Optimal spatial control of biological invasions

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Abstract

This study examines the spatial nature of optimal bioinvasion control. We develop a spatially explicit two-dimensional model of species spread that allows for differential control across space and time, and we solve for optimal spatial–dynamic control strategies. The qualitative nature of optimal strategies depends in interesting ways on aspects of landscape and invasion geometry. For example, reducing the extent of exposed invasion edge, through spread, removal, or strategically employing landscape features, can be optimal because it reduces long-term containment costs. Optimal invasion control is spatially and temporally “forward-looking” in the sense that strategies should be targeted to slow or prevent the spread of an invasion in the direction of greatest potential long-term damages. These spatially explicit characterizations of optimal policies contribute insights and intuition to the largely nonspatial literature on controlling invasions and to understanding control of spatial–dynamic processes in general.

Introduction

Much of the economic research on bioinvasion management frames the issue as a pest control problem, in which the population density of the invader is controlled. This literature has generally focused on the aggregate pest population, without consideration of its spatial characteristics [1], [2], [3]. But a critical feature of invasion problems is that they unfold over time and space and are thus driven by spatial–dynamic processes, rather than by simpler dynamic processes. Existing analytical work generally abstracts away from the spatial features of invasions, focusing on when and how much to control [4], [5], [6]. There is less understanding about where to optimally allocate control efforts or the effect of spatial characteristics of the invasion or landscape on optimal control choices.

This paper develops a bioeconomic model of bioinvasions that incorporates a spatial–dynamic spread process and that allows various aspects of space to be characterized explicitly. We examine optimal policies over a range of bioeconomic parameters, spatial configurations, and initial invasion types. The more interesting results show how the geometry of the initial invasion and landscape influences the qualitative characteristics of optimal policies. Optimal solutions often utilize landscape features or alter the shape of the initial invasion in order to reduce the length of exposed invasion front, thereby reducing long term control costs. Optimal policies also exhibit classic forward-looking behavior that not only anticipates impacts over time, but also looks forward over space to slow and steer the invasion front away from the direction of greatest potential damages, or in the direction where the costs of achieving control are low.

Section snippets

Related literature

In its most general form, a spatial spread process may be characterized with a partial differential equation (PDE) over continuous time and space. There is a paucity of literature in economics, mathematics or optimization theory on characteristics of optimally controlled PDE-based state equation systems. The most general is the elegant work by Brock and Xepapadeas [7], [8] who derive modified Pontryagin conditions for the optimal control of a renewable resource governed by continuous PDE state

A spatial–dynamic model of bioinvasions

We develop and solve a spatially explicit, deterministic, discrete space–time model that allows for growth and spread of a species and differential control over both time and space. We focus on the situation in which an invasion has arrived, established itself, and been discovered within the focal landscape. Upon discovery, the initial invasion has some arbitrary character (e.g., size, shape, location) that may depend upon seeds having been introduced by animals (e.g., birds), wind, or other

Results

Optimal control strategies for invasions vary dramatically across invasion, landscape, and economic characteristics, ranging from no control to complete eradication depending upon parameters. Between these two extremes, optimal policies include: eradication of part of the invasion and containment or abandonment of the rest, immediate complete containment, partial containment that allows some spread prior to complete containment, partial containment followed by abandonment of control efforts,

Synthesis and discussion

The novel parts of our findings are those that explore the manner in which the topology of an invasion and the landscape determine the optimal policy, in addition to basic economic factors. Invasions that are identical in relative size can have dramatically different optimal control policies if they differ in shape and location. While this appears to militate against deriving simple rules of thumb, we are able to synthesize the intuition behind many results.

Conclusions

This paper has two purposes. The first is to provide understanding of economically optimal spatial control of bioinvasions. Optimal solutions for spatially explicit optimization problems generate a far richer set of solution characteristics than work that treats space only implicitly. In addition to the control principles we have derived, our approach could be applied to specific invasion problems to guide on the ground management. Data requirements include estimates of expected damages from

Acknowledgments

The authors gratefully acknowledge NSF-funded Biological Invasions IGERT (NSF DGE 0114432 PI Strauss) and USDA's PREISM program (58-7000-7-0088 PI Wilen) for financial support and four anonymous reviewers for their comments and suggestions.

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