Editing Theoretical ecology (section)

===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 />