Competitive Lotka Volterra model analyzing insecticide impact on Aphids and Lady beetles


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



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:

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