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.

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**

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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.

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).

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The following file contains activities and problems associated with the simulation 2.

- 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