



Estimating pest insect population density
from trap counts
( MSci project)

Prerequisites:
Basic knowledge of numerical methods, good programming skills, solid knowledge of the calculus and the probability theory (the undergraduate level)

Description:
In ecological studies, populations are usually described in terms of the population density or population size. Having these values known over a period of time, conclusions can be made about a given species, community, or ecosystem as a whole. In
particular, in pest management, the information gained about pest abundance in a given field or area is then used to make a decision about
pesticide application. To avoid
unjustified decisions and unnecessary losses, the quality of the information about the population density is therefore a matter of primary
importance. However, the population density is rarely measured straightforwardly, e.g.~by direct counting of the individuals. In the case of insects, their density is often estimated based on trap counts. The problem is that, once the trap counts are collected, it is not always clear how to use them in order to obtain an estimate
of the population density in the field. The aim of this project is to overcome current limitations of trapping methods used in ecological studies through developing a
theoretical and computational framework that enables a direct estimate of populations from trap counts. A `meanfield' diffusion model will be considered to investigate if
it is capable of revealing the generic relationship between trap catches and population density.

References:
Crank, J., 1975. The mathematics of diffusion (2nd edition). Oxford University press, Oxford.
Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, Cambridge.
Okubo, A., Levin SA, 2001. Diffusion and Ecological Problems: Modern Perspecives. Springer, Berlin.






Reconstructing pest insect population density function in a 3D domain
( MSci project)

Prerequisites:
Basic knowledge of numerical methods, good programming skills, solid knowledge of the calculus and the probability theory (the undergraduate level)

Description:
In reallife ecological problems the pest population size is normally reconstructed from the density function obtained via field measurements.
In the case of pest insects, their population density is often estimated based on trap counts obtained for traps randomly installed in the domain of interest. Once the trap counts have been collected, it is possible to use interpolation methods to reconstruct the density function at any given point of the domain.
There are many methods of interpolating randomly spaced point data. Some of
these methods are global and utilize all the
known values to evaluate an unknown value, while in local methods only a
specified number of nearest neighbors are used to evaluate an unknown value.
The aim of this project is to investigate global and local methods of interpolation when the data available for interpolation (that is, trap counts) are sparse as it often occurs in ecological applications.
First, trap catches will be transformed into values of the population density function at given locations by generating a nonuniform covering partition of the domain where the trap counts are collected. Then the interpolation methods will be investigated for a discrete (sparse) density function and the conclusions about their accuracy will be provided.

References:
de Berg, M et al., 2008.
Computational Geometry: Algorithms and Applications. SpringerVerlag.
Atkinson, K., 1985. Elementary Numerical Analysis. John Wiley & Sons.
Kot, M, 2001. Elements of Mathematical Ecology. Cambridge University Press, Cambridge.






Evaluation of pest insect abundance in the presence of noise
( MRes project)

Prerequisites:
Basic knowledge of numerical methods, good programming skills, solid knowledge of the calculus and the probability theory (the undergraduate level)

Description:
The project deals with the problem of evaluation of pest insect population size under the assumption that sampled data available to us
are randomly perturbed.
The evaluation is done by a spatially discrete method of numerical integration where the sampled data (e.g., obtained by trapping as in
pest insect monitoring) are considered as values of the population density.
Numerical integration is a computational technique that allows one to evaluate the pest population size when a discrete set
of sampled data is available. Integration of the pest population density function should give us an estimate of the pest population size,
the accuracy of the estimate depending on the number of traps installed in the agricultural field to collect the data.
A widespread approach in numerical integration is to assume that data are precise, so that a random error is zero when data are collected.
This assumption, however, does not hold in ecological applications. An inherent random error always presents in field measurements and it
can strongly affect the accuracy of an estimate. Hence,
the project will study the impact of a random error in density measurements on an estimate of pest insect abundance. Particular attention will be paid
to the case when perturbed data are sparse.

References:
N.B.Petrovskaya, S.V.Petrovskii, A.K.Murchie. Challenges of Ecological Monitoring: Estimating Population Abundance From Sparse Trap Counts.
J.R.Soc.Interface, 2012, vol.9(68), pp.420435.
N.B.Petrovskaya, E.Venturino. Numerical Integration of Sparsely Sampled Data. Simulat. Modell. Pract. Theory, 2011, vol.19(9), pp.18601872.






A numerical study of diffusive predatorprey system with autotaxis and nonlinear convection
(PhD project)

Prerequisites:
Basic knowledge of numerical methods, good programming skills, solid knowledge of the theory of ordinary and partial differential equations

Description:
This project will apply advanced numerical methods to study a realworld ecological problem of high theoretical
and practical importance, such as mechanisms of spatiotemporal pattern formation in ecological communities.
Spatial distribution of ecological populations is very rarely homogeneous. Species heterogeneity is a common phenomenon
observed on various spatial scales, and it has profound implications for ecosystems dynamics. However, its origin is poorly
understood and the corresponding underlying mechanisms often remain obscure.
The aim of the project is to enhance understanding of the role that directional motion of individuals
(e.g., autotaxis or convection) plays in ecological pattern formation. While considerable progress has
been made in identification scenarios of pattern formation in diffusive systems, i.e. under approximation of
random individual motion, the question about possible impact of directional motion is largely open. One reason
for that is that systems of partial differential equations with convective terms are much more difficult for
numerical study. Effective numerical methods have appeared only recently and in this project they will be applied
to the problem of pattern formation in relevant mathematical models of population dynamics.

References:
Okubo, A., Levin SA, 2001. Diffusion and Ecological Problems: Modern Perspecives. Springer, Berlin.






The `single field’ problem in ecological monitoring programmes
(PhD project)

Prerequisites:
Good programming skills, solid knowledge of the probability theory, partial differential equations, and numerical methods

Description:
Ecosystems are under pressure due to anthropogenic impact which has increased considerably over the last few decades.
This pressure can significantly affect the structure of ecological communities, often enhancing population outbreaks of harmful species.
Comprehensive ecological monitoring of pest species is therefore necessary in order to provide detailed and timely information about species
that can potentially cause problems.
It has been increasingly recognized that ecosystems and agroecosystems dynamics is essentially multiscale and its comprehensive understanding
is not possible unless the interaction between the processes going on different spatial and temporal scales is taken into account.
As far as the data collection is concerned, there are several spatial scales in the pest monitoring problem. The first and smallest spatial scale
is related to a single trap. The next spatial scale arises when the local information about the pest density obtained by trapping.
A system of N traps is installed in an agricultural field in order to estimate the pest abundance over the field and we refer to this problem as
a ‘single field’ problem. The project is to investigate the potential importance of the results obtained on the other spatial scales
(i.e., the data from the single trap and the data from the landscape scale) for the accurate pest population size evaluation when a single
agricultural field is concerned.
The approach is based on ideas of numerical integration where one essentially new feature of this evaluation technique is
that the characteristic size of the pest species aggregation can be smaller that the distance between neighbouring traps,
in which case a probabilistic approach should be used.

References:
N.B.Petrovskaya, S.V.Petrovskii, A.K.Murchie. Challenges of Ecological Monitoring: Estimating Population Abundance From Sparse Trap Counts.
J.R.Soc.Interface, 2012, vol.9(68), pp.420435.

S.V.Petrovskii, N.B.Petrovskaya, D.Bearup. Multiscale Approach to Pest Insect Monitoring:
Random Walks, Pattern Formation, Synchronization,
and Networks.
Phys Life Rev, 2014, vol.11, pp.467525, doi: 10.1016/j.plrev.2014.02.001




