Editing Theoretical ecology (section)

===Structured population growth===
{{See also|Matrix population models}}

Another assumption of the exponential growth model is that all individuals within a population are identical and have the same probabilities of surviving and of reproducing.  This is not a valid assumption for species with complex life histories.  The exponential growth model can be modified to account for this, by tracking the number of individuals in different age classes (e.g. one-, two-, and three-year-olds) or different stage classes (juveniles, sub-adults, and adults) separately, and allowing individuals in each group to have their own survival and reproduction rates.
The general form of this model is
:<math>\mathbf{N}_{t+1} = \mathbf{L}\mathbf{N}_t</math>
where '''N<sub>t</sub>''' is a [[Euclidean vector|vector]] of the number of individuals in each class at time ''t'' and '''L''' is a [[Matrix (mathematics)|matrix]] that contains the survival probability and fecundity for each class.  The matrix '''L''' is referred to as the [[Leslie matrix]] for [[Age class structure|age-structured]] models, and as the Lefkovitch matrix for [[Ontogeny|stage-structured]] models.<ref>{{cite book |author=Hal Caswell |title= Matrix Population Models: Construction, Analysis, and Interpretation |year= 2001 |publisher=Sinauer}}</ref>

If parameter values in '''L''' are estimated from demographic data on a specific population, a structured model can then be used to predict whether this population is expected to grow or decline in the long-term, and what the expected [[population pyramid|age distribution]] within the population will be.  This has been done for a number of species including [[loggerhead sea turtle]]s and [[right whale]]s.<ref>{{cite journal |author=D.T.Crouse, L.B. Crowder, H.Caswell |year=1987 |title=A stage-based population model for loggerhead sea turtles and implications for conservation  |journal=Ecology |pages=1412–1423 |doi=10.2307/1939225 |volume=68 |issue=5|jstor=1939225 |bibcode=1987Ecol...68.1412C |s2cid=16608658 }}</ref><ref>{{cite journal |author1=M. Fujiwara |author2=H. Caswell |year = 2001 |title = Demography of the endangered North Atlantic right whale  |journal = Nature  |volume = 414 |issue = 6863 |pages = 537–541 |doi=10.1038/35107054 |pmid = 11734852 |bibcode = 2001Natur.414..537F|s2cid=4407832 }}</ref>