Fractional Calculus and Dynamic Approach to Complexity Page: 39
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lower part of the cycle. Now using the fact that x(O) = x(T) we can combine the integrals
on the right-hand side of Eq. (92) to get the result
(93) Xeq = dxxpeq(x).
where the equilibrium pdf depends inversely on the velocity through
(94) peq(x) (=V1y) Vx y
+T x (x, y-)v(x, +
Using this ensemble averaging methodology, the equilibrium point reached by the fractional
trajectory can be understood intuitively through the equivalent subordination process. Imag-
ine an ensemble of ordinary Lotka-Volterra systems all identically prepared with the initial
condition V(O) =_(xo, yo) at time t = 0. Each system then progresses along the cycle, but
making irregular steps in time according to its own internal clock. The probability density
p(V, t) to be at the coordinate V = (x, y) at time t disperse from the initial delta function,
eventually covering all possible points on the cycle, as some systems step ahead while others
get stuck for extended periods. After a adequately long time an equilibrium pdf peq (V),
that is proportional to the inverse of the x and y velocities, is reached. The pdf for a single
subordinated trajectory to have a prey level x at four precise experimental times is shown
in Fig. 4.3. The initial delta function spreads to cover the entire range of possible x-values,
and tends to the equilibrium pdf peq (x). The first moment of the prey distribution in the
asymptotic time limit tends toward the fixed point as predicted by Eq. (93). An equivalent
analysis can be repeated for the predator population. The time required for this equilibrium
condition to be reached depends on the index a of the fractional derivative. For ca= 1 there
is no decorrelation between the equally alike systems, and all travel together along the cycle.
Therefore we can intuitively anticipate that for cv - 1 the time to reach equilibrium becomes
practically infinitely extended as the decorrelation among the systems is an extremely slow
process. Moving from a discrete operational time to a continuous time representation [1],
Eq. (60) becomes39
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Beig, Mirza Tanweer Ahmad. Fractional Calculus and Dynamic Approach to Complexity, dissertation, December 2015; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc822832/m1/48/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .