Mathematics, Department of


Date of this Version



Deng, B. Acta Biotheor (2018) 66: 213.



The Canadian lynx and snowshoe hare pelt data by the Hudson’s Bay Company do not fit the classical predator-prey theory. Rather than following the peak density of the hares, the peak density of the lynx leads it, creating the hares-eat-lynx (HEL) paradox. Although trappers were suspected to be a cause, no mathematical models in the literature have demonstrated the HEL effect. In this paper we consider various multitrophic models with interactions in vegetation-hare-lynx, hare-lynx-coyote, hare-lynx-trapper, and hare-lynx-coyote-trapper, and use Newton’s gradient search method to best fit each model to the data and then select the one with the least error as the benchmark model for the data. We will conclude from the benchmark model and its sensitivity analysis the following: (a) the trappers as a participant rather than an observer of the system changed the observed; (b) the lynx and hare populations in the wild follow the lynx-eats-hares (LEH) cycle but the harvested animals follow the HEL cycle; (c) the benchmark fit is more sensitive to changes in all lynx-trapper interactions than the respective hare-trapper interactions; (d) trappers did not interfere each other’s trapping activities a century ago; (e) the Hudson’s Bay Company’s hare pelt number was severely fewer than it should be. These results together dispel a long-held hypothesis that the pelt data is a proxy of the hare and lynx populations in the wild. It also shows that theoretical ecology must move beyond the classical Lotka-Volterra paradigm into a contemporary framework in which Holling’s predation theory is central to population modeling. It also demonstrates to the ecologists that we do not have to collect data from all dimensions in order to gain a good understanding of complex systems, but instead we can systematically fit models to empirical data of any dimensions with identifiable uncertainties and a complete understanding of parameter sensitivities to the best fit.