Rev.Acad.Canar.Cienc ., III (Num . 1) , 75 - 85 (1991)
LOGISTIC - STOCHASTIC GROWTH AND PATCHY DISTRIBUTIONS
IN THE SEA
Joa~ M. Pacheco Caatelao and ceaar Rodriguez Mielgo
Facultad de Cienciaa del Mar
Universidad de Las Palmaa de Gran Canaria
35017 Las Pal maa, ESPA~A
Abstrac t Patchiness is the name given to a
heterogeneous, unequal, spatial distribution of many
populations. A simple mathematical model to explain the
temporal evolution of patch sizes is built. The model
introduces random parameters in a basic differential
equation which rules the logistic growth of patch
sizes thus obtaining a stochastic differential equation
whose associated Fokker - Planck equation is solved
afterwards .
1. INTRODUCTION
Logistic growth is one of the moat important tools
in the modelling of various problems in Ecology.
Roughly speaking, it is a variant of the classical
malthuai an growth law y"= ky where k is dependent on
the population size y and on some limiting factor. In
general , this factor represents the maximum
that can survive on the available resources,
population
although
short periods can happen where
greater than this maximum.
small-scale phenomena can modify
the population is
On the other. hand,
the logistic growth
path in such a way to render it difficult to recognize.
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The contribution of these must be added to the logistic
pattern, in order to obtain an equation where both
large and emall-ecale factors are represented.Thie can
be achieved by way of a stochastic differential
equation, whose solution is the probabilistic
distribution of the population eize rather than ite
actual eize.
2. PATCH SIZES IN THE OCEAN.
The study of patches of several substances or
living beings is of foremost importance in the field of
marine sciences, where estimation of patch sizes and of
their spreading mechanisms is an active research field.
When one deals with patches whose constituents are
passive, logistic growth is a rather adequate modelling
for the estimation of patch sizes. First of all, we
shall suppose that patches are elongated bodies whose
size can be described by the diameter L(t), where time
dependence shows the variability of L. Second, the
size of the patch depends on how energy is fed into it.
Two sources are available: a) large or medium-ecale
energy-containing eddies characteristic of the ocean
zone where the patch appears. b) small-scale eddies
responsible for very small variations in the patch
size. The scale of these is so small with respect to
the energy -containing eddies that they can be
considered as noise. In any definite ocean area one
can find a · typical scale for the energy-containing
eddies. It is natural to think that whenever the size
of a patch is greater than thi.s typical scale, the
patch will break into smaller patches. Thus, the
typical scale i~ a limiting factor for the patch size.
Now we write E for this typical scale and find the
logistic expression for L(t):
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dL
dt
= L A( t- -r ) L (1)
where >.. models features of the ocean ambient. A typical
interpretation for >.. is the stress tensor given by the
velocity gradient within the eddies. Now we turn to
small-scale energy transfers. If the scale is very far
from the typical scale, we find that fluctuations are
much faster than those due to the general logistic
pattern. Thus we model them as
ddLt = μ(t) L (2)
where exponential growth is prevented by the
pattern of μ(t). This μ(t) is thought of
changing
as a
stochastic process; thus equation (2) is a stochastic
differential equation. Physical considerations allow us
to write the expression
μ(t) = ~ tp(t) (3)
where tp(t) is white noise. The parameter k models the
intensity of randomness and can be interpreted in
various ways. One of them is the effect of shear
stresses and of molecular viscosity. By adding
equations (2) and (3) we find a simple model for patch
size in the ocean:
-5!!::._ = >.. ( t - _!::._ ) L + ( 2k)v2 4'( t) L ( 4)
dt E
This is a Langevin-type stochastic differential
equation whose solution is some stochastic process
L(t), instead of a deterministic function.
3. THE FOKKER-PLANCK EQUATION FOR PATCH SIZE.
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Equation (4) ia usually interpreted aa shorthand
for the physically formulated difference equation:
M = X. ( t - ~ ) L At + .;--;;;- L AW( t )
where AW(t) is the increment of the Wiener process.
Thie process ( alao called Brownian 'motion )ia defined
aa the stochastic Ito integral of white noise.
Equations (4) and (5) can be uaed to ahow that the
solution process L(t) is a Markov process.
In effect, we aee that the formal solution to
the Langevin equation (4), with the initial condition
L(O)=O is:
.;--;;;- J~ a( L( s» ds f ~ </J( s) ds
L(t) = e e e
from which we obtain
_ft+At a( L) ds t+At4>(' s) ds
L( t+At) = L( t) e e
Since </>( t) ia 6-correlated, its value a in the interval
[t,t+At] are independent of the previous history and
L(t+At) doea not depend on the history preceding L(t>.
Therefore the process L<t> is Markovian.
Homogeneity of the proceaa can be aaaumed on
physical grounds. Owing to the inertial ayatem that
prevails in the oceans, the joint probability denaitiee
depend only on the time difference between
obaervationa. In thia way, time homogeneity expreaaes
the invariance of the mechanism which generates
fluctuations.
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Thus, a Fokker-Planck equation (FPE) can be
formulated for our process, whose drift and diffusion
terms are calculated in the standard way:
iJ p(L. t )
iJ t
iJ [ L ] iJ
2
= - -- A.(t- -)L p + k --(L 2 p)
iJL E iJL2
The solution of this FPE is simply the probability
distribution p(L,t) of finding a size Lat time t. As
it is a continuous distribution, it must be interpreted
as the probability of finding L at time t between some
fixed values .
Now we proceed to find a solution of the FPE. It
is reasonable to think that the steady state is
natural in normal conditions, under the assumption that
the environmental eddies are in statistical
equilibrium within the tidal period.
The FPE (5) can be written as
iJ Dt p( L. t) = L fp p ( L. t) ( 6)
where
iJ
Lfp = - -ar:- A(L) + 8( L)
and A( L ) and B(L) are time-independent drift and
diffusion coefficients respectively.
Now equation (6) may be written in the form
iJp iJJ
-at+~=O (7)
where
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(5)
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J( L • t ) = [ A( L) - :L B( L) ] p( L • t ) (8)
Because (7) is a continuity equation for a probability
distribution, J may be interpreted as a probability
current, i.e. probability flow.
Therefore, we concentrate on solving the stationary
FPE with the supplementary normalization:
.r: p( L>dL = t
4. THE GAMMA DISTRIBUTION.
Solving the stationary FPE, we
foilowing results. For a stationary
probability current must be a constant.
the stochastic variable L cannot reach
obtain the
procces the
Nevertheless,
values smaller
than zero, so we require that the probability current
be zero at L 0. Thus, the probability current
vanishes for all L. Setting J = 0, we rewrite equation
(7) as
A( L) P,} L) = a"l 8( L) P~/ L)
for which the solution is obtained by a single integration
as
A(S)
B(S) dS]
where N is a normalisation constant such that
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We then obtain
A. A. L t -;;;- - --;;:£
p ( L> = N L e
s kL 2
with
N
A. = A.
(~) ~ - 2
r(~ - t ) E k A.
Finally, we write the expression for stationary p(L):
p (L)
s
t = r<b>
t
a
where we have introduced
H-r- t
a = Ek
-A.-- and
e
L
a
A.
b = ~ - t
This happens to be a gamma distribution
depending on two parameters: E and b. b models the
relative incidence of large and small-scale eddies in
the spreading of the patch. When b is large the
energy-containing eddies dominate, and if b is small,
the more chaotic behaviour of the small-scale eddies
prevents the patch from attaining a size similar to E.
This is shown in the accompanying graphs, where the
evolution of p(L> according to E and .b is represented.
It is interesting to note that the larger b is, the
closer is the mode distribution to the typical scale£.
5. ACKNOWLEDGEMENTS.
Both authors received financial support from grant
n° 81/02.06.87 of the Gobierno Aut6nomo Canario
Canary Islands Government )
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8)
C)
Figure 1: The Ga11111a distributions for b=5 and A) E=200, 8) E=500, C) E=lOOO.
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A>
8)
C>
Figure 2: The Gamma distributions for b=lO and A> E=200, 8) E=500, C) E=lOOO.
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A)
8)
Figure 3: The GalilTla distributions fo1 b=25 and A) E=200, 8) E=500, C) E=lOOO.
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6. REFERENCES AND BIBLIOGRAPHY.
[1] COX,D., MILLER, H. (1965), The theory of stochastic
processes. Chapman Hall, London.
[2] GARDINER, C. (1983) Handbook of stochastic methods.
Springer-Verlag, Berlin.
[3] HEATH, M., McLACHLAN, P. (1987) Dispersion and
mortality of yolk-sac herring larvae. J. PLankton Res.
9' ( 613-630) .
[4] MARK.ATOS, N. (1986) The mathematical modelling of
turbulent flows. AppL. Hath. HodeU .• to, (190-220)
[5] MAY, R.M. (1975) Biological population obeying
difference equations. J. Theor. BioLo~.51, (511-524).
[6] OKUBO, A. ( 1982) Critical patch size for plankton
and patchiness. Leet. N. Biomath .• 54, (456-477).
[7] RISKEN, H. (1984) The Fokker-PLanck equation.
Springer-Verlag, Berlin.
[ 8 J RODRIGUEZ, C. ( 1988) Tes is Doctoral, Facul tad de
Ciencias del Mar, Universidad de Las Palmas.
Recibido : 20 de Marzo de 1991
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