EXPLORABLES

# Lotka Martini

This explorable illustrates the dynamics of a ** predator-prey model** on a hexagonal lattice. In the model a prey species reproduces
spontaneously but is also food to the predator species. The predator requires the prey for reproduction. The system is an example of an

**system, in which two dynamical entities interact in such a way that the activator (in this case the prey) activates the inhibitor (the predator) that in turn down-regulates the activator in a feedback loop. Activator-inhibitor systems often exhibit oscillatory behavior, like the famous**

*activator-inhibitor***, a paradigmatic model for predator prey dynamics.**

*Lotka-Volterra System*First press the ** Play** button, watch and keep on reading ...

### This is how it works

Everything is happening on a hexagonal lattice. Each lattice site can be in one of three states:

**empty**, denoted by \(\emptyset\)**predator**, denoted by \(Y\) and**prey**, denoted by \(X\).

Prey dynamics is governed by simple reproduction and death. When a prey site and an empty site are adjacent the prey individual spontaneously reproduces at a ** prey-specific reproduction rate** and puts an offspring onto the empty site. Any prey individual can also die at a

**and vacate its site. So we have the two reactions:**

*prey-specific death rate*\[X+\emptyset\rightarrow 2X \qquad X\rightarrow \emptyset\]

Predator dynamics is a bit more complicated. Reproduction can occur when *two* adjacent predators have a common neighboring site that is prey. When this condition is met, the prey is replaced by a third *baby* predator at a ** predator-specific reproduction rate**. All the predators can also spontaneously die at a

**. This is summarized in the reactions**

*predator-specific death rate*\[2Y+X\rightarrow 3Y \qquad Y\rightarrow \emptyset\]

We thus have four reactions and four rate parameters. The initial setup is chose such that the prey's death rate is 0.

#### Reproduction steps for prey and predators

An isolated prey individual can spawn offspring to neighboring site. A pair of adjacent predators can *eat* prey in their neighborhood and replace it with their baby.

### Try this

For the predefined values of rates you can observe that the system approaches a dynamical quasi-equilibrium in which predators and prey coexist.

One of the key features of an activator-inhibitor system like this one is, that the interactions make the system more robust to parameter changes. For example, if you *increase the prey's reproduction rate* the overall fraction of prey doesn't change much, because the increased production of prey is followed by an increase in predators that regulate it.

#### Related Explorables

This particular predator prey model is related to the Explorable *Critical HexSIRSize* that describes contagion processes near eradication and the spatial Rock-Paper-Scissors game covered in *Cycledelic*.