# Lotka-Volterra model

The Lotka-Volterra equations $$\begin{split} x'&=a \,x-b \,xy \\ y'&=d \,xy-c\,y \end{split}\label{eq1}$$ 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

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## Lotka-Volterra variation

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: $$\begin{split} x' &= x(a - cx -dy) \\ y' & = -y(b - ex) - h \end{split}\label{eq2}$$ 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).

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## Worksheet exemplar

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

## References

• 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