Abstract
This short chapter presents the mathematics of matrix population models. The first section examines discrete linear systems using scalar notation and computer simulations. The simulations lead to the discovery of an asymptotic growth rate and stage structure, which we can determine by ad hoc methods for systems of only two or three components. The second section presents the matrix algebra that is needed as background for a more mathematical study of discrete linear systems. The two threads are united in the final section, which develops the ideas of eigenvalues and eigenvectors, methods for finding them, and interpretations for population models. The problem sets include population viability case studies for endangered falcons, cheetahs, and loggerhead turtles as well as population models for aphids and the teasel plant.
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Notes
- 1.
Of course there could be “growth” at a rate less than 1 for some models.
- 2.
In fact, much of the fostering and general population increase was due to the efforts of falconers, who then felt that their investment of time and money should be rewarded by being permitted to harvest baby falcons for their sport. This case serves as an example of the difficult issues involved in trying to create conservation policies that satisfy diverse interests.
- 3.
The work presented in the paper focuses on the reliability of population projections when parameter values are uncertain, which is beyond the scope of what we can do here. Nevertheless, our methods allow us to obtain population growth rate predictions for a variety of circumstances and to determine critical parameter values for population viability.
- 4.
This software was written when the author was co-teaching an interdisciplinary research course. The research involved population dynamics of aphids and coccinellids (ladybird beetles). Boxbug biology combines the biology of these two real insect types.
- 5.
The advantage of the formalism of vectors will only become apparent after we have defined arithmetic operations that allow for vector calculations to faithfully reproduce the corresponding scalar calculations.
- 6.
The reason for wanting to do this will become clear in Section 6.3.
- 7.
The reason for this interest will become clear in Section 6.3.
- 8.
Lightly copying the first two columns of the 3 ×3 determinant to the right of the matrix, giving the appearance of a 3 ×5 matrix, will help you to see this.
- 9.
These patterns do NOT hold in higher dimensional determinants. The reader who wants to work with higher dimensional matrices should consult a linear algebra book for a complete definition of the determinant.
- 10.
Note that what we have done is essentially equivalent to the part of Example 6.1.5 that followed the calculation of λ. The whole procedure will be made systematic in Section 6.3.
- 11.
The pronunciation is “eye-gen-value,” with a hard g as in “get.”
- 12.
See Section 6.2.
- 13.
This won’t work if you choose a scalar unknown whose value needs to be 0, but you can try again with that scalar unknown set to 0 instead. This never happens in stage-structured population models, because the stable structure of a viable population must have positive numbers for each stage. The corresponding mathematical property is guaranteed by the Perron-Frobenius theorem.
- 14.
These distinctions are largely arbitrary. Since size is a continuous variable, we could just as easily use two or four classes of rosettes. There are integral projection models that deal with continuous size structure and discrete time, but these models are far beyond the scope of this book. They are also impractical, unless there is an enormous amount of data on the effect of rosette size on the future of the plant.
- 15.
This life history is common among aphid species.
- 16.
Pea aphids are born pregnant, so the amount of time required for a newborn aphid to become a reproductive adult can be as little as 8 days.
- 17.
Experiments corresponding to this scenario consistently show the behavior predicted by the model.
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Ledder, G. (2013). Discrete Dynamical Systems. In: Mathematics for the Life Sciences. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7276-6_6
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