Isocline

 ( Ordinary Differential Equations ) 

Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains a given slope. The word comes from the Greek language 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 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.

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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.

• Hanski, I. (1999) Metapopulation Ecology. Oxford University Press, Oxford, pp. 43–46.
• Mathworld: Isocline

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