Abstract

Citation

Researchers should cite this work as follows:

• Burghes, D. (2023). 1975-David_Burghes-Population dynamics An introduction to differential equations. SIMIODE, QUBES Educational Resources. doi:10.25334/ZWYM-ZZ38

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Burghes, David N. 1975. Population dynamics An introduction to differential equations. International Journal of Mathematical Education in Science and Technology. 6(3): 265-276.

See https://www.tandfonline.com/doi/abs/10.1080/0020739750060302 . Accessed 19 March 2023.

Articles from this journal are FREEly available to members of the Mathematical Association of America through their member portal www.maa.org.

Abstract:  In this paper a number of population models, which lead to differential equations, are derived. First-order variables separable equations are formulated from the Malthusian population model and its extension to include crowding effects. Age structure effects are shown to lead to differential equations with a time lag and the dynamics of exploited fish populations are briefly examined. Two models of interacting species are examined, predictor-prey and competing species, both of which lead to simultaneous coupled non-linear differential equations but with solutions which have vastly differing characteristics.

From the text,

“Many differential equation courses tend to be rather dull and tedious. Students suffer them willingly because they usually end with a ' cookery book ' type of examination where marks are easy to pick up. Interest can be revived by a thorough investigation of the properties of the solutions obtained, and brighter students will be inspired by a close investigation of particular pitfalls and fallacies that can occur. But the ' average ' student will only enjoy differential equations if he has been sufficiently motivated towards them. I stress the word enjoy because all mathematics should be enjoyed, and I think that it is important that one teaches the appreciation of mathematics, no matter what topic is under study.

“In recent years I have sensed a growing frustration amongst students taking applied mathematics courses (including ' methods' courses). They desire the mathematics to be applied to real problems, and not just to rather abstract mechanical problems. Of course differential equations can be motivated through mechanics, but so much mechanics as taught in schools and colleges has little meaning or application in the real physical world. Also, many students now meet differential equations before mechanics. Where else can we look for motivation?

“Differential equations are used extensively in biology, economics, chemistry and the social sciences. Population dynamics, for example, can be fruitfully used to motivate many types of differential equations and has advantages in that population models are relatively easy to formulate, and the topic itself is currently topical. In the following sections I will outline a number of varied population models, which can be used to motivate differential equations ranging from simple first-order to phase plane stability.”

Straightforward coverage is offered of exponential and logistic growth with food limitations and delays, exploited fish populations, and two species interaction predator-prey, addressing issues of stability.

Keywords: population, model, differential equation, delay, lag, fish