Agriculture Reference
In-Depth Information
where
r
H
and
K
H
are parameters representing the rate of host growth and maximum
host area, respectively. An analytical solution of disease incidence (
y
=
Y/H
) is
possible for this set of two equations (Waggoner, 1986). The LDE approach is very
flexible. For example, one could assume that absolute change in host size depends
on the amount of disease-free tissues (Jeger, 1987b), then
⎛
⎝
⎞
⎠
dH
dt
K
H
−
H
(8.5)
(
)
=
r
H
H
−
Y
⎜
⎟
K
H
Sometimes, it is possible that
dH/dt
may be negative due to root loss or defoliation
resulting from the disease. This was taken into consideration for the potato-
Verticillium
dahliae
system (Johnson, 1988). In cases where linked equations cannot
be readily integrated, iterative numerical methods may be used to estimate parameter
values (Buwalda
et al.
, 1982; Gilligan and Kleczkowski, 1997).
Lalancette and Hickey (1986) took a different approach for incorporating host
growth into disease progress models; instead of expressing disease and host growth
as a function of time, they expressed disease as a function of host growth. One key
area concerned with the effects of the host on disease development that still needs
further research is ontogenic resistance, where host tissues become gradually
resistant to diseases as they age. This ontogenic resistance has been observed in
many diseases, for example apple scab (MacHardy, 1996; Li and Xu, 2002) and
powdery mildew (Rogers, 1959; Mence and Hildebrandt, 1966). Equally, host
tissues may become more susceptible with increasing age.
The effect of root growth on soil-borne disease development, particularly in the
contact rate between inoculum (primary and secondary) and root, has received
considerable interests. Jeger (1987a) showed that including root growth and
inoculum density into a simple monomolecular model can result in a sigmoid, a
monomolecular, a Gompertz-type or even an asymptotically exponential disease
progress curve with the shape depending on the product of root growth and
inoculum density. Gilligan and his co-workers developed various forms of models to
investigate the effects of root growth, primary and secondary inoculum (infections),
and biocontrol on soil-borne disease development (Gilligan and Bailey, 1997;
Gilligan
et al.
, 1997; Gilligan and Kleczkowski, 1997). One of the important
conclusions was the importance of host density and the rate of root production on
disease development.
(b) Variable rate parameter
The rate of disease change depends on many factors, such as cultivar, environmental
conditions and crop husbandry practices. Of these factors, environmental conditions
are the most variable in terms of the amplitude and frequency of their variation
within a season. In addition, environmental conditions not only directly but also
indirectly affect disease development. For example, the apple scab fungus needs
rainfall to discharge ascospores and free water or moisture to infect while the rate of
infection depends critically on temperature (MacHardy, 1996; see Chapter 18). The
