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

An isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions. 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, 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.


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

Mathworld: Isocline