13th International Conference on Membrane Computing - MTA Sztaki

P. Ramón, A. Troina
Fig. 1. Metapopulation dynamics.
describes a selective pressure driving populations evolution through the logistic
model [47]:
dN
(1
dt = r · N · − N )
K
where N represents the number of individuals in the population.
CWC Modelling 3 (Logistic Model) The logistic model with growth rate r
and carrying capacity K, for an environment modelled by a compartment with
label l, can be encoded within CWC using two stochastic rewrite rules describing
(i) a reproduction event for a single individual at the given rate and (ii) a
death event modelled by a fight between two individuals at a rate that is inversely
proportional to the carrying capacity:
r
l : a ↦−→ a a
2·r
K−1
l : a a ↦−→ a
If N is the number of individuals of species a, the number of possible reactants
for the first rule is N and the number of possible reactants for the second rule
is, in the exact stochastic model, ( )
N
2 =
N·(N−1)
2
, i.e. the number of distinct
pairs of individuals of species a. Multiplying this values by the respective rates
we get the propensities of the two rules and can compute the value of N when
the equilibrium is reached (i.e., when the propensities of the two rules are equal):
r · N = 2·r
K−1 · N·(N−1)
2
, that is when N = 0 or N = K.
For a given species, this model allows to describe different growth rates and
carrying capacities in different ecological regions. Identifying a CWC compartment
type (through its label) with an ecological region, we can define rules
describing the growth rate and carrying capacity for each region of interest.
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