Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration

Abstract

Monitoring of pest insects is an important part of the integrated pest management. It aims to provide information about pest insect abundance at a given location. This includes data collection, usually using traps, and their subsequent analysis and/or interpretation. However, interpretation of trap count (number of insects caught over a fixed time) remains a challenging problem. Firstly, an increase in either the population density or insects activity can result in a similar increase in the number of insects trapped (the so called "activity-density" problem). Secondly, a genuine increase of the local population density can be attributed to qualitatively different ecological mechanisms such as multiplication or immigration. Identification of the true factor causing an increase in trap count is important as different mechanisms require different control strategies. In this paper, we consider a mean-field mathematical model of insect trapping based on the diffusion equation. Although the diffusion equation is a well-studied model, its analytical solution in closed form is actually available only for a few special cases, whilst in a more general case the problem has to be solved numerically. We choose finite differences as the baseline numerical method and show that numerical solution of the problem, especially in the realistic 2D case, is not at all straightforward as it requires a sufficiently accurate approximation of the diffusion fluxes. Once the numerical method is justified and tested, we apply it to the corresponding boundary problem where different types of boundary forcing describe different scenarios of pest insect immigration and reveal the corresponding patterns in the trap count growth.

Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks

“…For the sake of simplicity, the analysis above was restricted to the 1D case. The 2D case is technically much more complicated and will be considered elsewhere [14]. Here we mention that, as we have observed in numerous numerical simulations (not shown here), the results described in this section are generic and most of them remain valid, at least qualitatively, in the 2D case.…”

The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

“…Our interest is primarily based on understanding the underlying mechanisms that govern movement; it is sufficient to focus on a 1D conceptual scenario. Despite the fact that this case is hardly realistic in terms of modeling movement in a real field setting, however it does provide a theoretical background for the more realistic 2D case [56]. Furthermore, any unnecessary additional complexity that would arise due to the effects of trap and field geometries is then avoided.…”

“…A linear approximation is used in (A7); alternatively, a more accurate way to compute the flux uses a quadratic polynomial [56], which yields an error in line with the numerical scheme, i.e., O(∆x 2 ). For our simulations, the linear approximation suffices, and we choose the spatial step ∆x to be sufficiently small to ensure that any accumulated errors are negligible.…”

The “Lévy or Diffusion” Controversy: How Important is the Movement Pattern in the Context of Trapping?

“…Since our interest is primarily based on understanding the underlying mechanisms which govern movement, it is sufficient to focus on a 1D conceptual scenario. Despite the fact that this case is hardly realistic in terms of modelling movement in a real field setting, it does provide a theoretical background for the more realistic 2D case [55]. Also, any unnecessary additional complexity which would arise due to the effects of trap and field geometries are then avoided.…”

“…To compute this we approximate the derivative Here, we take the absolute value instead of omitting the '−' sign which would be required since the flux is in the opposite direction of the positive x-axis. A linear approximation is used in (A.7), alternatively, a more accurate way to compute the flux uses a quadratic polynomial [55], which yields an error in line with the numerical scheme i.e O(∆x 2 ). For our simulations the linear approximation suffices, and we choose the spatial step ∆x to be sufficiently small to ensure that any accumulated errors are negligible.…”