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Dynamics of Populations in Extended Systems
Michel Droz
1
and Andrzej Pekalski
2
1
Insitut de Physique Theorique, Universite de Geneve,
quai E. Ansermet 24, 1211 Geneve 4, Switzerland,
Michel.Droz@physics.unige.ch
,
http://theory.physics.unige.ch/˜droz
2
Institute of Theoretical Physics, University of Wroclaw,
pl. M. Borna 9, 50-204 Wroclaw, Poland,
apekal@ift.uni.wroc.pl
Abstract.
Two models of spatially extended population dynamics are
investigated. Model A describes a lattice model of evolution of a predator
- prey system. We compare four different strategies involving the prob-
lems of food resources, existence of cover against predators and birth.
Properties of the steady states reached by the predator-prey system are
analyzed. Model B concerns an individual-based model of a population
which lives in a changing environment. The individuals forming the pop-
ulation are subject to mutations and selection pressure. We show that,
depending on the values of the mutation rate and selection, the popula-
tion may reach either an active phase (it will survive) or an absorbing
phase (it will become extinct). The dependence of the mean time to
extinction on the rate of mutations will also be discussed. These two
problems illustrate the fact that cellular automata or Monte-Carlo simu-
lations, which take completely the spatial fluctuations into account, are
very useful tools to study population dynamics.
1
Introduction
The dynamics of interacting species has attracted a lot of attention since the
pioneering works of Lotka [1] and Volterra [2]. In their independent studies, they
showed that simple prey-predator models may exhibit limit cycles during which
the populations of both species have periodic oscillations in time. However, this
behavior depends strongly on the initial state, and is not robust to the addition
of more general non-linearities or to the presence of more than two interacting
species [3]. In many cases the system reaches a simple steady-state.
In such mean-field type models, it is assumed that the populations evolve
homogeneously, which is obviously an oversimplification. An important question
inmodelingpopulationdynamicsconsistsinunderstandingtheroleplayedbythe
local environment on the dynamics [4]. There are many examples in equilibrium
and nonequilibrium statistical physics showing that, in low enough dimensions,
the local aspects (fluctuations) play a crucial role and have some dramatic effects
on the dynamics of the system.
