1,604 research outputs found
Self-organized patterns of coexistence out of a predator-prey cellular automaton
We present a stochastic approach to modeling the dynamics of coexistence of
prey and predator populations. It is assumed that the space of coexistence is
explicitly subdivided in a grid of cells. Each cell can be occupied by only one
individual of each species or can be empty. The system evolves in time
according to a probabilistic cellular automaton composed by a set of local
rules which describe interactions between species individuals and mimic the
process of birth, death and predation. By performing computational simulations,
we found that, depending on the values of the parameters of the model, the
following states can be reached: a prey absorbing state and active states of
two types. In one of them both species coexist in a stationary regime with
population densities constant in time. The other kind of active state is
characterized by local coupled time oscillations of prey and predator
populations. We focus on the self-organized structures arising from
spatio-temporal dynamics of the coexistence. We identify distinct spatial
patterns of prey and predators and verify that they are intimally connected to
the time coexistence behavior of the species. The occurrence of a prey
percolating cluster on the spatial patterns of the active states is also
examined.Comment: 19 pages, 11 figure
Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos
We develop a formalism to describe the discrete-time dynamics of systems
containing an arbitrary number of interacting species. The individual-based
model, which forms our starting point, is described by a Markov chain, which in
the limit of large system sizes is shown to be very well-approximated by a
Fokker-Planck-like equation, or equivalently by a set of stochastic difference
equations. This formalism is applied to the specific case of two species: one
predator species and its prey species. Quasi-cycles --- stochastic cycles
sustained and amplified by the demographic noise --- previously found in
continuous-time predator-prey models are shown to exist, and their behavior
predicted from a linear noise analysis is shown to be in very good agreement
with simulations. The effects of the noise on other attractors in the
corresponding deterministic map, such as periodic cycles, quasiperiodicity and
chaos, are also investigated.Comment: 21 pages, 12 figure
Non-standard discretization of biological models
We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volume-preserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan's method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of Lotka-Volterra type
Hungry Volterra equation, multi boson KP hierarchy and Two Matrix Models
We consider the hungry Volterra hierarchy from the view point of the multi
boson KP hierarchy. We construct the hungry Volterra equation as the
B\"{a}cklund transformations (BT) which are not the ordinary ones. We call them
``fractional '' BT. We also study the relations between the (discrete time)
hungry Volterra equation and two matrix models. From this point of view we
study the reduction from (discrete time) 2d Toda lattice to the (discrete time)
hungry Volterra equation.Comment: 13 pages, LaTe
On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols
In this work we focus on a natural class of population protocols whose
dynamics are modelled by the discrete version of Lotka-Volterra equations. In
such protocols, when an agent of type (species) interacts with an agent
of type (species) with as the initiator, then 's type becomes
with probability . In such an interaction, we think of as the
predator, as the prey, and the type of the prey is either converted to that
of the predator or stays as is. Such protocols capture the dynamics of some
opinion spreading models and generalize the well-known Rock-Paper-Scissors
discrete dynamics. We consider the pairwise interactions among agents that are
scheduled uniformly at random. We start by considering the convergence time and
show that any Lotka-Volterra-type protocol on an -agent population converges
to some absorbing state in time polynomial in , w.h.p., when any pair of
agents is allowed to interact. By contrast, when the interaction graph is a
star, even the Rock-Paper-Scissors protocol requires exponential time to
converge. We then study threshold effects exhibited by Lotka-Volterra-type
protocols with 3 and more species under interactions between any pair of
agents. We start by presenting a simple 4-type protocol in which the
probability difference of reaching the two possible absorbing states is
strongly amplified by the ratio of the initial populations of the two other
types, which are transient, but "control" convergence. We then prove that the
Rock-Paper-Scissors protocol reaches each of its three possible absorbing
states with almost equal probability, starting from any configuration
satisfying some sub-linear lower bound on the initial size of each species.
That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a
distributed system. Some of our techniques may be of independent value
Mass Energy and Flow in closed ecosystems
The general equations of biomass and energy transfer for an n-species, closed ecosystem are written. It is demonstrated how in "ecological time" the parameters describing the dynamics of biomass transfer are related to the parameters of energy transfer, such as respiration, fixation, and energy content. This relationship is determinate for the straight-chain ecosystem, and a simple example is worked out. The results show how the density dependent terms in population dynamics arise naturally, and how the stable
system exhibits a hierarchy in energy per unit biomass. A procedure is proposed for extending the theory to include webbed systems, and the particular difficulties involved in the extension are brought before the scientific community for discussion
Applying allometric scaling to predator-prey systems
In population dynamics, mathematical models often contain too many parameters
to be easily testable. A way to reliably estimate parameters for a broad range
of systems would help us obtain clearer predictions from theory. In this paper,
we examine how the allometric scaling of a number of biological quantities with
animal mass may be useful to parameterise population dynamical models. Using
this allometric scaling, we make predictions about the ratio of prey to
predators in real ecosystems, and we attempt to estimate the length of animal
population cycles as a function of mass. Our analytical and numerical results
turn out to compare reasonably to data from a number of ecosystems. This paves
the way for a wider usage of allometric scaling to simplify mathematical models
in population dynamics and make testable predictions.Comment: 9 pages, 3 figure
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