Question

Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the expression 2 Dr , where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate v'(r), the derivative of the wave speed with respect to the reproductive rate r.

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The derivative is  $v(r)' = \sqrt{\frac{D}{r} }$

The correct option is option 1

Explanation:

From  the question we are told that

   The equation of the speed of the wave of invasion is  $v(r) = 2 \sqrt{Dr}$

=>  $v(r) = 2 (Dr)^{\frac{1}{2} }$

=>  $v(r) = 2 * D^{\frac{1}{2} } r^{\frac{1}{2} }$

    Here r is the reproductive rate and the D is the parameter qualifying dispersal

Generally the derivative of this speed is mathematically represented as

         $v(r)' = \frac{2}{2} * D^{\frac{1}{2} } * r^{-\frac{1}{2} }$

=>     $v(r)' = \sqrt{\frac{D}{r} }$  

This derivative of the speed represents the rate of change of the invasive speed with respect to the the reproductive rate of an individual