Runs of Healthy and Diseased Trees in Transects Through an Infected Forest

Abstract

SUMMARY Narrow transects through an infected population of trees must contain alternating runs of healthy and diseased individuals. The frequency distributions of the lengths of these runs are examined. It is postulated that healthy trees are, conceptually, of two kinds: those growing in infected 'patches' that could be, but happen not to be, diseased; and those growing in uninfected 'gaps' that are not exposed to infection. Thus three kinds of tree make up the sequences observed in transects. It is then assumed that the manner in which one tree follows another in a transect may be likened to state transitions in a regular three-state Markov chain. Of the assumptions that may be made concerning the distribution of diseased trees with respect to healthy ones in a patchily infected forest, the most straightforward appears to be this: that the whole area of the population consists of a mosaic of 'clean' sub-areas (within which none of the trees is diseased) and of infected sub-areas (within which a fixed proportion, p, of the trees is diseased). It is further assumed that within infected sub-areas, the diseased and healthy trees are randomly mingled, or unsegregated. In an earlier paper (Pielou [1963]), a probability distribution was derived that would graduate the frequencies of the number of diseased trees per randomly thrown quadrat in such a population. It was also shown how the two parameters of the distribution, p and ir, might be estimated, where p is defined as above and 7r denotes the proportion of the total area occupied by infected sub-areas. Results obtained from randomly thrown quadrats, however, do not permit the drawing of any conclusions as to the sizes and arrangement of the individual infected sub-areas. All one can say is that, together, they occupy a proportion 7r of the total area of the population. It is impossible to determine whether these sub-areas are small and numerous, or large and sparse; and whether they occur at random or not. The work described in the present paper is an attempt to solve this problem by an examination of the alternating runs of diseased and