Allee Effect and Control of Lake System Invasion

Abstract

Abstract. We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that, in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N -lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.

Optimal spatial control of biological invasions

“…Albers et al [14], Bhat et al [15], Blackwood et al [16], Ding et al [17], Finnoff et al [18], Hof [19], Huffaker et al [20], Potapov and Lewis [21], and Sanchirico et al [22] find optimal solutions for special cases, often by focusing only on the steady state, simple landscapes, and interior (non-eradication) solutions, or by tackling reduced dimension problems. We are unaware of work that solves for fully optimal spatial-dynamic solutions (including transition paths and a full range of control options) in large dimension problems that allow for general characterizations of spatial features of the landscape and invasion.…”

Allee threshold and stochasticity in biological invasions: Colonization time at low propagule pressure

“…3b, however there is also a threshold. For small j there are still three steady states close to 0, a, and 1, and qualitatively the dynamics remains the same: if the initial population is small the invasion does not occur (see Keitt et al, 2001;Potapov and Lewis, 2008 for more details). For big j the first two steady states disappear, and only the fully invaded state remains.…”

“…According to practical examples (Veit and Lewis, 1996;Taylor and Hastings, 2005;Potapov and Lewis, 2008), typical values of a are about a few percent of carrying capacity, that is less than 0.1. Below we assume that a⪡1.…”

Optimal Control of Spatial-Dynamic Processes: The Case of Biological Invasions

“…These assumptions simplify the problem but also mandate maximal control in all time periods. Other studies simplify by solving only for the equilibrium optimality conditions, rather than the path by which that equilibrium is optimally achieved (Albers et al 2010;Potapov and Lewis 2008;Sanchirico et al 2010b). Some approaches incorporate realistic features of control options and the characterization of space but make the problem tractable by reducing the dimension.…”

“…4 In more general spatial-dynamic problems, high dimensionality limits the size of the problem that can be solved, thus limiting the questions that can be addressed (Konoshima et al 2008;Potapov and Lewis 2008). A second benefit of the model's simplicity is that the intuition is more readily apparent from results despite the inability to derive an analytical solution to this complex problem.…”