The Mathematics Behind Biological Invasions by Mark A. Lewis Sergei V. Petrovskii & Jonathan R. Potts

Author:Mark A. Lewis, Sergei V. Petrovskii & Jonathan R. Potts
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.41)

A similar integration process gives

(5.42)

[174, 289]. This dispersal kernel has interesting limiting distributions. Setting in (5.42) yields a Cauchy distribution,

(5.43)

If instead with , the familiar normal distribution (5.5) is recovered; see Appendix A.18.

The mechanistic approach to modeling dispersal can be extended in a variety of ways [267]. For example, it can include the case where there is ballistic motion with settling. Here, propagules move outward at a constant speed c and settle with failure rate h(t). Many patterns of dispersal can arise, depending upon the time-dependent settling rate. It is also possible to model multistage dispersal processes. For example, seeds that initially diffuse in the air may be redistributed on the ground by ants or rodents. Finally, explosive spore dispersal can be modeled using the laws of physics to give an intriguing kernel (Fig. 5.5), one that has singularities at the furthest possible dispersal distance. Details are given in [267].

Fig. 5.5A spore dispersal model. The model yields singular dispersal kernels, the mode being the maximum possible dispersal distance. Figure reproduced from [267]

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