300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. The solution curves are .
Given a
family of curves, assumed to be
differentiable, an isocline for that family is formed by the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of points at which some member of the family attains a given
slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
. The word comes from the
Greek words
ἴσος (isos), meaning "same", and the
κλίνειν (klenein), meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the
union of a small number of curves.
Isoclines are often used as a graphical method of solving
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
. In an equation of the form ''y' = f''(''x'', ''y''), the isoclines are lines in the (''x'', ''y'') plane obtained by setting ''f''(''x'', ''y'') equal to a constant. This gives a series of lines (for different constants) along which the solution curves have the same gradient. By calculating this gradient for each isocline, the
slope field can be visualised; making it relatively easy to sketch approximate solution curves; as in fig. 1.
Other uses
In
population dynamics, the term "zero-growth isocline" refers to the set of population sizes at which the rate of change for one population in a pair of interacting populations is zero.
However, this is rare and a more common term is
nullcline.
References
{{Reflist
*Hanski, I. (1999) Metapopulation Ecology. Oxford University Press, Oxford, pp. 43–46.
Mathworld: Isocline
Ordinary differential equations