Editing Theoretical ecology (section)

==Population ecology==
{{main|Population ecology}}
'''Population ecology''' is a sub-field of [[ecology]] that deals with the dynamics of [[species]] [[population]]s and how these populations interact with the [[natural environment|environment]].<ref name="Odum1959">{{cite book | last =Odum | first =Eugene P. | author-link =Eugene Odum | title =Fundamentals of Ecology | url =https://archive.org/details/fundamentalsofec0000odum/page/546 | url-access =registration | edition =Second | publisher =W. B. Saunders Co. | year =1959 | location =Philadelphia and London | isbn =9780721669410 | oclc =554879 | page =[https://archive.org/details/fundamentalsofec0000odum/page/546 546 p] }}</ref> It is the study of how the [[population size]]s of species living together in groups change over time and space, and was one of the first aspects of ecology to be studied and modelled mathematically.

===Exponential growth===
{{Main|Exponential growth}}

The most basic way of modeling population dynamics is to assume that the rate of growth of a population depends only upon the population size at that time and the per capita growth rate of the organism. In other words, if the number of individuals in a population at a time t, is N(t), then the rate of population growth is given by:
:<math> \frac{dN(t)}{dt}=rN(t) </math>
where r is the per capita growth rate, or the intrinsic growth rate of the organism.  It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This [[Linear differential equation#First order equation|first order]] [[linear differential equation]] can be solved to yield the solution
:<math> N(t) = N(0) \ e^{rt} </math>,
a trajectory known as [[Malthusian growth]], after [[Thomas Malthus]], who first described its dynamics in 1798. A population experiencing Malthusian growth follows an exponential curve, where N(0) is the initial population size. The population grows when r > 0, and declines when r < 0. The model is most applicable in cases where a few organisms have begun a colony and are rapidly growing without any limitations or restrictions impeding their growth (e.g. bacteria inoculated in rich media).

===Logistic growth===
{{Main|Logistic growth}}

The exponential growth model makes a number of assumptions, many of which often do not hold. For example, many factors affect the intrinsic growth rate and is often not time-invariant. A simple modification of the exponential growth is to assume that the intrinsic growth rate varies with population size. This is reasonable: the larger the population size, the fewer resources available, which can result in a lower birth rate and higher death rate. Hence, we can replace the time-invariant r with r’(t) = (b –a*N(t)) – (d + c*N(t)), where a and c are constants that modulate birth and death rates in a population dependent manner (e.g. [[intraspecific competition]]). Both a and c will depend on other environmental factors which, we can for now, assume to be constant in this approximated model. The differential equation is now:<ref name=Moss>Moss R, Watson A and Ollason J (1982) [https://books.google.com/books?id=l9YOAAAAQAAJ&dq=%22Logistic+growth%22&pg=PA52 ''Animal population dynamics''] Springer, page 52–54. {{ISBN|978-0-412-22240-5}}.</ref>
:<math> \frac{dN(t)}{dt}=((b-aN(t))-(d-cN(t)))N(t) </math>
This can be rewritten as:<ref name=Moss />
:<math> \frac{dN(t)}{dt}=rN(t) \left(1-\frac{N}{K}\right) </math>
where r = b-d and K = (b-d)/(a+c).

The biological significance of K becomes apparent when stabilities of the equilibria of the system are considered. The constant K is the [[carrying capacity]] of the population.  The equilibria of the system are N = 0 and N = K. If the system is linearized, it can be seen that N = 0 is an unstable equilibrium while K is a stable equilibrium.<ref name=Moss />

===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>