Lotka Volterra

The Lotka Volterra equations are a pair of first order differential equations that model predator prey populations. Without a predator, we would expect the prey population to grow exponentially. The increasing prey population means lots of food for our predator population, leading to a positive growth rate. At some point, the predator food demand surpasses the available resources, leading to a rapid fall in prey population. Due to the lack of prey, the predators starve to death and their population decreases as well. This concludes a cycle. The equations can easily be adapted for a model with multiple predators, which will produce to more complex periodic solutions. As you can see, the model is heavily simplified, in reality there are many other limiting factors, like food and space. Nonetheless, the Lotka Volterra model produces useful results that can for example be observed in nature and market economies.

Lotka Volterra equations

The parameters describe natural birth and deathrates of both species. Slight modifications of the Lotka Volterra ODEs can also be used to model how viruses spread when hosts get immunity for some time after an infection. Since immune individuals aren't available as hosts, this can be treated as "prey dying". The longer the immunity lasts, the longer the "birth rate". If you want to learn more, here's the wikipedia article.