Avoiding Extinction in a Managed Single Species Population Model by Means of Anticipative Control

Abstract

Population models with a high intrinsic growth rate that are subject to an Allee effect are known to exhibit chaotic transients which end in a population collapse and consequent extinction. In a managed environment, it might be possible to anticipate this event and affect the outcome by a carefully designed strategy. In this paper, the dynamics of a managed single species are modelled using an Anticipatory System and possible control strategies resulting from this are analysed and evaluated.

Anticipative chaos control in population models with the Allee effect

“…3. There is a globally attracting fixed point at x = 0 for r < e. For r ≥ e there are two additional fixed points, of which the smaller one, A, is always unstable and forms the threshold for the Allee effect (see also [18]). The immediate basin of attraction for the original fixed point…”

“…This discrete-time weak anticipatory model must run quicker than real time in order to generate predictions of future states. In [17], such a model is detailed for several common population maps, and in [18] the weak anticipatory model is applied in controlling extinction for Ricker map case. The control strategy, which is outlined in [19], is based on analyzing the states of the system and initiating preventative actions only when some undesirable state is predicted.…”

“…The control strategy, which is outlined in [19], is based on analyzing the states of the system and initiating preventative actions only when some undesirable state is predicted. Let's consider a version of Ricker map which may exhibit the Allee effect, and was usually investigated in this kind of studies [18][20] [21]:…”

An Investigation of the Effect of Using Anticipation in a Technological Substitution Model

“…Rather than use the approach of [6] where the anticipated values of variables are used continuously to implement strategy, the strategy considered in this section will mirror that originally envisaged by Rosen [23] and used in [4] where the anticipated value of a variable will trigger a control action if it is needed and leave the system unaltered otherwise. From Section 3, it is known that an invasion will automatically succeed, either when intraspecific competition is structurally dominant or the size of the invading technology is initially large.…”

“…This paper uses the particular weak anticipatory paradigm described in [3] and developed in [4] to investigate how invading strategies may be modified and implemented with anticipatory knowledge. In this approach, the anticipatory system consists of a discrete time system S and an internal model M (of S and possibly the environment).…”