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Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y'=xy

An Isocline is a series of lines with the same slope. The word comes from the Greek words Isos (ισος) meaning "same" and Klisi (κλίση) meaning "slope".

It is 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.

In population dynamics refers to the set of population sizes at which the rate of change, or partial derivative, for one population in a pair of interacting populations is zero.


Hanski, I. (1999) Metapopulation Ecology. Oxford University Press, Oxford, pp. 43-46.

Mathworld: Isocline