Transient Propagation of the Invasion Front in the Homogeneous Landscape and in the Presence of a Road.

Understanding the propagation of invasive plants at the beginning of invasive spread is important as it can help practitioners eradicate harmful species more efficiently. In our work the propagation regime of the invasive plant species is studied at the short-time scale before a travelling wave is established and advances into space at a constant speed. The integro-difference framework has been employed to deal with a stage-structured population, and a short-distance dispersal mode has been considered in the homogeneous environment and when a road presents in the landscape. It is explained in the paper how nonlinear spatio-temporal dynamics arise in a transient regime where the propagation speed depends on the detection threshold population density. Furthermore, we investigate the question of whether the transient dynamics become different when the homogeneous landscape is transformed into the heterogeneous one. It is shown in the paper how invasion slows down in a transient regime in the presence of a road.

Propagation of invasive plant species in the presence of a road.

Invasive plant species pose a significant threat to biodiversity and the economy, yet their management is often resource-intensive and expensive, and further research is required to make control measures more efficient. Evidence suggests that roads can have an important effect on the spread of invasive plant species, although little is known about the underlying mechanisms at play. We have developed a novel mathematical model to analyse the impact of roads on the propagation of invasive plants. The integro-difference equation model is formulated for stage-structured population and incorporates a road sub-domain in the spatial domain. The results of our study reveal, that, depending on the definition of the growth function in the model, there are three distinct types of behaviour in front of the road. Roads can act as barriers to invasion, lead to a formation of a beachhead in front of the road, or act as corridors allowing the invasive species to invade the domain in front of the road. Analytical and computational findings on how roads can impact the spread of invasive species show that a small change in conditions of the environment favouring the invasive species can change the case for the road, allowing the invasive species to invade the domain in front of the road where it previously could not spread.

A predictive model and a field study on heterogeneous slug distribution in arable fields arising from density dependent movement.

Factors and processes determining heterogeneous ('patchy') population distributions in natural environments have long been a major focus in ecology. Existing theoretical approaches proved to be successful in explaining vegetation patterns. In the case of animal populations, existing theories are at most conceptual: they may suggest a qualitative explanation but largely fail to explain patchiness quantitatively. We aim to bridge this knowledge gap. We present a new mechanism of self-organized formation of a patchy spatial population distribution. A factor that was under-appreciated by pattern formation theories is animal sociability, which may result in density dependent movement behaviour. Our approach was inspired by a recent project on movement and distribution of slugs in arable fields. The project discovered a strongly heterogeneous slug distribution and a specific density dependent individual movement. In this paper, we bring these two findings together. We develop a model of density dependent animal movement to account for the switch in the movement behaviour when the local population density exceeds a certain threshold. The model is fully parameterized using the field data. We then show that the model produces spatial patterns with properties closely resembling those observed in the field, in particular to exhibit similar values of the aggregation index.

Towards Building a Sustainable Future: Positioning Ecological Modelling for Impact in Ecosystems Management.

As many ecosystems worldwide are in peril, efforts to manage them sustainably require scientific advice. While numerous researchers around the world use a great variety of models to understand ecological dynamics and their responses to disturbances, only a small fraction of these models are ever used to inform ecosystem management. There seems to be a perception that ecological models are not useful for management, even though mathematical models are indispensable in many other fields. We were curious about this mismatch, its roots, and potential ways to overcome it. We searched the literature on recommendations and best practices for how to make ecological models useful to the management of ecosystems and we searched for 'success stories' from the past. We selected and examined several cases where models were instrumental in ecosystem management. We documented their success and asked whether and to what extent they followed recommended best practices. We found that there is not a unique way to conduct a research project that is useful in management decisions. While research is more likely to have impact when conducted with many stakeholders involved and specific to a situation for which data are available, there are great examples of small groups or individuals conducting highly influential research even in the absence of detailed data. We put the question of modelling for ecosystem management into a socio-economic and national context and give our perspectives on how the discipline could move forward.

Stability of Patches of Higher Population Density within the Heterogenous Distribution of the Gray Field Slug Deroceras reticulatum in Arable Fields in the UK.

Exploitation of heterogenous distributions of Deroceras reticulatum, in arable fields by targeting molluscicide applications toward areas with higher slug densities, relies on these patches displaying sufficient spatio-temporal stability. Regular sampling of slug activity/distribution was undertaken using 1 ha rectangular grids of 100 refuge traps established in 22 commercial arable field crops. Activity varied significantly between the three years of the study, and the degree of aggregation (Taylor's Power Law) was higher in fields with higher mean trap catches. Hot spot analysis detected statistically significant spatial clusters in all fields, and in 162 of the 167 individual assessment visits. The five assessment visits in which no clusters were detected coincided with low slug activity (≤0.07 per trap). Generalized Linear Models showed significant spatial stability of patches in 11 fields, with non-significant fields also characterized by low slug activity (≤1.2 per trap). Mantel's permutation tests revealed a high degree of correlation between location of individual patches between sampling dates. It was concluded that patches of higher slug density were spatio-temporally stable, but detection using surface refuge traps (which rely on slug activity on the soil surface) was less reliable when adverse environmental conditions resulted in slugs retreating into the upper soil horizons.

Movement patterns of the grey field slug (Deroceras reticulatum) in an arable field.

We report the results of an experiment on radio-tracking of individual grey field slugs in an arable field and associated data modelling designed to investigate the effect of slug population density in their movement. Slugs were collected in a commercial winter wheat field in which a 5x6 trapping grid had been established with 2m distance between traps. The slugs were taken to the laboratory, radio-tagged using a recently developed procedure, and following a recovery period released into the same field. Seventeen tagged slugs were released singly (sparse release) on the same grid node on which they had been caught. Eleven tagged slugs were released as a group (dense release). Each of the slugs was radio-tracked for approximately 10 h during which their position was recorded ten times. The tracking data were analysed using the Correlated Random Walk framework. The analysis revealed that all components of slug movement (mean speed, turning angles and movement/resting times) were significantly different between the two treatments. On average, the slugs released as a group disperse more slowly than slugs released individually and their turning angle has a clear anticlockwise bias. The results clearly suggest that population density is a factor regulating slug movement.

A computational study of density-dependent individual movement and the formation of population clusters in two-dimensional spatial domains.

The patterns of collective behaviour in a population emerging from individual animal movement have long been of interest to ecologists, as has the emergence of heterogeneous patterns among a population. In this paper we will consider these phenomena by using an individual-based modelling approach to simulate a population whose individuals undergo density-dependent movement in 2D spatial domains. We first show that the introduction of density-dependent movement in the form of two parameters, a perception radius and a probability of directed movement, leads to the formation of clusters. We then show that the properties of the clusters and their stability over time are different between populations of Brownian and non-Brownian walkers and are also dependent on the choice of parameters. Finally, we consider the effect of the probability of directed movement on the temporal stability of clusters and show that while clusters formed by Brownian and non-Brownian walkers may have similar properties with certain parameter sets, the spatio-temporal dynamics remain different.

Locomotor behaviour promotes stability of the patchy distribution of slugs in arable fields: Tracking the movement of individual Deroceras reticulatum.

BACKGROUND: The distribution of the grey field slug (Deroceras reticulatum Müller) in arable fields is characterised by patches containing higher slug densities dispersed within areas of lower densities. Behavioural responses that lead to the spatial/temporal stability of these patches are poorly understood, thus this study investigated behavioural mechanisms underpinning slug distribution using a new method for long-term tracking of individual slug movement in the field. RESULTS: A technique for implanting radio frequency identification (RFID) tags (each with a unique identification code) beneath the body wall of slugs was developed. Laboratory tests indicated no consistent detrimental effect on survival, feeding, egg laying or locomotor behaviour (velocity, distance travelled). Movement of individual slugs above and below the soil surface was recorded for >5 weeks (in spring and autumn) in winter wheat fields. Most (~80%) foraged within a limited area; and at the end of the observation period were located at a mean distance of 78.7 ± 33.7 cm (spring) or 101.9 ± 24.1 cm (autumn) from their release point. The maximum detected distance from the release point was 408.8 cm. The remaining slugs (~20%) moved further away and ultimately were lost. CONCLUSIONS: RFID tagging allowed continuous tracking of individual slugs, even below the soil surface. Localised movement of 80% of tracked slugs over 5 weeks offers a mechanism promoting stable slug patches in arable crops. Rapid dispersal of the remaining slugs facilitates exchange of individuals between patches. Precision targeting of pesticides at such stable slug patches may facilitate reduced usage. © 2020 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

When seeing is not believing: Comparative study of various spatial distributions of invasive species.

We address the problem of pattern recognition and comparison when spatial patterns of biological invasions are studied. A model of biological invasion is employed to simulate spatio-temporal dynamics of invasive species and generate a variety of spatial patterns including so called 'no front' patchy spatial distributions. We introduce several topological indices to understand whether various spatial distributions of invasive species can be compared to each other based on information about their topology. We also investigate how topological indices used to make conclusions about the spatial pattern are related to controlling parameters in the underlying process of biological invasion. Our analysis reveals that a small increment in the model parameters results in a small increment in topological indices when the topology of continuous front spatial pattern with no patches behind the front is considered. Meanwhile, 'no front' patchy spatial distributions present a different case where a small change in the model parameters results in random fluctuations of topological indices. The 'random' behaviour of patchy patterns is further studied to understand whether a patchy spatial structure can transform itself into a continuous front spatial distribution over time. In the paper it will be argued that apart from the topological quantities used to classify spatial distributions, the transition time required to establish topological properties of the spatial pattern must be taken into account in pattern recognition and analysis. Furthermore, it will be demonstrated that for some parameter values it is impossible to conclude about the topological type of spatial pattern, i.e. continuous front spatial distributions cannot be distinguished from 'no front' patchy distributions of invasive species, no matter what their topological indices are.

Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights.

Individual animal movement has been a focus of intense research and considerable controversy over the last two decades, however the understanding of wider ecological implications of various movement behaviours is lacking. In this paper, we consider this issue in the context of pattern formation. Using an individual-based modelling approach and computer simulations, we first show that density dependence ("auto-taxis") of the individual movement in a population of random walkers typically results in the formation of a strongly heterogeneous population distribution consisting of clearly defined animal clusters or patches. We then show that, when the movement takes place in a large spatial domain, the properties of the clusters are significantly different in the populations of Brownian and non-Brownian walkers. Whilst clusters tend to be stable in the case of Brownian motion, in the population of Levy walkers clusters are dynamical so that the number of clusters fluctuates in the course of time. We also show that the population dynamics of non-Brownian walkers exhibits two different time scales: a short time scale of the relaxation of the initial condition and a long time scale when one type of dynamics is replaced by another. Finally, we show that the distribution of sample values in the populations of Brownian and non-Brownian walkers is significantly different.

Towards the Development of a More Accurate Monitoring Procedure for Invertebrate Populations, in the Presence of an Unknown Spatial Pattern of Population Distribution in the Field.

Studies addressing many ecological problems require accurate evaluation of the total population size. In this paper, we revisit a sampling procedure used for the evaluation of the abundance of an invertebrate population from assessment data collected on a spatial grid of sampling locations. We first discuss how insufficient information about the spatial population density obtained on a coarse sampling grid may affect the accuracy of an evaluation of total population size. Such information deficit in field data can arise because of inadequate spatial resolution of the population distribution (spatially variable population density) when coarse grids are used, which is especially true when a strongly heterogeneous spatial population density is sampled. We then argue that the average trap count (the quantity routinely used to quantify abundance), if obtained from a sampling grid that is too coarse, is a random variable because of the uncertainty in sampling spatial data. Finally, we show that a probabilistic approach similar to bootstrapping techniques can be an efficient tool to quantify the uncertainty in the evaluation procedure in the presence of a spatial pattern reflecting a patchy distribution of invertebrates within the sampling grid.

Catching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization.

Synchronization of population dynamics in different habitats is a frequently observed phenomenon. A common mathematical tool to reveal synchronization is the (cross)correlation coefficient between time courses of values of the population size of a given species where the population size is evaluated from spatial sampling data. The corresponding sampling net or grid is often coarse, i.e. it does not resolve all details of the spatial configuration, and the evaluation error-i.e. the difference between the true value of the population size and its estimated value-can be considerable. We show that this estimation error can make the value of the correlation coefficient very inaccurate or even irrelevant. We consider several population models to show that the value of the correlation coefficient calculated on a coarse sampling grid rarely exceeds 0.5, even if the true value is close to 1, so that the synchronization is effectively lost. We also observe 'ghost synchronization' when the correlation coefficient calculated on a coarse sampling grid is close to 1 but in reality the dynamics are not correlated. Finally, we suggest a simple test to check the sampling grid coarseness and hence to distinguish between the true and artifactual values of the correlation coefficient.

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal.

Understanding of spatiotemporal patterns arising in invasive species spread is necessary for successful management and control of harmful species, and mathematical modeling is widely recognized as a powerful research tool to achieve this goal. The conventional view of the typical invasion pattern as a continuous population traveling front has been recently challenged by both empirical and theoretical results revealing more complicated, alternative scenarios. In particular, the so-called patchy invasion has been a focus of considerable interest; however, its theoretical study was restricted to the case where the invasive species spreads by predominantly short-distance dispersal. Meanwhile, there is considerable evidence that the long-distance dispersal is not an exotic phenomenon but a strategy that is used by many species. In this paper, we consider how the patchy invasion can be modified by the effect of the long-distance dispersal and the effect of the fat tails of the dispersal kernels.

Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration.

Monitoring of pest insects is an important part of the integrated pest management. It aims to provide information about pest insect abundance at a given location. This includes data collection, usually using traps, and their subsequent analysis and/or interpretation. However, interpretation of trap count (number of insects caught over a fixed time) remains a challenging problem. First, an increase in either the population density or insects activity can result in a similar increase in the number of insects trapped (the so called "activity-density" problem). Second, a genuine increase of the local population density can be attributed to qualitatively different ecological mechanisms such as multiplication or immigration. Identification of the true factor causing an increase in trap count is important as different mechanisms require different control strategies. In this paper, we consider a mean-field mathematical model of insect trapping based on the diffusion equation. Although the diffusion equation is a well-studied model, its analytical solution in closed form is actually available only for a few special cases, whilst in a more general case the problem has to be solved numerically. We choose finite differences as the baseline numerical method and show that numerical solution of the problem, especially in the realistic 2D case, is not at all straightforward as it requires a sufficiently accurate approximation of the diffusion fluxes. Once the numerical method is justified and tested, we apply it to the corresponding boundary problem where different types of boundary forcing describe different scenarios of pest insect immigration and reveal the corresponding patterns in the trap count growth.

Multiscale approach to pest insect monitoring: random walks, pattern formation, synchronization, and networks.

Pest insects pose a significant threat to food production worldwide resulting in annual losses worth hundreds of billions of dollars. Pest control attempts to prevent pest outbreaks that could otherwise destroy a sward. It is good practice in integrated pest management to recommend control actions (usually pesticides application) only when the pest density exceeds a certain threshold. Accurate estimation of pest population density in ecosystems, especially in agro-ecosystems, is therefore very important, and this is the overall goal of the pest insect monitoring. However, this is a complex and challenging task; providing accurate information about pest abundance is hardly possible without taking into account the complexity of ecosystems' dynamics, in particular, the existence of multiple scales. In the case of pest insects, monitoring has three different spatial scales, each of them having their own scale-specific goal and their own approaches to data collection and interpretation. In this paper, we review recent progress in mathematical models and methods applied at each of these scales and show how it helps to improve the accuracy and robustness of pest population density estimation.

A novel approach to evaluation of pest insect abundance in the presence of noise.

Evaluation of pest abundance is an important task of integrated pest management. It has recently been shown that evaluation of pest population size from discrete sampling data can be done by using the ideas of numerical integration. Numerical integration of the pest population density function is a computational technique that readily gives us an estimate of the pest population size, where the accuracy of the estimate depends on the number of traps installed in the agricultural field to collect the data. However, in a standard mathematical problem of numerical integration, it is assumed that the data are precise, so that the random error is zero when the data are collected. This assumption does not hold in ecological applications. An inherent random error is often present in field measurements, and therefore it may strongly affect the accuracy of evaluation. In our paper, we offer a novel approach to evaluate the pest insect population size under the assumption that the data about the pest population include a random error. The evaluation is not based on statistical methods but is done using a spatially discrete method of numerical integration where the data obtained by trapping as in pest insect monitoring are converted to values of the population density. It will be discussed in the paper how the accuracy of evaluation differs from the case where the same evaluation method is employed to handle precise data. We also consider how the accuracy of the pest insect abundance evaluation can be affected by noise when the data available from trapping are sparse. In particular, we show that, contrary to intuitive expectations, noise does not have any considerable impact on the accuracy of evaluation when the number of traps is small as is conventional in ecological applications.

Challenges of ecological monitoring: estimating population abundance from sparse trap counts.

Ecological monitoring aims to provide estimates of pest species abundance-this information being then used for making decisions about means of control. For invertebrate species, population size estimates are often based on trap counts which provide the value of the population density at the traps' location. However, the use of traps in large numbers is problematic as it is costly and may also be disruptive to agricultural procedures. Therefore, the challenge is to obtain a reliable population size estimate from sparse spatial data. The approach we develop in this paper is based on the ideas of numerical integration on a coarse grid. We investigate several methods of numerical integration in order to understand how badly the lack of spatial data can affect the accuracy of results. We first test our approach on simulation data mimicking spatial population distributions of different complexity. We show that, rather counterintuitively, a robust estimate of the population size can be obtained from just a few traps, even when the population distribution has a highly complicated spatial structure. We obtain an estimate of the minimum number of traps required to calculate the population size with good accuracy. We then apply our approach to field data to confirm that the number of trap/sampling locations can be much fewer than has been used in many monitoring programmes. We also show that the accuracy of our approach is greater that that of the statistical method commonly used in field studies. Finally, we discuss the implications of our findings for ecological monitoring practice and show that the use of trap numbers 'smaller than minimum' may still be possible but it would result in a paradigm shift: the population size estimates should be treated probabilistically and the arising uncertainty may introduce additional risk in decision-making.

Computational ecology as an emerging science.

It has long been recognized that numerical modelling and computer simulations can be used as a powerful research tool to understand, and sometimes to predict, the tendencies and peculiarities in the dynamics of populations and ecosystems. It has been, however, much less appreciated that the context of modelling and simulations in ecology is essentially different from those that normally exist in other natural sciences. In our paper, we review the computational challenges arising in modern ecology in the spirit of computational mathematics, i.e. with our main focus on the choice and use of adequate numerical methods. Somewhat paradoxically, the complexity of ecological problems does not always require the use of complex computational methods. This paradox, however, can be easily resolved if we recall that application of sophisticated computational methods usually requires clear and unambiguous mathematical problem statement as well as clearly defined benchmark information for model validation. At the same time, many ecological problems still do not have mathematically accurate and unambiguous description, and available field data are often very noisy, and hence it can be hard to understand how the results of computations should be interpreted from the ecological viewpoint. In this scientific context, computational ecology has to deal with a new paradigm: conventional issues of numerical modelling such as convergence and stability become less important than the qualitative analysis that can be provided with the help of computational techniques. We discuss this paradigm by considering computational challenges arising in several specific ecological applications.