Perverse Consequences of Infrequently Culling a Pest

Abstract

There are potentially many situations in which creatures will be subject to infrequent but regular culling. In terms of controlling crop pests, some farmers may only be able to afford to apply pesticides occasionally. Alternatively, pesticides may be applied only occasionally to limit their unwelcome side effects, which include pesticide resistance, chemical poisoning of agricultural workers, and environmental degradation. In terms of conservation, some species (such as the red deer in the UK) may be culled occasionally to maintain balances within their ecosystem. However, in this paper we discover, as the culmination of an exploration of adult-stage culling of a creature with juvenile and adult life stages, that, in certain circumstances, regular but infrequent culling will, perversely, increase the average population of the creature.

Delayed logistic population models revisited

“…The first phenomenon has been observed in fishing populations [21,47], and the second one is commonly known as the hydra effect, after the nice review of Abrams [1]. For related results in delay-differential equations, see [2,50,51].…”

A population model with birth pulses, age structure, and non-overlapping generations

“…There are two such assumptions. The first of these is w > 2, and the second is a lower bound on b in (33). This lower bound on b is well-defined and positive in view of the assumption that w > 2.…”

“…In the first row of data in the table, the first column shows the parameter choices used in Fig. 3(a) and (b), whilst the second, third, and fourth columns shows the ranges of values for the parameter b which these parameter choices produce for, respectively, the interval in (32), the interval in (36), and the interval in (33). By parts (ii)(D), (ii)(F), and (ii)(E) of Theorem 3, if b lies in the ranges of values in the second, third, and fourth columns Table 2 Data relevant to Fig.…”

“…Whilst our model is new, as far as we are aware, we note that a number of other models of populations subject to birth pulses and age structure have been studied in recent years [7,[22][23][24]. Furthermore, hybrid systems have been investigated in many other contexts, including (but not limited to) vaccination strategies [2,20,28], chemotherapeutic treatment of diseases [12,18], and pest control [16,17,26,27,30,33,36].…”

Dynamics of a structured population on two patches

“…The result one deduces is similar, but not identical to, Proposition 3.4 in [21] where the birth function is assumed to be of Ricker form. The study of invariant intervals can be used not only to establish endemicity on one or two patches, as in Theorem 7, but also to prove, for example, that infrequent regular culling of a pest can, perversely, increase its average population (Sections 8 and 9, [22]). …”