Abstract
Accurate identification of spatial patterns remains a challenging problem in many ecological applications. One example is a problem of biological invasion where distinguishing between patchy spatial density pattern and continuous front spatial density pattern is important for monitoring and control of the invasive species. In this paper we address the problem of pattern recognition in biological invasion in terms of a biologically meaningful mathematical model consisting of two coupled integro-difference equations. The model allows for generating topologically different spatial structures and we employ several topological characteristics of spatial pattern to investigate various spatial density distributions. It is argued that, among the other topological quantities, the number of objects in the visual image of a spatial distribution gives us the most reliable conclusion about spatial pattern when it is required to distinguish between continuous and discontinuous (patchy) spatial structures. Furthermore, sensitivity of the pattern classification above to the definition of a monitoring protocol is discussed in the paper. Two basic properties of the monitoring protocol (i.e. the threshold density value and the number of sampling locations) are investigated and it is demonstrated how their variation affects correct reconstruction of spatial density pattern.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)
Garnier, G., Roques, L., Hamel, F.: Success rate of a biological invasion in terms of the spatial distribution of the founding population. Bull. Math. Biol. 74, 453–473 (2012)
Harary, F., Harborth, H.: Extremal animals. J. Comb. Inf. Syst. Sci. 1, 1–8 (1976)
Hargis, C.D., Bissonette, J.A., David, J.L.: The behavior of landscape metrics commonly used in the study of habitat fragmentation. Landsc. Ecol. 13, 167–186 (1998)
Jankovic, M., Petrovskii, S.V.: Gypsy moth invasion in North America: a simulation study of the spatial pattern and the rate of spread. Ecol. Compl. 14, 132–144 (2013)
Kolmogorov, A.N., Petrovskiy, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Moscow Univ. Bull. Math. 1, 1–25 (1937)
Kot, M., Schaffer, W.M.: Discrete-time growth-dispersal models. Math. Biosci. 80, 109–136 (1986)
Lewis, M.A., Petrovskii, S.V., Potts, J.: The Mathematics Behind Biological Invasions, vol. 44. Springer, Berlin
Liebhold, A.M., Gurevitch, J.: Integrating the statistical analysis of spatial data in ecology. Ecography 25, 553–557 (2002)
Mistro, D.C., Rodrigues, L.A.D., Petrovskii, S.V.: Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect. Ecol. Compl. 9, 16–32 (2012)
Morozov, A.Y., Petrovskii, S.V., Li, B.L.: Spatiotemporal complexity of patchy invasion in a predator–prey system with the Allee effect. J. Theor. Biol. 238, 18–35 (2006)
Petrovskaya, N.B., Embleton, N.L.: Evaluation of peak functions on ultra-coarse grids. Proc. R. Soc. A 469, 20120665 (2013). https://doi.org/10.1098/rspa.2012.0665
Petrovskaya, N.B., Embleton, N.L.: Computational methods for accurate evaluation of pest insect population size. In: Godoy, W.A.C., Ferreira, C.P. (eds.) Ecological Modelling Applied to Entomology, pp. 171–218. Springer, Berlin (2014)
Petrovskaya, N.B., Petrovskii, S.V.: The coarse-grid problem in ecological monitoring. Proc. R. Soc. A 466, 2933–2953 (2010)
Petrovskaya, N.B., Petrovskii, S.V., Murchie, A.K.: Challenges of ecological monitoring: estimating population abundance from sparse trap counts. J. R. Soc. Interface 9, 420–435 (2012)
Petrovskaya, N.B., Petrovskii, S.V., Zhang, W.: Patchy, not patchy, or how much patchy? Classification of spatial patterns appearing in a model of biological invasion. Math. Model. Nat. Phenom. 12, 208–225 (2017)
Petrovskii, S.V., Morozov, A.Y., Venturino, E.: Allee effect makes possible patchy invasion in a prey–predator system. Ecol. Lett. 5, 345–352 (2002)
Petrovskii, S.V., Malchow, H., Hilker, F.M., Venturino, E.: Patterns of patchy spread in deterministic and stochastic models of biological invasion and biological control. Biol. Invasions 7, 771–793 (2005)
Petrovskii, S.V., Petrovskaya, N.B., Bearup, D.: Multiscale approach to pest insect monitoring: random walks, pattern formation, synchronization, and networks. Phys. Life Rev. 11, 467–525 (2014)
Rodrigues, L.A.D., Mistro, D.C., Petrovskii, S.V.: Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect. Theor. Ecol. 5, 341–362 (2012)
Rodrigues, L.A.D., Mistro, D.C., Cara, E.R., Petrovskaya, N.B., Petrovskii, S.V.: Patchy invasion of stage-structured alien species with short-distance and long-distance dispersal. Bull. Math. Biol. 77, 1583–1619 (2015)
Rosenberg, M., Anderson, C.: Spatial pattern analysis. In: Gibson, D. (ed.) Oxford Bibliographies in Ecology. Oxford University Press, New York (2016)
Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Petrovskaya, N., Zhang, W. (2020). Accurate Recognition of Spatial Patterns Arising in Spatio-Temporal Dynamics of Invasive Species. In: Aguiar, M., Braumann, C., Kooi, B., Pugliese, A., Stollenwerk, N., Venturino, E. (eds) Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-41120-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-41120-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-41119-0
Online ISBN: 978-3-030-41120-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)