Introducing herbivores into Daisyworld

 

In order to reflect fundamental ecosystems dynamics based on trophic interactions as well, we extend the 2D model even further and add herbivores to our toy planet (see also, e.g., Harding and Lovelock [15]). These vegetarian ``lattice animals'' move on the grid as random walkers. As in reality, the herbivores are capable of reproduction, but the latter capacity depends heavily on their nutritional state.

Formally, the herbivore dynamics can be conveniently described by an occupation map , defined on a second lattice layer parallel to the daisy lattice, and a small set of non-deterministic behavioural rules. If the site is occupied by an animal at time , i.e., , then those rules allow for four different events:

Death:

The herbivore deceases with a probability , thus .

Ingestion:

If the herbivore survives and if the cell is covered by a daisy, i.e. , then the animal consumes the plant. The number of ingestion acts that have been performed by the herbivore considered so far is stored as a state variable N. Thus

in this case. Evidently, we also have and .

Reproduction:

The herbivore in question may give birth to a ``lattice child'' at a randomly chosen next-neighbour cell with probability

The temperature-dependent factor is chosen similar to the one which governs the growth of daisies, while is assumed to behave like a discrete Heaviside function, i.e.,

The latter assumption reflects the conception that only ``grown-up'' lattice animals will be capable of replication. Clearly, the ingestive status of the new-born herbivore is zero.

Move:

The herbivore may also walk to a randomly chosen next-neighbour cell if this cell is not already occupied by another herbivore.

As a matter of fact, consistent combinations of all those events may occur according to the probabilistic set-up. Note that the presence of herbivores is not assumed to change the local albedo. So it is only the act of daisy consumption that affects the radiative properties of our virtual environment.

Note that the interaction between plants and animals is realized by mainly two processes, namely (i) temperature-dependent reproduction of herbivores and (ii) ingestion of daisies. Note also that our herbivores are rather dumb as their dietary strategy consists in random walking only.