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.

Using Probabilistic Approach to Evaluate the Total Population Density on Coarse Grids

Evaluation of the population density in many ecological and biological problems requires a satisfactory degree of accuracy. Insufficient information about the population density, obtained from sampling procedures negatively, impacts on the accuracy of the estimate. When dealing with sparse ecological data, the asymptotic error estimate fails to achieve a reliable degree of accuracy. It is essential to investigate which factors affect the degree of accuracy of numerical integration methods. When the number of traps is less than the recommended threshold, the degree of accuracy will be negatively affected. Therefore, available numerical integration methods cannot guarantee a satisfactory degree of accuracy, and in this sense the error will be probabilistic rather than deterministic. In other words, the probabilistic approach is used instead of the deterministic approach in this instance; by considering the error as a random variable, the chance of obtaining an accurate estimation can be quantified. In the probabilistic approach, we determine a threshold number of grid nodes required to guarantee a desirable level of accuracy with the probability equal to one.

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.

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.