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.