Model Variables:

x0: Number of Aphids before the simulation
y0 : Number of Lady beetles before the simulation
t: current time
ta : time at which insecticide was added.
z : insecticide application.

Constants:

K: Carrying capacity of Aphids
C: Carrying capacity scalar of Lady beetles
R1: Intrinsic growth rate of Aphids
R2: Intrinsic growth rate of Lady beetles
G: Scales the insecticide's rate of decay
D1: Intrinsic mortality rate - Lady beetles [in the absence of Aphids]
D2: Intrinsic mortality rate - Aphids [in the absence of Lady beetles]
A1: Rate at which insecticides impact Aphids
A2: 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:

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