Abstract
This chapter introduces the reader to the basic ideas underlying ordinary differential equations. Knowledge of the basic rules of differentiation and integration is assumed; however, one of the main objectives of the chapter is to convey to the biologist reader, in an intuitive way, the basic idea of infinitesimal change and differentiation. The second main aim of this chapter is to provide an introduction to ordinary differential equations. The emphasis of this chapter is on usability. By the end of the chapter the reader will be able to formulate basic, but practically useful, differential equations; have a grasp of some basic concepts, including stability and steady states); and will also have an understanding of some basic methods to solve them. Furthermore, the reader will be able to critically evaluate differential equation models she may encounter in the research literature. The more general theoretical introduction into this topic is accompanied by fully worked case studies, including a differential equation model of the spread of malaria and a stability analysis of Cherry and Adler’s bistable switch.
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- 1.
The reader who is not familiar with the rules of differentiation, is encouraged to consult Appendix A.1 for a refreshment of the rules.
- 2.
Of course there are indirect ways to measure the curvature of the Earth. Based on triangulation the ancient Greeks already knew that the Earth is (at least approximately) round.
- 3.
Incidentally, while we have an intuitive understanding for the fact that bacterial colonies cannot sustain exponential growth for a very long time, economic growth follows a similar law. Suppose a country experienced a trend growth rate of about 2 % per year. Year after year the 2 % rate means a higher absolute growth, because the economy of which we take the 2 % has grown from the year before. Hence, (at least to a first approximation) the economy grows according to the differential (4.17), a law that we have just seen to be unsustainable for bacteria because of resource depletion.
References
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Casti, J.: Reality Rules: II The Frontier. Wiley, New York (1992)
Cherry, J., Adler, F.: How to make a biological switch. J. Theor. Biol. 203(2), 117–133 (2000). doi:10.1006/jtbi.2000.1068
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Barnes, D.J., Chu, D. (2015). Differential Equations. In: Guide to Simulation and Modeling for Biosciences. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-6762-4_4
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DOI: https://doi.org/10.1007/978-1-4471-6762-4_4
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