The Lotka-Volterra equations
\begin{equation}
\begin{split}
x'&=a \,x-b \,xy \\
y'&=d \,xy-c\,y
\end{split}\label{eq1}
\end{equation}
also known as the predator-prey equations, are a pair of first-order, non-linear, 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. In this equations we have that:
$x$ and $y$ are the respective populations of the prey and predator species;
$x'$ and $y'$ represent the instantaneous growth rates of the two populations;
The constants $a, b, c$ and $d$ are positive real parameters describing the interaction of the two species. These constants are based on empirical observations and depend on the
particular species being studied.
Simulation 1
The following simulation demonstrates the solutions to the Lotka-Volterra equations for the values $a=0.1,$ $b=0.002,$ $c = 0.2$ and $d = 0.0025$.
Things to try:
Change the initial conditions $x_0$, $y_0$.
Drag sliders and then click the Update parameters button to make changes.
Explore the populations of each specie, at a particular time, by dragging the slider Time
Sorry, the simulation is not supported for small screens.
The Lotka-Volterra equations (\ref{eq1}) are unrealistic because they do not include the effect of limited resources on the food supply of the prey. Also, in the modern environment, prey are often culled or harvested. A more realistic system includes two extra terms:
\begin{equation}
\begin{split}
x' &= x(a - cx -dy) \\
y' & = -y(b - ex) - h
\end{split}\label{eq2}
\end{equation}
where all of $x, y, a, b, c, d, e, h$ are positive, and $a/c>b/e.$ In this model we have that
$x$ represents the number of prey
$y$ represents the number of predators.
$a$ is the growth rate of the prey.
$b$ is the death rate of the predators independent of the prey.
$d$ is the is the rate of consumption of the prey per predator.
$a / c$ is the carrying capacity of the prey independent of the predators.
$e$ is the growth rate of the predator per prey consumed,
$h$ is prey harvesting.
Simulation 2
The following simulation demonstrates the solutions to equations (\ref{eq2}) for the fixed values $a=1,$ $b=0.25,$ $c = 0.01,$ $d = 0.02$ and $e=0.02$.
Things to try:
Change the initial conditions $x_0$, $y_0$.
Change the value of harvesting $h$.
Explore the populations of each specie, at a particular time, by dragging the Slider (right-side screen).
Sorry, the simulation is not supported for small screens.
W. E. Boyce and R. C. DiPrima. Elementary Differential Equations and Boundary Value Problems, 9th ed. New York: Wiley, 2009.
A. J. Lotka. Elements of physical biology. Williams and Wilkins, Baltimore, 1925.
V. Volterra. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Memoria della Reale Accademia Nazionale dei Lincei, 2(1926) 31-113
The University of Queensland