Time Dependent Diffusion as a Mean Field Counterpart of Lévy Type Random Walk

Abstract

Abstract. Insect trapping is commonly used in various pest insect monitoring programs as well as in many ecological field studies. An individual is said to be trapped if it falls within a well defined capturing zone, which it cannot escape. The accumulation of trapped individuals over time forms trap counts or alternatively, the flux of the population density into the trap. In this paper, we study the movement of insects whose dynamics are governed by time dependent diffusion and Lévy walks. We demonstrate that the diffusion model provides an alternative framework for the Cauchy type random walk (Lévy walk with Cauchy distributed steps). Furthermore, by calculating the trap counts using these two conceptually different movement models, we propose that trap counts for pests whose dynamics may be Lévy by nature can effectively be predicted by diffusive flux curves with time-dependent diffusivity.

Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks

“…What is an appropriate choice of D(t) is a subtle issue [2] and will be considered in detail elsewhere [3]. Here we only consider two tentative examples:…”

“…Let us consider the single-trap scale of the pest monitoring problem. In this section, we show that, even in case of a genuine Levy walk, the problem of trap counts interpretation can still be addressed based on the diffusion equation [2,3].…”

The “Lévy or Diffusion” Controversy: How Important is the Movement Pattern in the Context of Trapping?

“…In the context of insect pest monitoring, diffusion models have shown to provide a good theoretical framework and the means for trap count interpretation [12,27,35]. In particular, time dependent diffusion provides an adequate description for more complicated behaviour, at least where standard diffusion fails [26,27,35,64]. The notion of insect movement with time dependent diffusivity is not new, and has been observed in field studies [19].…”

“…Obviously, the issue becomes more difficult if a combination of these features are present. In the context of insect pest monitoring, diffusion models have shown to provide a good theoretical framework and the means for trap count interpretation [12,27,35]. In particular, time dependent diffusion provides an adequate description for more complicated behaviour, at least where standard diffusion fails [26,27,35,64].…”

“…Traps are usually installed in the field under controlled experimental conditions as a means to estimate population abundance [13][14][15]. They are then exposed for the duration of study, insects are caught, traps are normally emptied at regular intervals and the total number which fall into the trap Petrovskii [27]. Our aim is two-fold; that is to propose a diffusion model which consists of parameters that are of ecological significance and can be interpreted with reference to the diffusive properties.…”