Paying Our Dues: The Role of Professional Societies in the Evolution of Mathematical Biology Education.

Mathematical biology education provides key foundational underpinnings for the scholarly work of mathematical biology. Professional societies support such education efforts via funding, public speaking opportunities, Web presence, publishing, workshops, prizes, opportunities to discuss curriculum design, and support of mentorship and other means of sustained communication among communities of scholars. Such programs have been critical to the broad expansion of the range and visibility of research and educational activities in mathematical biology. We review these efforts, past and present, across multiple societies-the Society for Mathematical Biology (SMB), the Symposium on Biomathematics and Ecology Education and Research (BEER), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM). We then proceed to suggest ways that professional societies can serve as advocates and community builders for mathematical biologists at all levels, noting that education continues throughout a career and also emphasizing the value of educating new generations of students. Our suggestions include collecting and disseminating data related to biomath education; developing and maintaining mentoring systems and research communities; and providing incentives and visibility for educational efforts within mathematical biology.

Model selection for integrated pest management with stochasticity.

In Song and Xiang (2006), an integrated pest management model with periodically varying climatic conditions was introduced. In order to address a wider range of environmental effects, the authors here have embarked upon a series of studies resulting in a more flexible modeling approach. In Akman et al. (2013), the impact of randomly changing environmental conditions is examined by incorporating stochasticity into the birth pulse of the prey species. In Akman et al. (2014), the authors introduce a class of models via a mixture of two birth-pulse terms and determined conditions for the global and local asymptotic stability of the pest eradication solution. With this work, the authors unify the stochastic and mixture model components to create further flexibility in modeling the impacts of random environmental changes on an integrated pest management system. In particular, we first determine the conditions under which solutions of our deterministic mixture model are permanent. We then analyze the stochastic model to find the optimal value of the mixing parameter that minimizes the variance in the efficacy of the pesticide. Additionally, we perform a sensitivity analysis to show that the corresponding pesticide efficacy determined by this optimization technique is indeed robust. Through numerical simulations we show that permanence can be preserved in our stochastic model. Our study of the stochastic version of the model indicates that our results on the deterministic model provide informative conclusions about the behavior of the stochastic model.

On impulsive integrated pest management models with stochastic effects.

We extend existing impulsive differential equation models for integrated pest management (IPM) by including stage structure for both predator and prey as well as by adding stochastic elements in the birth rate of the prey. Based on our model, we propose an approach that incorporates various competing stochastic components. This approach enables us to select a model with optimally determined weights for maximum accuracy and precision in parameter estimation. This is significant in the case of IPM because the proposed model accommodates varying unknown environmental and climatic conditions, which affect the resources needed for pest eradication.

Integrated pest management with stochastic birth rate for prey species.

Song and Xiang (2006) developed an impulsive differential equations model for a two-prey one-predator model with stage structure for the predator. They demonstrate the conditions on the impulsive period for which a globally asymptotically stable pest-eradication periodic solution exists, as well as conditions on the impulsive period for which the prey species is permanently maintained under an economically acceptable threshold. We extend their model by including stage structure for both predator and prey as well as by adding stochastic elements in the birth rate of the prey. As in Song and Xiang (2006), we find the conditions under which a globally asymptotically stable pest eradication periodic solution exists. In addition, we numerically show the relationship between the stochastically varying birth rate of the prey and the necessary efficacy of the pesticide for which the probability of eradication of the prey species is above 90%. This is significant because the model recognizes varying environmental and climatic conditions which affect the resources needed for pest eradication.