This model demonstrates the relationship between Aphids and Lady beetles, and analyzes the impact insecticides have on that system.

__ Model Variables: __

x_{0}: Number of Aphids before the simulation

y_{0} : Number of Lady beetles before the simulation

t: current time

t_{a} : time at which insecticide was added.

z : insecticide application.

__ Constants: __

K: Carrying capacity of Aphids

C: Carrying capacity scalar of Lady beetles

R_{1}: Intrinsic growth rate of Aphids

R_{2}: Intrinsic growth rate of Lady beetles

G: Scales the insecticide's rate of decay

D_{1}: Intrinsic mortality rate - Lady beetles [in the absence of Aphids]

D_{2}: Intrinsic mortality rate - Aphids [in the absence of Lady beetles]

A_{1}: Rate at which insecticides impact Aphids

A_{2}: Rate at which insecticides impact Lady beetles

The original first order, non linear differential equations, Lotka - Volterra equations are often used to help explain dynamics of biological systems. The equations are as follows:

$$ \frac{dx}{dt}= R_1x - D_1xy$$
$$ \frac{dy}{dt}= R_2yx - D_2y$$

In our model, we modified the Lotka-Volterra equations to incorporate the impact insecticides would have, if introduced in a natural biological habitat consisting of Aphids and Lady beetles. The model assumes a natural setting and makes simple assumptions about species growth, mortality, and predation. Modified equations are as follows:

$$ \frac{dx}{dt}= R_1 x (1 - \frac {x}{K_1}) - B_1xy - A_1xz $$
$$ \frac{dy}{dt}= R_2y(\frac {D_1x} {(C+x)}) - D_2y - A_1A_2yz $$
\[ \begin{cases}
\ &
\ if t\leq t_a & , z = e ^ {G(t_a - t)}\\
\ &
\ else & , z = 0 \\
\end{cases}
\]