Q. Assertion : A population growing in a habitat with limited resources shows initially a lag phase, followed by phases of acceleration and deceleration and finally an **asymptote**, when the population density reaches the carrying capacity. Reason : In **Verhulst-Pearl Logistic growth**, plot of N (population density) at time (t) results in a **sigmoid** curve. From Wikipedia: "A **sigmoid** **function** is a bounded, differentiable, real **function** that is defined for all real input values and has a non-negative derivative at each point." In plainer English, **sigmoid** **functions** are 'S' shaped, smooth, and remain finite for all values of c. A primary example of a **sigmoid** **function** is called the Logistic **Function**: f.

Such **functions** are written in the form f(x - h), where h represents the horizontal shift. The numbers in this **function** do the opposite of what they look like they should do. For example, if you have the equation g ( x ) = ( x - 3) 2 , the graph of f(x)=x 2 gets moved to the right three units; in h ( x ) = ( x + 2) 2 , the graph of f(x)=x 2.

Properties. In general, a **sigmoid** **function** is real -valued and differentiable, having either a non-negative or non-positive first derivative which is bell shaped. There are also a pair of horizontal **asymptotes** as . The logistic **functions** are sigmoidal and are characterized as the solutions of the differential equation [1].