By Paul H. Rabinowitz
The papers during this quantity symbolize the complaints of the complicated Seminar on purposes of Bifurcation thought held in Madison on October 27-29, 1976. aside from the survey through M. G. Crandall, the papers are released within the order within which they're offered.
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Extra resources for Applications of bifurcation theory: proceedings of an advanced seminar
10 Plots of the logistics model of population growth. The left plot shows the per capita growth rate vs. population, and the right plot shows the growth rate vs. population. Both plots give important interpretations of the model. 11). ) But, there are qualitative properties of solutions that can be exposed without actually ﬁnding the solution. Often, all we may want are qualitative features of a model. 11), p(t) = 0 and p(t) = K, corresponding to no animals (extinction) and to the number of animals represented by the carrying capacity, respectively.
Scientiﬁc exactness is sometimes sacriﬁced for mathematical tractability. Model predictions depend strongly on the assumptions, and changing the assumptions changes the model. If some assumptions are less critical than others, we say the model is robust to those assumptions. The best strategy to learn modeling is to begin with simple examples and then graduate to more diﬃcult ones. The reader is already familiar with some models. In an elementary science or calculus course we learn that Newton’s second law, force equals mass times acceleration, governs mechanical systems such as falling bodies; Newton’s inverse-square law of gravitation describes the motion of the planets; Ohm’s law in circuit theory dictates the voltage drop across a resistor in terms of the current; or the law of mass action in chemistry describes how fast chemical reactions occur.
14) where populations are measured relative to the carrying capacity and time is measured relative to how fast the population is growing. For example, if the carrying capacity is K = 300, 000, and the dimensioned p varies between 0 < p < 300, 000, it is much simpler to have dimensionless populations P with 0 < P < 1. 14) it is easy to see that the equilibria are P = 0 and P = 1, the latter corresponding to the carrying capacity p = K. We have pointed out that an autonomous model can be easily analyzed qualitatively without ever ﬁnding the solution.
Applications of bifurcation theory: proceedings of an advanced seminar by Paul H. Rabinowitz