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2 Model A
2.1 Definition
We consider square lattice
L × L
, with periodic boundary conditions. On each
cell of the lattice there may be a predator (e.g. a wolf), a prey (e.g. a rabbit),
both of them, or it may be empty. It is forbidden for two or more animals of
the same species to occupy the same cell at the same time. The animals are
moving from one cell to another following the rules specific for a given strategy.
A part of the cells serves as a shelter for rabbits against wolves. On such cells
(
rabbit-holes
) a wolf cannot eat a rabbit. The remaining cells are covered with
grass which is eaten by rabbits. If a wolf and a rabbit are at a same time at
the same grassy cell, the wolf eats the rabbit. The distribution of rabbit-holes
is random and remains unchanged in time. The grass is always available for the
rabbits on grassy cells.
Each animal has to feed at least once every
k
Monte Carlo steps (MCS). This
is realized in our model by attributing to each animal a counter containing
k
“food rations”. The counter is increased by one after each ”meal” (eating grass
by a rabbit or eating rabbit by a wolf) and decreased after completing one MCS.
For simplicity we assume the same value of
k
for predators and prey. An animal
which did not eat for
k
MCS dies.
If an animal has at least one nearest neighbor of the same species, the pair
produce, at most,
M
offspring.
M
is the physiological birth rate. In order to
breed, apart from finding a partner in its neighborhood, the animal must be
strongenough,i.e.itmusthaveatleast
k
min
foodrations.Eachoffspringreceives
at birth
k
of
food rations. The offspring are located, using the blind ant rule (i.e.
only one search is made for an empty cell for each progeny), within the Moore
neighborhood [8], which in the case of the square lattice contains 8 cells. If the
density is high, there is room for only a fraction of
M
, hence less progeny is born.
This procedure takes care of the unrealistic unlimited growth of a population
found in the classic Lotka-Volterra models and replaces, in a natural way, the
phenomenological Verhulst factor.
If an animal moves into a cell with a rabbit-hole - nothing happens. A rabbit
cannot neither breed nor eat, but it cannot also be eaten by a wolf.
Weshallinvestigatetheroleofthefivestrategies(fouractiveandonepassive)
for the animals.
1. Passive strategy. It is simply no strategy at all, denoted in the following as
NO
. This is the case when the animals move randomly through the lattice.
2. Food for wolves (
W
). A wolf moves into a cell on which there is a rabbit but
not a rabbit-hole. If there is more than one such cell, or there is none, the
choice is made at random. The rabbits are moving randomly.
3. Food for rabbits (
R
). A rabbit moves into a cell on which there is grass (with
or without a wolf). As above, if there is more than one such site the choice
is random. Wolves have no strategy.
4. Food for both species (
RW
). Wolves and rabbits search for their food. Si-
multaneous application of the rules 2 and 3.
