### Abstract

In many invasive species the number of invading individuals is proportional to the time since the population has established (its `age`), not its density. Examples include plant diseases which spread via lesions, which grow on leaves with time and produce ever-increasing amounts of infective material. In this paper, a Leslie matrix model is developed to represent the age structure and reproductive potential due to lesions, particularly for mycelial colonies associated with fungal plant pathogens. Lesion size (and therefore age) is bounded by leaf size, which can be quite large, leading to large matrices. The production of new mycelial colonies is affected by dispersal of spores from the reproductive age-classes of existing colonies, so that dispersal must be included in the matrix model by convolution operators. The infinite-dimensional version of the model is more tractable than the large, finite models, and is used to determine an upper bound on rates of invasion. The model is applied to model the life history of the oömycete Phytophthora infestans, causal agent of potato late blight disease. It is shown that the infinite-dimensional model closely predicts behavior of finite-dimensional models, cut off at certain age-classes of lesions because of finite leaf size. Surprisingly, the infinite-dimensional model is more tractable than finite-dimensional model versions, yielding robust results for practical situations

Original language | English |
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Pages (from-to) | 117-140 |

Journal | Linear Algebra and Its Applications |

Volume | 398 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- focus expansion
- linear determinacy
- invasion speed
- dispersal
- disease
- competition
- demography
- examples

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## Cite this

Powell, J. A., Slapnicar, I., & van der Werf, W. (2005). Epidemic spread of a lesion-forming plant pathogen - analysis of a mechanistic model with infinite age structure.

*Linear Algebra and Its Applications*,*398*, 117-140. https://doi.org/10.1016/j.laa.2004.10.020