Research

Professor Gourley's main research areas are population dynamics
and epidemiology. He is especially interested in delay equations, which arise very commonly indeed in numerous branches of mathematical ecology and medicine. For example, many insects go through a larval stage before becoming mature reproducing adults, and in disease modelling the incubation time of the disease (which can be as long as several decades for some illnesses) needs to be taken care of. These considerations lead to the study of delay differential equations. Partial differential equations can have delay terms too. Most of Prof Gourley's publications are listed here.

Prof Gourley is an associate editor of Mathematical Biosciences and Engineering (manuscripts should be submitted to the editorial office).

Recent and ongoing research projects

Mosquito borne diseases

With J. Wu (York University, Toronto) and R. Liu (University of Wyoming),
Prof Gourley has been working on the development of mathematical models
to assess the effectiveness of culling as a tool to eradicate mosquito-borne
diseases such as West Nile virus. Mosquitoes can be culled either at the larval stage using
larvicides, or as adults using adulticides sprayed into the atmosphere
as fine droplets. The mathematical model becomes either a
system of autonomous delay differential equations with impulses (if
the adult mosquitoes are culled) or a system
of nonautonomous delay differential equations where the time-varying
coefficients are determined by the culling times and rates (in the
case when only the larval mosquitoes are culled). Sufficient conditions have been
found to ensure eradication of the disease and comparisons have been made
between the effectiveness of larvicides and adulticides for the control of West Nile
virus.
Birds, as well as mosquitoes, play an important role in the spread of
West Nile virus. The American crow is an especially vulnerable species. The age structure of birds is important as immature fledgling birds are more vulnerable to attack by
mosquitoes. Incorporation of the age structure of birds has been a
recent extension of our work, as has the calculation of the spatial
speed of spread of the disease on a large scale.

The widespread use of adulticide sprays has, in fact, been criticised. There are concerns about the implications for public health of using toxic chemicals, and it is known that some of the chemicals can kill other wildlife such as dragonflies (see picture above) which are beneficial to us because they prey on larval and adult mosquitoes. Bats are natural predators of mosquitoes, but there are concerns about increasing the number of bats because they can carry rabies. It seems likely that a combination of the use of pesticides and encouraging natural predators is best and it is exactly the kind of situation in which mathematical modelling is beneficial. Public health authorities usually emphasize larviciding as the chemicals come in pellet or granular form and are placed into standing water where the larvae develop. They offer little risk to fish and other organisms and are not sprayed into the air. Adulticide spraying is usually considered only if larvicide alone is not sufficient.

Models of bacteriophage infection

Bacteriophage infection can be a significant
mechanism of mortality in marine prokaryotes and the constituents released
by cell lysis can be an important pathway of nutrient recycling. This has direct
bearing on issues such as global warming and topics of geochemical
cycles. Viral infection also has direct implications for genetic
exchange in the sea.

With Y. Kuang at Arizona State University, Prof Gourley has been working on
delay differential equation models and partial differential equation models with
delay that model the dynamics of such infections and particularly the transport of bacteria
during the period between infection and lysis, when virus particles are released back into
the sea.

The Asian Longhorned Beetle

International trade, travel and climate change increase the risk of insect infestations by exotic species and new plant diseases. A notable example has been the Asian Longhorned Beetle, pictured above, which has found its way into the USA and Canada in wooden packing cases used for imports from China where the beetle causes substantial damage to poplar plantations as shown above.

Like many beetles it cannot fly long distances and so infestations tend to spread slowly. The larvae tunnel deep into the trees and prefer the upper canopy making detection difficult. Currently, the only means of control is to cut down infested trees and chip or burn them, and replace with non-host species. The beetle has no known natural predator in the USA.

With X. Zou at the University of Western Ontario, Prof Gourley has been working on the development of a mathematical model of the infestation by this beetle of a wooded area, and the control of the infestation. So far in the USA and Canada the beetle has affected only urban and suburban areas, and the aim of our work is principally to make predictions about the implications of the beetle taking hold in a forested area. The model takes account of the developmental stages of the beetle and the destruction of a fraction of trees detected as infested, and therefore also of any larvae in them. The model is a four dimensional system of delay differential equations for the numbers of adult beetles, larval beetles, susceptible and infested trees. Two time delays are present in the model: one of these is the developmental time from egg to adult, and the other is the mean time between infestation of a tree and its subsequent destruction. There is a delicate interplay between these two delays. The model yields insights into the fraction of infested trees that must be removed if the infestation is to be eradicated, and estimates for the number of susceptible trees that will escape infestation.

Winning strategies in competition models

This work, in collaboration with Y. Kuang at Arizona State University, was
motivated by the issue of how
diffusion affects the competition outcome of two competing species
that are identical in all respects other than their strategies on
how they spatially distribute their birth rates. Our findings suggest that, in a fast
diffusion environment, where different species have the same birth
rates on average, those that do well are those that have
greater spatial variation in their birth rates. This suggests a
possible explanation for the evolution of grouping behaviour in many species.

Prof Gourley's current and former PhD students:

David Schley (completed 1999)
Jafar Al-Omari (completed 2003)
Debbie Bennett (completed 2004)
Yuliya Kyrychko (completed 2004) (supervised jointly with Dr MV Bartuccelli)
Robin Simons (completed 2005) (supervised jointly with Prof R Hoyle)
John Rayman (completed 2008)
Alan Terry (completed 2009)
Hayley O'Farrell (started 2011) (supervised jointly with Dr. J. Godolphin)