Question

Explain the Lotka-Volterra model: Show your work.

1. The basic Assumption
2. The competition dynamics for both:
   1. dN_B/dt = r_BN_B(K_B-N_B- a_BGN_G)/K_B
   2. dN_G/dt = r_GN_G(K_G-N_G- a_GBN_B)/K_G

Answer 1

The Lotka-Volterra competition model describes the outcome of competition between two species over ecological time. Because one species can competitively exclude another species in ecological time, the competitively-inferior species may increase the range of food types that it eats in order to survive.

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy,\\{\frac {dy}{dt}}&=\delta xy-\gamma y,\end{aligned}}}{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy,\\{\frac {dy}{dt}}&=\delta xy-\gamma y,\end{aligned}}}
where

x is the number of prey (for example, rabbits);
y is the number of some predator (for example, foxes);
{\displaystyle {\tfrac {dy}{dt}}}{\tfrac {dy}{dt}} and {\displaystyle {\tfrac {dx}{dt}}}{\tfrac {dx}{dt}} represent the instantaneous growth rates of the two populations;
t represents time;
α, β, γ, δ are positive real parameters describing the interaction of the two species.
The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.