 involves parameters of both host infection dynamics and the vector's stage-structured dynamics.     

The importance of disturbance by fire and other abiotic and biotic factors in driving cheatgrass invasion varies based on invasion stage

Disturbances create fluctuations in resource availability that alter abiotic and biotic constraints. Exotic invader response may be due to multiple factors related to disturbance regimes and complex interactions between other small- and large-scale abiotic and biotic processes that may vary across invasion stages. We explore how cheatgrass responds to both frequency and season of prescribed burning for a 10-year period in ponderosa pine forested stands. To understand interactions of fire disturbance, other abiotic factors, biotic resistance, and propagule pressure, we use long-term data from different spatial scales representing different invasion stages (local establishment or spread and broader scale extent/impact) to model cheatgrass dynamics. We found that after 10 years, cheatgrass cover increased with fall burning regardless of burn frequency (1 burn vs. 3 burns). There was no evidence that cheatgrass invasion is decreasing through time even in areas burned only once. Factors important for explaining local fine-scale cheatgrass establishment and spread, and broader scale extent/impact varied. The spatial extent of the first burns facilitated fine-scale cheatgrass establishment while bare soil cover constrained establishment. Biotic resistance, in the form of native annual forb cover, constrained fine-scale cheatgrass spread. Initial cheatgrass abundance in 2002, a factor related to propagule pressure, was key for explaining the broader scale extent/impact of cheatgrass by 2012. Biotic resistance, in the form of native perennial grass cover, constrained extent/impact but only when initial cheatgrass abundance was low. Our findings regarding factors affecting invasion dynamics may be useful to consider for future restoration and conservation efforts in burned ponderosa pine forests.     

Density dependence slows invader spread in fragmented landscapes

Patchiness is a defining characteristic of most natural and anthropogenic habitats, yet much of our understanding of how invasions spread has come from models of spatially homogeneous environments. Except for populations with Allee effects, an invader's growth rate when rare and dispersal determine its spread velocity; intraspecific competition has little to no influence. How this result might change with landscape patchiness, however, is poorly understood. We used simulation models and their analytical approximations to explore the effect of density dependence on the spread of annual plant invaders moving through heterogeneous landscapes with gaps in suitable habitat. We found that landscape patchiness and discrete invader population size interacted to generate a strong role for density dependence. Intraspecific competition greatly slowed the spread of invasions through patchy landscapes by regulating how rapidly a population could produce enough seeds to surpass habitat gaps. Populations with continuously varying density showed no such effect of density dependence. We adapted a stochastic dispersal model to approximate spread when gap sizes were small relative to the mean dispersal distance and a Markov chain approximation for landscapes with large gaps. Our work suggests that ecologists must consider reproduction at both low and high densities when predicting invader spread.     

Influence of foraging behavior and host spatial distribution on the localized spread of the emerald ash borer, <Emphasis Type="Italic">Agrilus planipennis</Emphasis>

Management programs for invasive species are often developed at a regional or national level, but physical intervention generally takes place over relatively small areas occupied by newly founded, isolated populations. The ability to predict how local habitat variation affects the expansion of such newly founded populations is essential for efficiently targeting resources to slow the spread of an invasive species. We assembled a coupled map lattice model that simulates the local spread of newly founded colonies of the emerald ash borer (Agrilus planipennis Fairmaire), a devastating forest insect pest of ash (Fraxinus spp.) trees. Using this model, we investigated the spread of A. planipennis in environments with different Fraxinus spp. distributions, and explored the consequences of ovipositional foraging behavior on the local spread of A. planipennis. Simulations indicate that increased larval density, resulting from lower host tree density or higher initial population sizes, can increase the spread rate during the first few years after colonization by increasing a density-dependent developmental rate and via host resource depletion. Both the radial spread rate and population size were greatly influenced by ovipositional foraging behavior. Two known behaviors of ovipositing A. planipennis females, attraction towards areas with high ash tree density and attraction to stressed trees, had opposing effects on spread. Results from this model illustrate the significant influence of resource distribution and foraging behavior on localized spread, and the importance of these factors when formulating strategies to monitor and manage invasive pests.     

How predation can slow, stop or reverse a prey invasion

Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predator-prey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.     

An overview of NMR-based metabolomics to identify secondary plant compounds involved in host plant resistance

Secondary metabolites provide a potential source for the generation of host plant resistance and development of biopesticides. This is especially important in view of the rapid and vast spread of agricultural and horticultural pests worldwide. Multiple pests control tactics in the framework of an integrated pest management (IPM) programme are necessary. One important strategy of IPM is the use of chemical host plant resistance. Up to now the study of chemical host plant resistance has, for technical reasons, been restricted to the identification of single compounds applying specific chemical analyses adapted to the compound in question. In biological processes however, usually more than one compound is involved. Metabolomics allows the simultaneous detection of a wide range of compounds, providing an immediate image of the metabolome of a plant. One of the most universally used metabolomic approaches comprises nuclear magnetic resonance spectroscopy (NMR). It has been NMR which has been applied as a proof of principle to show that metabolomics can constitute a major advancement in the study of host plant resistance. Here we give an overview on the application of NMR to identify candidate compounds for host plant resistance. We focus on host plant resistance to western flower thrips (Frankliniella occidentalis) which has been used as a model for different plant species.     

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.