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Pattern Formation in a Model for Mountain Pine Beetle Dispersal: Linking Model Predictions to Data

Bulletin of Mathematical Biology volume 75, pages 1778–1797 (2013)Cite this article

Abstract

Pattern formation occurs in a wide range of biological systems. This pattern formation can occur in mathematical models because of diffusion-driven instability or due to the interaction between reaction, diffusion, and chemotaxis. In this paper, we investigate the spatial pattern formation of attack clusters in a system for Mountain Pine Beetle. The pattern formation (aggregation) of the Mountain Pine Beetle in order to attack susceptible trees is crucial for their survival and reproduction. We use a reaction-diffusion equation with chemotaxis to model the interaction between Mountain Pine Beetle, Mountain Pine Beetle pheromones, and susceptible trees. Mathematical analysis is utilized to discover the spacing in-between beetle attacks on the susceptible landscape. The model predictions are verified by analysing aerial detection survey data of Mountain Pine Beetle Attack from the Sawtooth National Recreation Area. We find that the distance between Mountain Pine Beetle attack clusters predicted by our model closely corresponds to the observed attack data in the Sawtooth National Recreation Area. These results clarify the spatial mechanisms controlling the transition from incipient to epidemic populations and may lead to control measures which protect forests from Mountain Pine Beetle outbreak.

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Acknowledgements

Funding for this project was provided by NSERC (SS,RT), PIMS IGTC (SS), MITACS (SS,RT), UBC Okanagan (SS,RT), NSF DEB 0918756 (JP), and WWETAC (JP). We would like to acknowledge the assistance of Mary Reid and Kurt Trzcinski in developing the mathematical model and Daniel Coombs, Sylvie Desjardins, and the Tyson lab group for insightful comments. Further, we would like to thank the reviewers for their valuable suggestions.

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Affiliations

  1. University of British Columbia Okanagan Campus, Kelowna, Canada

    S. Strohm & R. C. Tyson

  2. Mathematics and Statistics, Utah State University, Logan, UT, USA

    J. A. Powell

Authors
  1. S. Strohm
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  2. R. C. Tyson
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  3. J. A. Powell
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Correspondence to S. Strohm.

Appendix

Appendix

A.1 Estimating Allee Attack Threshold, k p

The parameter k p is the density of flying MPB required for 50 % nesting beetle success. Assuming each female beetle makes a single gallery, we estimate the density of flying MPB required for a 50 per cent success of mass attack to be 40 beetles/m2 (Raffa and Berryman 1983). Since our model uses area in ha instead of m2, we must multiply this quantity by a conversion factor

$$ B_h = B_m \frac{\mathit{SA}}{\mathit{TA}}, $$
(21)

where B h is the number of beetles per hectare, B m is the number of beetles per m2, SA is the surface area of the tree attacked, and TA is the area within which a beetle is considered to be “attacking” a tree.

Assuming that the basal 7.5 m of a tree can be attacked (Raffa and Berryman 1983), we find the surface area as SA=πdh, where the tree diameter, d, can range from 0.1874 to 0.3456 m and the height h is taken to be 7.5 m. Therefore, SA can range from 4.42 to 8.14 m2. We assume that beetles will attack within a 10 m2 area of the tree, TA=10 m2=10−3 ha, which makes k p =176800–325600 beetles/ha. We initially assume the value of k p =250000 MPB/ha.

A.2 FFT Analysis of MPB Data

An example of the FFT analysis of the data (in 2007) is shown in Fig. 4. Each year the landscape was analysed for regions of incipient epidemic densities of MPB. The size and number of regions chosen in each year is displayed in Fig. 5.

Fig. 4
figure 4

The SNRA (top left) is plotted for 2007, where white areas denote regions of MPB attack. The red highlighted region (left bottom) is analysed using Discrete fast Fourier transforms to determine the power spectral density (right). In this graph, the dominant wavenumber (ν d ) is represented by a solid red dot, and the upper (ν u ) and lower (ν l ) bounds on the dominant wavenumber are displayed with vertical green and black lines, respectively (Color figure online)

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Fig. 5
figure 5

The size (km2) (a) and number of regions (b) over the time period 1991–2009. Multiple regions of the same size in the same year are represented by a single dot for clarity

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Strohm, S., Tyson, R.C. & Powell, J.A. Pattern Formation in a Model for Mountain Pine Beetle Dispersal: Linking Model Predictions to Data. Bull Math Biol 75, 1778–1797 (2013). https://doi.org/10.1007/s11538-013-9868-8

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Keywords

  • Mountain Pine Beetle
  • Dendroctonus ponderosae
  • Pattern formation
  • Aerial detection survey
  • Management