Pareto-efficient biological pest control enable high efficacy at small costs
Introduction
Pests are major concerns in agriculture. Local outbreaks cause financial losses and regional outbreaks threaten the food security of entire populations. This is of particular concern in developing nations where agriculture constitutes a larger share of the economy but in which agricultural practices have not yet reached the same technical and procedural standards as in developed nations. In India, for example, the “Army worm” Spodoptera litura (Fabr.) has defoliated many economically important crops including cotton, sunflower, and soybean (Dhaliwal et al., 2010). Farmers have traditionally resorted to pesticides to prevent and mitigate pest outbreaks, but their use may have unwanted consequences including insect resistance, resurgence, outbreak of secondary pests, and pesticide residues affecting human health and the environment. Indeed, heavy usage of synthetic pesticides has been linked to pest resistance, pest resurgence, health risks from exposure, and food contamination (Khooharo et al., 2008, Yadav, 2010).
Biological pest control is an alternative to chemical pest control in which naturally occurring enemies such as predators, parasitoids, or pathogens rather than pesticides are used to control the pests. The use of naturally occurring enemies to suppress insect pests has several advantages over chemical pest control, in particular safety for farmers, consumers, and non-targeted organisms. Biological pest control can potentially be efficacious at low cost and should not normally pose any danger for either farmers or consumers. They can be host-specific, they preserve natural enemies, and they may beneficially impact biodiversity (Lacey et al., 2001). Unlike the use of pesticides, there is little consensus on how to apply biological control for maximal efficiency. One reason for this is the complex interplay of non-linear interactions between the crop, the pest, and the natural enemy. The potential benefits of improved biological pest-control strategies are particularly large for inundative and augmentative applications, in which large numbers of natural enemies are released, as the timing of the release may significantly affect the total cost and efficacy.
To the authors knowledge, a handful of studies have explored design of biological pest-control strategies from the perspective of mathematical analysis and/or optimal control theory. These studies have considered problems of bioeconomic equilibrium, demographic stability, and optimal-release strategies (Getz and Gutierrez, 1982, Grasman et al., 2001, Bhattacharyya and Bhattacharya, 2006, Rafikov et al., 1993, Cardoso et al., 2009). While these studies have furthered our understanding of biological pest control, the proposed pest-control strategies may not easily be communicated to agriculture professionals as they typically lack a regular pattern and sometimes require continuous release of natural enemies. Moreover, with Cardoso et al. (2009) as an important exception, only single-objective optimization is usually considered. Finally, to the authors knowledge, the studies to date have not explicitly modeled the crop, which as a third dynamic state variable could potentially impact the results. Developing simple but efficient rules for biological pest control in agricultural systems with crop–pest–enemy interactions thus remains an important challenge from both a theoretical and applied perspective.
In this paper, we suggest a simple method for developing strategies for biological pest control that are easy to apply, efficacious, and simultaneously near optimal in terms of profit. The strategies are Pareto efficient in that they optimally trade off between profit and efficacy. We demonstrate our method on a dynamic model of the Spodoptera litura worm defoliating soybean crops while being controlled by a natural enemy, the Spodoptera polyhedrosis virus (Cherry et al., 1997, Fuxa, 2004). Specifically, we investigate one-off control strategies and periodic-control strategies. Using our measures of efficacy and profit, we find Pareto-efficient one-off and periodic control strategies that are close to optimal in the sense of profit and simultaneously not sensitive to perturbations. We show that one-off control strategies are preferable when immigration of pests is relatively low to intermediate. We also show that, for high immigration rates, one-off control can be replaced by simple periodic controls to achieve even better results.
Section snippets
Model
In this section, we first present the sample model on which we will demonstrate our method for deriving simple control strategies. This model consist of a pest–pathogen–crop system in which the pest is controlled biologically through the release of individuals that are infected with a virus. The infection spreads into the susceptible pest population and thus control the growth of pest biomass in the field. Second, we give basic results on the dynamics of the model considering equilibria and
Control strategies
Next, we introduce and give a precise definition of the control strategies that we consider for the release of infected individuals. We chose our class of control strategies for conceptual simplicity, though as we will show these strategies are capable of achieving near-optimal profits. Before describing the control strategies, we briefly note that the sample model in (2.1)–(2.3) considers only the number of pests as well as biomass of crop, no spatial dependence is involved. Consequently, we
Dual-objective approach
We here define our measures of profit and efficacy, after which we describe the concept of Pareto efficiency used for dual-objective optimization.
Besides trying to optimize profit we also consider maximizing efficacy, i.e. minimizing sensitivity to perturbations on the profit. We will now define our profit function, our measure of efficacy, i.e. half-biomass time, and also recall the economic concept of Pareto efficiency, which we will use to trade-off between the two objectives profit and
Results
Having introduced our modeling framework, we now demonstrate how to find preferable control strategies. First, we conclude that only a small reduction in profit allows for efficacious control strategies. Second, we show that one-off control strategies are sufficient when immigration rates of susceptible pests are low to intermediate, while periodic control strategies are recommended for high immigration rates. Finally, we conclude that the determined control strategies are not far from optimal
Discussion
We have considered biological control of agricultural pests. Using a dual-objective approach and the economic concept of Pareto efficiency, we have determined one-off and periodic-control strategies that are stable to perturbations and simultaneously nearly optimal in terms of profit. Our optimization approach as well as our measure of efficacy are general and can be applied to effectively any crop–pest–pathogen system. Depending on the immigration rate of pests from nearby fields, we recommend
Acknowledgements
The work of HZ was supported by the Swedish Research Council, the National Natural Science Foundation of China, Grant ID 11201187 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars and the China Scholarship Council. The authors thank Daniel Simpson for useful discussions and comments.
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