resource management, consumer-resource model, predator-prey model, phase line, whale populations, bifurcation
Mathematics | Physical Sciences and Mathematics | Population Biology | Science and Mathematics Education
Observed whale dynamics show drastic historical population declines, some of which have not been reversed in spite of restrictions on harvesting. This phenomenon is not explained by traditional predator prey models, but we can do better by using models that incorporate more sophisticated assumptions about consumer-resource interaction. To that end, we derive the Holling type 3 consumption rate model and use it in a one-variable differential equation obtained by treating the predator population in a predator-prey model as a parameter rather than a dynamic variable. The resulting model produces dynamics in which low and high consumption levels lead to single high and low-level stable resource equilibria, respectively, while intermediate consumption levels result in both high and low stable equilibria. The phase line analysis is made more transparent by applying a particular structure to the function that gives the derivative in terms of the state. By positing a consumption level that starts low, gradually increases through technological change and human population growth, and decreases as a result of public policy, we are able to tell a story that explains the unexpectedly rapid decline of some resources, such as whales, followed by limited recovery in response to conservation. The analysis also offers guidelines for how to establish sustainable harvesting for restored populations. We include a bifurcation analysis and suggestions for how to teach the material with three different levels of focus on the modeling aspect of the study.
"Qualitative Analysis of a Resource Management Model and Its Application to the Past and Future of Endangered Whale Populations,"
Vol. 14, Article 3.
Available at: https://scholarship.claremont.edu/codee/vol14/iss1/3
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