To switch or taper off: the dynamics of saturation

Many situations in population biology involve a rate – typically a contact or recruitment rate – which increases linearly for small populations but reaches a maximum value (saturates) for large populations. Models for populations of variable size need to incorporate both characteristics to predict behavior accurately. This can be done by defining the rate as a continuous, piecewise linear function with a switch point, or via a Verhulst-type (smooth) saturation function. This paper presents several examples of both approaches and draws some conclusions about the differences from a modeling perspective.