The finite element implementation of a K.P.P. equation for the simulation of tsetse control measures in the vicinity of a game reserve

Abstract

An equation, strongly reminiscent of Fisher's equation, is used to model the response of tsetse populations to proposed control measures in the vicinity of a game reserve. The model assumes movement is by diffusion and that growth is logistic. This logistic growth is dependent on an historical population, in contrast to Fisher's equation which bases it on the present population. The model therefore takes into account the fact that new additions to the adult fly population are, in actual fact, the descendents of a population which existed one puparial duration ago, furthermore, that this puparial duration is temperature dependent. Artificially imposed mortality is modelled as a proportion at a constant rate. Fisher's equation is also solved as a formality. The temporary imposition of a 2% day(-1) mortality everywhere outside the reserve for a period of 2years will have no lasting effect on the influence of the reserve on either the Glossina austeni or the G. brevipalpis populations, although it certainly will eradicate tsetse from poor habitat, outside the reserve. A 5km-wide barrier with a minimum mortality of 4% day(-1), throughout, will succeed in isolating a worst-case, G. austeni population and its associated trypanosomiasis from the surrounding areas. A more optimistic estimate of its mobility suggests a mortality of 2% day(-1) will suffice. For a given target-related mortality, more mobile species are found to be more vulnerable to eradication than more sedentary species, while the opposite is true for containment.

An improved temporal formulation of pupal transpiration in Glossina

“…These would be obvious places in which to concentrate control measures and one immediate application of this research. Barriers of the type modelled in [7] might be far more efficacious if deployed in the immediate vicinity of pupal sites, rather than for the purposes of containment. While early stage mortality is considered to be the most significant, by far, in any model of tsetse population dynamics it is of even greater relevance when in the context of control measures.…”

A model of teneral dehydration in Glossina

“…For the first time it becomes clear why G. austeni is a relatively sedentary (Childs, 2010) denizen of low-lying, often coastal, vleis and estuaries, whereas the highly mobile G. brevipalpis (Childs, 2010) is also associated with drainage lines in the atmospherically-drier regions of the hinterland. G. austeni only needs to larviposit where it feeds while G. brevipalpis can strike out into drier, surrounding country to feed, in spite of its pupa requiring the most humid substrate of all.…”

“…On the other hand, G. austeni eclodes much earlier, leaving G. brevipalpis to bear the brunt of density-dependent predation and parasitism (Rogers and Randolph, 1990). In this way G. austeni may compete with G. brevipalpis in the many environments which facilitate a sympatric, G. brevipalpis-G. austeni population (Childs, 2010).…”

A set of discrete formulae for the performance of a tsetse population during aerial spraying

“…By far the biggest threat to any aerial spraying operation on mainland Africa is re-invasion from adjacent, untreated areas. Closed populations need to be created by temporary barriers of odour-baited targets such as the one used successfully by Kgori et al, 2006. Childs (2010 and Esterhuizen et al (2006) comprehensively researched the design of such odour-baited, target barriers for G. austeni and G. brevipalpis; albeit mostly from a point of view of a control in its own right.…”