Pest insects pose a significant threat to food production worldwide resulting in annual losses worth hundreds of billions of dollars. Pest control attempts to prevent pest outbreaks that could otherwise destroy a sward. It is good practice in integrated pest management to recommend control actions (usually pesticides application) only when the pest density exceeds a certain threshold. Accurate estimation of pest population density in ecosystems, especially in agro-ecosystems, is therefore very important, and this is the overall goal of the pest insect monitoring. However, this is a complex and challenging task; providing accurate information about pest abundance is hardly possible without taking into account the complexity of ecosystems' dynamics, in particular, the existence of multiple scales. In the case of pest insects, monitoring has three different spatial scales, each of them having their own scale-specific goal and their own approaches to data collection and interpretation. In this paper, we review recent progress in mathematical models and methods applied at each of these scales and show how it helps to improve the accuracy and robustness of pest population density estimation.

Ecological monitoring aims to provide estimates of pest species abundance-this information being then used for making decisions about means of control. For invertebrate species, population size estimates are often based on trap counts which provide the value of the population density at the traps' location. However, the use of traps in large numbers is problematic as it is costly and may also be disruptive to agricultural procedures. Therefore, the challenge is to obtain a reliable population size estimate from sparse spatial data. The approach we develop in this paper is based on the ideas of numerical integration on a coarse grid. We investigate several methods of numerical integration in order to understand how badly the lack of spatial data can affect the accuracy of results. We first test our approach on simulation data mimicking spatial population distributions of different complexity. We show that, rather counterintuitively, a robust estimate of the population size can be obtained from just a few traps, even when the population distribution has a highly complicated spatial structure. We obtain an estimate of the minimum number of traps required to calculate the population size with good accuracy. We then apply our approach to field data to confirm that the number of trap/sampling locations can be much fewer than has been used in many monitoring programmes. We also show that the accuracy of our approach is greater that that of the statistical method commonly used in field studies. Finally, we discuss the implications of our findings for ecological monitoring practice and show that the use of trap numbers 'smaller than minimum' may still be possible but it would result in a paradigm shift: the population size estimates should be treated probabilistically and the arising uncertainty may introduce additional risk in decision-making.

Obtaining information about pest-insect population size is an important problem of pest monitoring and control. Usually, this problem has to be solved based on scarce spatial data about the population density. The problem of monitoring can thus be linked to a more general mathematical problem of numerical integration on a coarse grid. Numerical integration on coarse grids has rarely been considered in literature as it is usually assumed that the grid can be refined. However, this is not the case in ecological monitoring where fine grids are not available. In this paper, we introduce a method of numerical integration that allows one to accurately evaluate an integral on a coarse grid. The method is tested on several functions with different properties to show its effectiveness. We then use the method to obtain an estimate of the population size for different population distributions and show that an ecologically reasonable accuracy can be achieved on a very coarse grid consisting of just a few points. Finally, we summarize our mathematical findings as a protocol of ecological monitoring, thus sending a clear and practically important message to ecologists and pest-control specialists.

It has long been recognized that numerical modelling and computer simulations can be used as a powerful research tool to understand, and sometimes to predict, the tendencies and peculiarities in the dynamics of populations and ecosystems. It has been, however, much less appreciated that the context of modelling and simulations in ecology is essentially different from those that normally exist in other natural sciences. In our paper, we review the computational challenges arising in modern ecology in the spirit of computational mathematics, i.e. with our main focus on the choice and use of adequate numerical methods. Somewhat paradoxically, the complexity of ecological problems does not always require the use of complex computational methods. This paradox, however, can be easily resolved if we recall that application of sophisticated computational methods usually requires clear and unambiguous mathematical problem statement as well as clearly defined benchmark information for model validation. At the same time, many ecological problems still do not have mathematically accurate and unambiguous description, and available field data are often very noisy, and hence it can be hard to understand how the results of computations should be interpreted from the ecological viewpoint. In this scientific context, computational ecology has to deal with a new paradigm: conventional issues of numerical modelling such as convergence and stability become less important than the qualitative analysis that can be provided with the help of computational techniques. We discuss this paradigm by considering computational challenges arising in several specific ecological applications.

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