Volume 10, Issue 2

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Individual-Based Models of the Spread of Disease, Weeds, and Insects

In the examples of seeds and fungal ascospores, the endpoint of each spread event is independent of the suitability of the environment for survival. Movement of insects often differs in that insects are attracted to suitable environments. Hence the endpoint of a movement event is influenced by the suitability of some or all of the cells of the map. European House Borer is a flying beetle that is attracted to pine wood on which it lays eggs and in which the larvae of the insect feed and grow. We have approximated the spread of a beetle as a series of random flights with a probability that the beetle will find a suitable environment at the end of each flight. The probability of finding a suitable environment is a function of the density of pine in the cell where the flight terminates. If unsuccessful, the insect will fly repeatedly up to a maximum number of flights after which it dies. As for the earlier examples the distance travelled in each flight is chosen from a Cauchy distribution.

A function using NestWhileList that produces a sequence of flights as a list of realCoordinatePair endpoints can be implemented as follows; the function fly returns a realCoordinatePair endpoint from a starting point and a median spread distance, and the function habitatNotFound returns True if a habitat is not found and False otherwise.

Figure 4 shows a graph of the flight path for a beetle that starts at a random location in a 50m square cell of origin and finds a suitable habitat in the neighbouring cell after 10 flights.

Figure 4. Simulated flight path of European House Borer.

New infestations reduce the number of uninfested habitats in the cell and consequently reduce the probability that future infestations will occur in the cell.

In the following example, spread of European House Borer has been modelled in the region surrounding Perth, Western Australia. The median flight distance used was 5m, with a maximum of 10 flights per series. Figures 5 through 9 were generated using ListDensityPlot. Dark blue cells are in pine plantations and have a high density of suitable environments. Light blue cells have a low density of suitable environments, and white cells have no suitable environments. Black dots indicate locations of infested habitats and red dots indicate endpoints of a series of flights. The dots were superimposed on the figure as an Epilog. Labels, ovals, and arrows were added outside of Mathematica.

The simulation was initialised with a single beetle laying eggs into a suitable habitat at the indicated location (Figure 5).

Figure 5. Site of initial infestation for simulated spread of European House Borer near Perth, Western Australia.

In early generations the insect population built up gradually. Most series of flights ended near the site of the original infestation, but in a few cases beetles travelled many kilometres. In Generation 5 one beetle landed on the edge of the Gnangara pine plantation (Figure 6). Total flights, Unsuccessful flights, and Flights off the map refer to series of flights rather than individual flights within a series.

Figure 6. Endpoints of flights in Generation 5 of simulated spread of European House Borer near Perth, Western Australia.

The landing on the edge of the Gnangara pine plantation produced the first successful infestation remote from the initial infestation (Figure 7).

Figure 7. Sites of infestation after Generation 5 of simulated spread of European House Borer near Perth, Western Australia.

The insect population built up rapidly in the plantation and by Generation 9 most flights originated from the plantation (Figure 8).

Figure 8. Endpoints of flights in Generation 9 of simulated spread of European House Borer near Perth, Western Australia.

By Generation 9 the borer was widely established within the plantation (Figure 9). New infestations had occurred in cells with a low density of suitable habitats near the plantation, and an infestation had occurred in one of the small plantations to the south of the site of the original infestation.

Figure 9. Sites of infestation after Generation 9 of simulated spread of European House Borer near Perth, Western Australia.