Lotka Martini

by Dirk Brockmann

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 activator-inhibitor 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 Lotka-Volterra System, a paradigmatic model for predator prey dynamics.

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This is how it works

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

  1. empty, denoted by \(\emptyset\)
  2. predator, denoted by \(Y\) and
  3. 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 prey-specific death rate and vacate its site. So we have the two reactions:

\[X+\emptyset\rightarrow 2X \qquad X\rightarrow \emptyset\]

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 predator-specific death rate. This is summarized in the reactions

\[2Y+X\rightarrow 3Y \qquad Y\rightarrow \emptyset\]

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.

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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.

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.