Systems of Partial Differential Equations

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.

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“…see Thomas 1995) and is therefore consistent with the accuracy of approximation (18). Similarly, we have…”

Section: Methodssupporting

confidence: 88%

“…see Thomas (1995). However, we mention here that that their applicability and efficiency depend on the geometry of the domain.…”

Section: Introductionmentioning

confidence: 95%

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

Bearup

Petrovskaya

Petrovskii

2015

Mathematical Biosciences

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“…Finite difference methods on structured grids (see [44] and [46]). These methods are very popular, and a great deal of work has been devoted to adaptive grid finite difference methods (see, for example, [36]) to deal with boundary conditions defined along irregular interfaces.…”

Section: Survey Of Numerical Algorithmsmentioning

confidence: 99%

Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

Sethian¹,

Shan²

2008

Journal of Computational Physics

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We present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces. Instead of the typical approach of solving in a larger rectangular domain, our approach performs most calculations only in the desired domain. To do so efficiently, we have developed a one-sided multigrid method to solve the corresponding large sparse linear systems.Our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions. Our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling.We begin by motivating the set of equations which model the electrodeposition process. Building on existing models for superconformal electrodeposition, we develop a model which naturally arises from a conservation law form of surface additive evolution. We then introduce several numerical algorithms, including a conservative material transport level set method and our multigrid method for one-sided diffusion equations. We then analyze the accuracy of our numerical methods. Finally, we compare our result with experiment over a wide range of physical parameters.

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“…To understand these concepts we must first clarify some ideas from mathematical analysis. Here we follow [31] and [32]. Throughout K denotes the fields R and C.…”

Section: Elements Of Numerical Analysismentioning

confidence: 99%

“…For a proof of consistency and stability in case that G and m are both equal to zero, refer to [32]. …”

Section: Finite-difference Schemementioning

confidence: 99%

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

Macías‐Díaz

Puri

2005

Numerical Methods Partial

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In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions (1)in an open sphere D around the origin, where the internal and external damping coefficients-β and γ, respectively-are constant, and the nonlinear term has the form G ′ (w) = w p , with p > 1 an odd number. The functions φ and ψ are radially symmetric in D, and φ, ψ, rφ and rψ are assumed to be small at infinity. We prove that our scheme is consistent order O(∆t 2 ) + O(∆r 2 ) for G ′ identically equal to zero, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to compare the physical effects of β and γ.

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Systems of Partial Differential Equations

Cited by 78 publications

References 0 publications

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

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

Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

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