300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. Given a family of curves, assumed to be

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

, 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 is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...

. The word comes from the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

words ἴσος (isos), meaning "same", and the κλίνειν, 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 whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

. 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 Slope fields (also called direction fields) are a graphical representation of the solutions to a first-order differential equation of a scalar function. Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope ...

can be visualised; making it relatively easy to sketch approximate solution curves; as in fig. 1.

Other uses

population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a ...

, 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