Question

Populations of aphids and ladybugs are modeled by the equations dtdA​=3A−0.01ALdtdL​=−0.4L+0.001AL​ (a). Find the equilibrium solutions. Enter your answer as a list of ordered pairs (A,L), where A is the number of aphids and L the number of ladybugs. For example, if you found three equilibrium solutions, one with 100 aphids and 10 ladybugs, one with 200 aphids and 20 ladybugs, and one with 300 aphids and 30 ladybugs, you would enter (100,10),(200,20),(300,30). Do not round fractional answers to the nearest integer. Answer = (b). Find an expression for dL/dA. dAdL​=

Answer 1

The equilibrium solutions are: (0, 300), (400,0).

An expression for dL/dA is dL/dA = 0.001L - 0.4

What is an expression?

At equilibrium dA/dt = 0 and dL/dt = 0 and solve for A and L

3A - 0.01AL = 0 -----------------1

-0.4L + 0.001AL = 0-------------2

3A - 0.01AL = 0

A(3 - 0.01L) = 0

This gives two solutions: A = 0 or

L = 300/1

-0.4L + 0.001AL = 0

L(-0.4 + 0.001A) = 0

This gives two solutions: L = 0 or A = 400

So, the equilibrium solutions are:

(0, 300), (400,0)

find an expression for dL/dA, take the derivative of the second equation with respect to A

dL/dA = d/dA(-0.4L + 0.001AL)

Now, differentiate each term:

dL/dA = 0.001L - 0.4

So, dL/dA = 0.001L - 0.4

An expression for dL/dA is dL/dA = 0.001L - 0.4