Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 233

and eigenvectors for the matrix. If you would like to see all the eigenvalues and
eigenvectors for your matrix, use the command

eigen (A)

Exercise 7.4. Calculate the dominant eigenvalue and the dominant right eigenvector
for the transition matrix you generated in Exercise 7.1 , and compare it with the values
from the example above. How do the two populations differ in their growth rate? How
do they differ in their stable distribution across stages?

7.10 FURTHER APPLICATIONS OF THE PROJECTION
MATRIX
In this chapter, we have introduced you to the basic structure and theory behind using
projection matrices in estimating population growth and structure. Biologists and
mathematicians have gone much further in exploring methods to account for natu-
ral variability in populations. Sensitivity analysis uses partial derivatives of matrix
elements to determine the relative impact of each element on

λ

and therefore exam-
ines the importance of each life stage on overall population growth [ 1 ]. Computer
simulations are used to generate multiple estimates of growth rates under different
environmental states, providing a better understanding of how robust a population is
to environmental change [ 8 ]. Models can also incorporate spatial data on patches or
multiple populations, providing information on the relative contribution of each patch
or population to overall growth [ 8 ]. The flexibility and power of the projection matrix
in population growth models has led to many important discoveries, contributions,
and practical applications in ecology and conservation biology. We encourage you to
explore the topic further through the literature cited here, or better yet, collect your
own data from a wild population and apply the tools you learned here to predict its
future growth.

References
[1] Caswell H. Matrix population models: construction, analysis, and interpretation.
2nd ed. Sunderland, MA: Sinauer Associates; 2001.
[2] Burns

JH, Blomberg SP, Crone EE, Ehrlen J, Knight TM, Pichan-

court J-B, et al. Empirical tests of life-history evolution theory

using phylogenetic analysis of plant demography. J Ecol 2010;98:

334-44.
[3] Steets JA, Knight TM, Ashman T-L. The interactive effects of herbivory and
mixed mating for the population dynamics of Impatiens capensis .AmNat
2007;170:113-27.