Skip to main content
SpringerLink
Log in

Numerical Study of Pest Population Size at Various Diffusion Rates

  • Chapter
  • First Online:
Book cover Dispersal, Individual Movement and Spatial Ecology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2071))

Abstract

Estimating population size from spatially discrete sampling data is a routine task of ecological monitoring. This task may however become challenging in the case that the spatial data are sparse. The latter often happens in nationwide pest monitoring programs where the number of samples per field or area can be reduced to just a few due to resource limitation and other reasons. In this rather typical situation, the standard (statistical) approaches may become unreliable. Here we consider an alternative approach to evaluate the population size from sparse spatial data. Specifically, we consider numerical integration of the population density over a coarse grid, i.e. a grid where the asymptotical estimates of numerical integration accuracy do not apply because the number of nodes is not large enough. We first show that the species diffusivity is a controlling parameter that directly affects the complexity of the density distribution. We then obtain the conditions on the grid step size (i.e. the distance between two neighboring samples) allowing for the integration with a given accuracy at different diffusion rates. We consider how the accuracy of the population size estimate may change if the sampling positions are spaced non-uniformly. Finally, we discuss the implications of our findings for pest monitoring and control.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    At least, for any t not too small, in order to avoid the effect of the initial conditions.

  2. 2.

    Note that the notation x 0, x 1, x 2 we use to discuss the hump approximation is not the same as the numeration of grid nodes we introduced in the previous section. In other words, the “endpoints” x 0 and x 2 are arbitrarily located interior points of the interval [0, 1].

References

  1. C.J. Alexander, J.M. Holland, L. Winder, C. Wooley, J.N. Perry, Performance of sampling strategies in the presence of known spatial patterns. Ann. Appl. Biol. 146, 361–370 (2005)

    Article  Google Scholar 

  2. W. Alt, Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147–177 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996)

    MATH  Google Scholar 

  4. F. Bartumeus, J. Catalan, G.M. Viswanathan, E.P. Raposo, M.G.E. da Luz, The influence of turning angles on the success of non-oriented animal searches. J. Theor. Biol. 252, 43–55  (2008)

    Article  Google Scholar 

  5. B. Boag, K. Mackenzie, J.W. McNicol, R. Neilson, Sampling for the New Zealand flatworm, in Crop Protection in Northern Britain 2010. Proceedings of the Conference held at the University of Dundee, Scotland (Page Bros, Norwich, 2010), pp. 45–50

    Google Scholar 

  6. G.D. Buntin, Developing a primary sampling program, in Handbook of Sampling Methods for Arthropods in Agriculture, ed. by L.P. Pedigo, G.D. Buntin (CRC Press, Boca Raton, 1994), pp. 99–115

    Google Scholar 

  7. P.J. Davis, Interpolation and Approximation (Dover, New York, 1975)

    MATH  Google Scholar 

  8. P.J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975)

    MATH  Google Scholar 

  9. S.M. Dunn, A. Constantinides, P.V. Moghe, Numerical Methods in Biomedical Engineering (Elsevier, New York, 2006)

    Google Scholar 

  10. M. Giona, H.E. Roman, Fractional diffusion equation for transport phenomena in random media. Phys. A 185, 87–97 (1992)

    Article  Google Scholar 

  11. J.M. Holland, J.N. Perry, L. Winder, The within-field spatial and temporal distribution of arthropods in winter wheat. Bull. Entomol. Res. 89, 499–513 (1999)

    Article  Google Scholar 

  12. P.M. Kareiva, N. Shigesada, Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)

    Article  Google Scholar 

  13. M. Kogan, Integrated pest management: historical perspectives and contemporary developments. Annu. Rev. Entomol. 43, 243–270 (1998)

    Article  Google Scholar 

  14. S.X. Liao, M. Pawlak, On image analysis by moments. IEEE Trans. Pattern Anal. Mach. Intell. 18(3), 254–266 (1996)

    Article  Google Scholar 

  15. H. Malchow, S.V. Petrovskii, E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations (Chapman & Hall/CRC Press, London, 2008)

    MATH  Google Scholar 

  16. J.D. Murray, Mathematical Biology (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  17. P. Northing, Extensive field based aphid monitoring as an information tool for the UK seed potato industry. Aspects Appl. Biol. 94, 31–34 (2009)

    Google Scholar 

  18. A. Okubo, S. Levin, Diffusion and Ecological Problems: Modern Perspectives (Springer, Berlin, 2001)

    Book  Google Scholar 

  19. M.A. Pascual, P. Kareiva, Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches. Ecology 77, 337–349 (1996)

    Article  Google Scholar 

  20. J.N. Perry, Simulating spatial patterns of counts in agriculture and ecology. Comput. Electron. Agric. 15, 93–109 (1996)

    Article  MathSciNet  Google Scholar 

  21. N.B. Petrovskaya, S. Petrovskii, The coarse-grid problem in ecological monitoring. Proc. R. Soc. A 466, 2933–2953 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. N.B. Petrovskaya, E. Venturino, Numerical integration of sparsely sampled data. Simul. Model. Pract. Theor. 19(9), 1860–1872 (2011)

    Article  Google Scholar 

  23. N.B. Petrovskaya, S.V. Petrovskiy, A.K. Murchie, Challenges of ecological monitoring: estimating population abundance from sparse trap counts. J. R. Soc. Interface 9, 420–435 (2012)

    Article  Google Scholar 

  24. S.V. Petrovskii, On the plankton front waves accelerated by marine turbulence. J. Mar. Syst. 21, 179–188 (1999)

    Article  Google Scholar 

  25. S.V. Petrovskii, B.-L. Li, Exactly Solvable Models of Biological Invasion (Chapman & Hall/CRC, Boca Raton, 2006)

    MATH  Google Scholar 

  26. S.V. Petrovskii, H. Malchow, Spatio-temporal chaos in an ecological community as a response to unfavorable environmental changes. Adv. Complex Syst. 4, 227–250 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. S.V. Petrovskii, B.-L. Li, H. Malchow, Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems. Bull. Math. Biol. 65, 425–446 (2003)

    Article  Google Scholar 

  28. G.A. Seber, The Estimation of Animal Abundance and Related Parameters (Charles Griffin, London, 1982)

    Google Scholar 

  29. L.A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32, 653–665 (1977)

    Article  MATH  Google Scholar 

  30. J.A. Sherratt, M. Smith, Periodic travelling waves in cyclic populations: field studies and reaction diffusion models. J. R. Soc. Interface 5, 483–505 (2008)

    Article  Google Scholar 

  31. G.W. Snedecor, W.G. Cochran, Statistical Methods (The Iowa State Iniversity Press, Ames, 1980)

    Google Scholar 

  32. V.M. Stern, Economic thresholds. Annu. Rev. Entomol. 18, 259–280 (1973)

    Article  Google Scholar 

  33. W.J. Sutherland (ed.), Ecological Census Techniques: A Handbook (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  34. L.R. Taylor, I.P. Woiwod, J.N. Perry, The negative binomial as a dynamic ecological model for aggregation, and the density dependence of k. J. Anim. Ecol. 48, 289–304 (1979)

    Article  MathSciNet  Google Scholar 

  35. P. Turchin, Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants (Sinauer, Sunderland, 1988)

    Google Scholar 

  36. L. Yaroslavsky, A. Moreno, J. Campos, Frequency responses and resolving power of numerical integration of sampled data. Opt. Express 13(8), 2892–2905 (2005)

    Article  Google Scholar 

Download references

Acknowledgements

This study was partially supported by The Leverhulme Trust through grant F/00-568/X.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

    Natalia Petrovskaya & Nina Embleton

  2. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Sergei V. Petrovskii

Authors
  1. Natalia Petrovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nina Embleton
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergei V. Petrovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natalia Petrovskaya .

Editor information

Editors and Affiliations

  1. Mathematical & Statistical Sciences, University of Alberta, Central Academic Building 632, Edmonton, T6G 2G1, Alberta, Canada

    Mark A. Lewis

  2. Mathematical Institute, Centre for Mathematical Biology, University of Oxford, St. Giles Street 24-29, Oxford, OX1 3LB, United Kingdom

    Philip K. Maini

  3. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Sergei V. Petrovskii

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Petrovskaya, N., Embleton, N., Petrovskii, S.V. (2013). Numerical Study of Pest Population Size at Various Diffusion Rates. In: Lewis, M., Maini, P., Petrovskii, S. (eds) Dispersal, Individual Movement and Spatial Ecology. Lecture Notes in Mathematics(), vol 2071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35497-7_13

Download citation

Publish with us

Policies and ethics

Search

Navigation