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A lot of activities have been devoted during the past years to the study of
extended prey-predator models (see [5,6] and references therein). It was shown
that the introduction of stochastic dynamics plays an important role [7], as
well as the use of discrete variables, which prevent the population to become
vanishingly small. These ingredients are included in the so called individual
agents based lattice models and it is now recognized that these models give a
better description of population dynamics than simple mean-field like models.
Both synchronous [8] and asynchronous dynamics may be be considered.
Generally speaking, there are two goals in studying the dynamics of the
predator-prey systems. One is the explanation of the possible oscillations in
the temporal evolution of the densities of prey and predators, as well as of the
correlations between them. This is the classic problem in the field [9,7,10]. The
second problem is the derivation and discussion of the long time, steady states
at which a predator-prey system finally arrives.
In this contribution we shall discuss two differents models of population dy-
namics, illustrating the importance of considering spatially extended systems.
The first one [11] (Model A), is a lattice model of evolution of a predator -
prey system. Both species in order to survive must eat, at least once during a
given time period. The prey eats the grass growing in unlimited quantities on
most of the cells of the lattice. On the remaining cells the prey find cover against
predators, but not food. We compare four active strategies - the prey move in
the direction of grassy cells, they move away from the predators, the predators
move in the direction of the prey and both species move in the direction of their
respectivefoodresources.Thesestrategiesarecomparedwithrandomwandering
of both species. We show that the character of the asymptotic stationary state
dependson theinitialconcentrationsofpredatorsand prey, density of the shelter
for the prey and on the strategy adopted. The predators are more vulnerable to
the proper choice of the strategy. The strategies which lead in most cases to the
extinction of predators and prey are those in which the predators are searching
the prey.
Thesecondmodel[12](ModelB)isanindividual-basedmodelofapopulation
which lives in a changing environment. The individuals forming the population
are subject to mutations and selection pressure. Using Monte Carlo simulations
we have shown that, depending on the values of the mutation rate and selection,
the population may reach either an active phase (it will survive) or an absorbing
phase (it will become extinct). We have determined that the transition between
the two states (phases) is continuous. We have shown that when the selection is
weakerthepopulationlivesinallavailablespace,whileiftheselectionisstronger,
it will move to the regions where the living conditions are better, avoiding those
with more di
?
cult conditions. The dependence of the mean time to extinction
on the rate of mutations has been determined and discussed.
In the following sections, we shall define the models and discuss their prop-
erties.
