Agriculture Reference
In-Depth Information
lifetime of host tissue is often only one or a few latent periods long, so this epidemic
phase will usually be short and will often start from a high initial inoculum level
represented by the standing population during vegetative growth.
7.8 CHANGES OVER TIME-SCALES LONGER THAN EITHER CROP
OR PATHOGEN LIFETIME
7.8.1 Possible dynamical patterns
It is helpful to distinguish internal and external forces acting on a population. This is
largely an artificial distinction but makes it possible to think about patterns of
change in a structured way. If mathematical models of how populations change over
time are used, internal forces are those that change only because of processes
described by the model. External forces are those that change in particular ways over
time or at specific times, in ways for which the model offers no explanation.
The simplest type of model includes only the pathogen population. In a constant
host population and a steady environment, if time lags are not considered, this kind
of model will always show the pathogen population tending to return to an
equilibrium level from arbitrary starting conditions. If environmental disturbances
alter the position of the equilibrium, the population will tend to track these
disturbances.
However, the state of the pathogen population at some past times may influence
how it now grows: this influence may arise from the age structure, or from changes
induced in the host. In this case, other, more complicated dynamical patterns can
arise. Even in a steady environment, a pathogen population may tend to oscillate
regularly between two or more distinct levels, or to oscillate irregularly in a quasi-
periodic or chaotic pattern. A quasi-periodic pattern is one in which there are two
fundamental frequencies of oscillation in population numbers present but the ratio of
the two frequencies is an irrational number, so no matter how long a time is
considered, no multiple of one frequency is ever equal to a multiple of the other. A
chaotic pattern is one in which very small changes in population numbers cause very
large changes in the population at later times and in which, even in a steady
environment, no regularly repeating sequence of population numbers ever appears.
In general, populations are more likely to show complicated behaviour if they
both (a) change very fast (on their natural time-scale as discussed in section 7.3) and
(b) are strongly influenced by their state or size a considerable time before. The
main ways in which the past state of a population is likely to influence its present
growing conditions are through interactions with its host, through interactions with
other populations which compete with it or consume it by parasitism or predation, or
through strong natural selection operating differently on different sizes or types of
population.
A model including the host population must have at least two variables, one each
to represent the host and the pathogen populations. If the pathogen has a free-living
stage, or the host can become immune, more complex models still may be
appropriate. Any model with more than two variables considered, or which includes
