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Unformatted text preview: QABE Lecture 19 Differential Equations & Growth II School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Limited Growth 2 2.1 A Problem of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Limiting Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Seeing it Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Logistic Growth 4 3.1 A Problem at the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Logistic Growth Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Applied to Australian Population Data . . . . . . . . . . . . . . . . . . . . 6 4 Appendix: An Integration Workout 7 1 Introduction We introduced the concept of growth last lecture. However, despite all the interest that we can gain from the exponential growth story, the reality is that it has some not very pleasing properties. The most significant of which is its limit behaviour. As time goes to infinity, we find the exponential curve skyrocketing off any graph paper in the world, itself reaching infinite heights. Now while some may think that this is true that populations, stocks, money can grow and grow infinitely others beg to differ! The more realistic view would suggest that just like money doesnt grow on trees, there are limits to growth . Whether in the form of capacity constraints for manu facturing plants (we just cant get any more inputs out of the ground), or as inbuilt restraint mechanisms as in yeast growth (they literally begin to die in their own waste), things eventually reach a limit. So first up, we will study a limited growth equation, and see how it can be used. In the second case, we will round off one more sharp edge to get a possibly even more realistic growth equation. Of course, we could keep going and going, creating richer and richer models of growth, but for now, it will suce to cover the main three models. They are relatively simple, but despite any failings they still describe some situations very well. Thus they arebut despite any failings they still describe some situations very well....
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