Patchy, not patchy, or how much patchy? Classification of spatial patterns appearing in a model of biological invasion

Abstract

Abstract. Good understanding of spatiotemporal patterns of species spread during biological invasion is needed for efficient monitoring and control of harmful alien pests. Various growth-dispersal-type models of population dynamics predict that invasive species spread can follow two qualitatively different scenarios such as the propagation of a continuous population front and the "no-front" patchy invasion. Distinguishing between these two patterns of spread is important, in particular because the patchy invasion poses a much greater challenge for monitoring and control. However, a mathematical theory of the patchy invasion is missing and it remains unclear what are the restrictions on parameter values and how much different this dynamical regime is from the continuous front propagation. In this paper, we address these issues in terms of a biologically meaningful mathematical model consisting of two coupled integral-difference equations. We show that the relevant domain of the parameter space has a complex intermittent structure. We also suggest a criterion that can be used to distinguish between the patchy invasion and the continuous front propagation: the patchy-invasion spatial pattern is shown to be much more sensitive to the cutoff at low densities.Mathematics Subject Classification. 92D40, 35B36, 35Q92, 37N25.

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

“…One reason we want to use the symmetric initial conditions is that the numerical solution remains symmetric at any time t > 0 and the solution symmetry can then be considered as a reliable means to control the accuracy of the numerical method. Meanwhile, it is worth mentioning here that basic topological properties of spatial patterns are not affected by the choice of asymmetric initial conditions although their qualitative estimates are, of course, different; see also the discussion in [43] where the sensitivity of the results to the choice of the initial conditions has been investigated.…”

“…The situation is further exacerbated by the fact that the formation of no front patchy structures is possible for many invasion regimes generated in simulation, yet in many cases transition from continuous front to patchy patterns and vice versa cannot be predicted when parameters in the model used to simulate biological invasion are varied (e.g. see [19,43]). Thus, a question arises about reliable criteria to allow one to compare the topology of continuous front and patchy spatial structures.…”

“…The methods of pattern recognition are of great demand in the problem of biological invasion [8,29,31,58], yet a universal protocol for spatial pattern analysis does not exist and criteria for comparison between continuous front spatial structures and discontinuous spatial distributions remain unclear. Primary topological analysis of 'no front' patchy spatial patterns in a specific model of patchy invasion has been carried out in [43,44]. In this paper we utilize the approach in [44] to investigate several topological characteristics that can potentially help us to compare various spatial structures.…”

“…The simplest, yet reliable, way to distinguish between the continuous front and patchy spatial patterns is based on counting the number of objects in the image of spatial distribution [43]. Any continuous front is considered as a single object, while a 'no front' patchy structure can be thought of as several separate objects in the spatial domain.…”

Improving the biological control of Persicaria perfoliata (Polygonaceae) using Rhinoncomimus latipes Korotyaev (Coleoptera: Curculionidae)

“…Highly invaded sites that have been infested by a dominant exotic plant for several years often are characterized by a mosaic of patches of various sizes and abundances of that dominant invasive plant (Kartzinel et al 2015, Petrovskaya et al 2017. If this patchy mosaic occurs in a physiographicallyhomogeneous environment, one may hypothesize that patches lacking in the dominant invader include plant species that make local spread by the invader more difficult; these species could include native species, exotic species, or other exotic invaders.…”