Sunday, February 5, 2012

Predator-Prey Networks as Nature's Food Supply Chains -- Where Economics, Operations Research, and Ecology All Meet

I have always loved economics and operations research but as an undergrad at Brown University almost majored in Applied Mathematics -- Biology. Hence, I have also taken a lot of science courses, including courses in ecology and botany, but until recently had never done research in ecological modeling.

I became interested in this area of research due to a truly original paper, entitled "NEATS: A Network Economics Approach to Trophic Systems," published in the journal Ecological Modelling, co-authored by a group of researchers based in France: Mullon, Shin, and Cury. The paper applies some of the results in my Network Economics: A Variational Inequality Approach book to formulate and determine equilibria in predator-prey complex webs. One reads regularly about the impact of science on economics but this paper demonstrates how economics and, especially, network economics, can be used to combine both biological constraints that couple biomass balance equations with complementarity principles using Walras' law. The authors investigate the solutions to simple food chains, bilayer networks, complex food webs, and even to cannibalism (the links loop back to the specific nodes in such networks)!

I read the paper and I was hooked. I also began a correspondence with Christian Mullon, the first author of the above paper.

I was intrigued by the possible relationships between ecological predator-prey networks in nature and supply chains. If we can rigorously establish the connections, we should be able to learn how to design more resilient and robust supply chain networks.

In a recent paper, Dynamics and Equilibria of Ecological Predator-Prey Networks as Nature’s Supply Chains, that I co-authored with the "other" Professor Nagurney, we were able to establish the equivalence between predator-prey ecological networks (think of food chains, for example, as in fisheries, to start) and multitiered supply chains consisting of "agents" who are manufacturers, retailers, or consumers, respectively. What I found fascinating is that in "product" or, shall I say, "corporate" supply chains, one explicitly considers various decision-makers' objective functions, which tend to include the maximization of profits, at least for the manufacturers and the retailers. Moreover, the various decision-makers "compete." In predator-prey ecological networks, on the other hand, the competition (for food and survival) is clear, but, until now, no-one really quantified prices or value in that context.

In our paper, we showed, using a dynamic model of predator prey interactions, that the stationary points or equilibria coincide precisely with those of the equilibria in supply chain networks! The general supply chain model that we used to show this equivalence was the supply chain network equilibrium model that I developed with Professors June Dong and Ding Zhang and which was published in Transportation Research E in 2002. That model has served as the foundation for numerous extensions; for just a few, click here.

The dynamics that we utilized was that of projected dynamical systems, since in operations research, as well as in economics, we need to be able to deal with constraints.

Hence, amazingly, predator-prey interactions have an underlying economics, whereas supply chain networks, in a sense, are ecological predator-prey systems.

Through operations research and network economics, we have built bridges to ecology.

Our paper, Dynamics and Equilibria of Ecological Predator-Prey Networks as Nature’s Supply Chains, was recently published in Transportation Research E. It expanded on our earlier work that showed the equivalence between bipartite predator-prey networks and classical spatial price equilibrium problems going back to the work of the Nobel Laureate, Paul Samuelson. That paper, "Spatial Price Equilibrium and Food Webs: The Economics of Predator-Prey Networks," was published in the Proceedings of the 2011 IEEE International Conference on Supernetworks and System Management, F. Y. Xu and J. Dong, Editors, IEEE Press, Beijing, China, pp 1-6.

I had the chance to meet Dr. Mullon when I was in Paris last October to give a plenary talk on supply chains at the NetGCoop conference. He and his wife treated me to a delicious lunch and we walked for miles.

I am delighted that Mullon and I have now collaborated and the result is a paper on game theory and big fish versus small fish, which combines some of my work in variational inequalities and human migration with punctuated equilibria and fish behavior and ecology. Appropriately, it has been submitted to an ecology journal.

For those of you who enjoy economics and operations research as much as I do, do read Professor Fred Murphy's wonderful essay. It first appeared in 1995 and has been updated (he graciously provided me with the latest version this past week) and published in the centennial edition of the Encyclopedia of Operations Research and Management Science, edited by Saul Gass and Michael Fu. And Fred cited my Network Economics book (thanks!) as well as the spatial price equilibrium paper of Samuelson noted above and quite a few of my other favorite citations.