EXPLORABLES

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*.