A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Otto Sarah P. & Day Troy

Author:Otto, Sarah P. & Day, Troy [Otto, Sarah P.]
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2011-09-19T00:00:00+00:00

Model (9.25) contains six parameters: r, the intrinsic growth rate of the prey when at low abundance; K, the carrying capacity of the prey; c, the contact rate of predator with prey; a, the probability that a predator consumes a prey, given it makes a contact; ε, the conversion coefficient of captured prey into new predators; and μ, the per capita mortality rate of the predator. To use the proportional transformation technique, begin by defining the new variables N* = N/, P* = P/ as well as the new measurement of time t* = t/, where , , and are constants (to be chosen shortly).

We now derive an equation for the dynamics of N* with respect to the new time variable t*, i.e., dN*/dt*. Applying the chain rule of calculus twice (Appendix 2, A2.14), we get

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