Download Coupled Logistic Map for Symbiotic Relations

Journal of the Korean Physical Society, Vol. 30, No. 1, February 1997, pp. 49∼53
Coupled Logistic Map for Symbiotic Relations
Gyu-Seung Shin
Department of Physics and Institute of Natural Sciences
Kyung Hee University, Yongin 449-701
(Received 11 July 1996, in final form 15 January 1997)
The symbiotic relations are formulated with a parametrically coupled logistic map. Three types
of interspecific relations (mutualism, competition and predation) are studied with maps with simple
sigmoidal relating functions. We investigate the stability condition and the basin of survival fixed
point for mutualism. It is shown that the degree of mutualism plays a key role for the stability of the
relation and the survival of the species. Also, for competition and predation, we obtain the phase
diagrams in which the parameter planes are divided into the regions to which the stable survival
point and limit cycles and the chaos correspond.
I. INTRODUCTION
These equations may seem to be much different from the
earlier models of the symbiotic relation; however, their
leading terms coincide with those of the earlier models
of mutualism or competition. Eq. (2) has a more restricted set of parameters (a, b). Also, x and y, the population variables of two species constitute a phase plane
{(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1} which is a two-dimensional
version of that of the original logistic map. Additionally, we must note that the functions µ(y) and ν(x) with
which we are dealing have sigmoidal forms which are
symmetric for the point (x, y) = (0.5, 0.5). The main
reasons that we take the functional forms of µ(y) and
ν(x) as in Eq. (2) are not the ecological realities but the
tractability which can be briefly stated as follows; (i)
they have only two parameters a, b, and (ii) from the
ecological point of view, the most important fixed point
is the stable survival one. With the symmetric relation
of Eq. (2), the point can be fixed at (0.5, 0.5) for any
values of the parameters; consequently, it can make successive considerations much simpler and can help our intuition for the symbiotic relations. We must admit that
the equations may be insufficient for general ecological
descriptions. However, we expect that Eq. (2) can show
us the main qualitative facets of symbiotic relations.
A logistic map which describes the population dynamics of a species would be the most famous nonlinear difference equation. The map would have dual significance
as an instructive example of the nonlinear dynamics and
as an ecological model [1]. In spite of its simple form, it
could show us the essential features of the nonlinear dynamics [2]. The symbiotic relations between species have
been very important fields in ecology. An interspecific interaction can have various forms which can be classified
largely as prey-predator type, competition and mutualism [3]. Like the population dynamics of one species, the
interacting populations of two species have been formulated in either differential equations or difference equations [4–8].
In this paper, we formulate these relations with a parametrically coupled logistic map as
xn+1 = µ(yn )xn (1 − xn ),
yn+1 = ν(xn )yn (1 − yn ).
(1)
In Eq. (1), µ or ν, which is the control parameter of
one species in a 1-dimensional map, is no longer a simple
parameter but a relating function depending on the population of the other species. µ(y) and ν(x) are functions
of y or x and must satisfy 0 ≤ µ(y) ≤ 4, 0 ≤ ν(x) ≤ 4 in
the phase plane {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. For general ecological considerations, µ(y) and ν(x) may have
various functional forms which need to have sets of large
numbers of parameters.
Now, we are to deal with a simple symmetric case. We
introduce the relation of sigmoidal functions as
xn+1 = {2 tanh[2a(yn − 12 )] + 2}xn (1 − xn ),
yn+1 = {2 tanh[2b(xn − 21 )] + 2}yn (1 − yn ).
II. STABILTY OF FIXED POINTS
The map of Eq. (2) can have five fixed points at most
and at least two in the phase plane; the number of fixed
points depends on the control parameters a, b. The
points and their characteristics are summarized in Table ??. The fixed points are solutions of the following
(2)
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Journal of the Korean Physical Society, Vol. 30, No. 1, February 1997
Table 1. Fixed points and their stability conditions.
Fixed
point
r1
r2 a
r3
r4
r5
Position
Characteristic
(0.5, 0.5)
survival
survival
species 2 extinctb
species 1 extinctd
all species extinct
a−1
( 22 tanh
, 0)
tanh a−2
2 tanh b−1
(0, 2 tanh b−2 )
(0, 0)
Existence
condition
always existing
√
√
a < ln 3, b > ln 3 bc
√
√
a > ln 3, b < ln 3
always existing
Stability
condition
|ab| < 1
a = b > 0, ab > 1
same as the existence condition
same as the existence condition
√
√
a > ln 3, b > ln 3
a The existence and the position of r are dependent on a, b and should be calculated numerically.
2
b The extinction of one species results in a 1-dimensional map. The existence condition is the same as the stability condition.
c a < ln √3, b > ln √3 is equivalent to 2 tanh a−1 > 0, 2 tanh b−1 < 0.
2 tanh a−2
2 tanh a−2
d When (a, b) is in the region a < ln √3, b < ln √3, the survival basin fills almost all of the phase plane.
equations;
x = {2 tanh[2a(y − 12 )] + 2}x(1 − x),
y = {2 tanh[2b(x − 12 )] + 2}y(1 − y).
(3)
Here, each equation can be defactorized as x = 0 or x =
1
1
1− 2 tanh[2a(y−
and y = 0 or y = 1− 2 tanh[2b(x−
.
1
1
2 )]+2
2 )]+2
Hence, the fixed points are points where the curves
of these functions cross each other or cross the x-axis
1
or the y-axis. Curves of x = 1 − 2 tanh[2a(y−
is
1
2 )]+2
plotted for several values of a in Fig. 1. Curves of
1
y = 1 − 2 tanh[2b(x−
can be obtained by symmet1
)]+2
2
1
ric transformation of those of x = 1 − 2 tanh[2a(y−
1
2 )]+2
for x = y. The number of fixed points varies from three
at least to five at most, depending on the values of a and
Fig. 1. Plots of x = 1 −
of a: (a) a = 3.5, (b) a =
(d) a = −3.5.
1
2 tanh[2a(y− 1
)]+2
2
1
ln
3
=
0.5493,
2
for several values
(c) a = 0.25 and
b. Two of the five points are the survival ones for which
neither of the populations of the species is zero. One of
the two is the point r 1 whose the position (0.5, 0.5) is
invariant, which is the consequence of the symmetry as
previously noted. The existence and the position of the
other survival point r 2 depend on the control parameters. However, we will not pay any more attention to
this point because it can not be stable for the whole region of the parameter plane, except for the symmetric
relation a = b for which the point r 2 is stable for ab > 1.
This condition has little significance from the biological
point of view. Unstable fixed points can have meaning
in physics, but have little meaning from the biological
point of view because instability can have no place in
nature. From the fact that the extinction of any one
of the species means breakdown of the symbiotic rela-
Fig. 2. The basins of attraction for a = 0.5, b = 0.8.
Outside of the contour is the basin of the fixed point (0.0704,
0), which is r3 in Table 1.
Coupled Logistic Map for Symbiotic Relations - Gyu-Seung Shin
Fig. 3. The basins of attraction for a = 0.7, b = 1.0. The
attractor on the outside of the contour is the point (0, 0),
which is r5 in the Table 1.
tion and brings about a one-dimensional map, the fixed
point r 1 is one of the main objects of attention in this
map because it is the only survival fixed point which has
biological significance.
The remaining three points are the points of extinction at which at least one of two species is exterminated.
The existence
conditon
√
√ of the fixed point r 3 (r 4 ) are
a < ln 3 (b < ln 3), and the stability condition is
equivalent to the existence condition. When these two
Fig. 4. The basins of attraction for a = 0.8, b = 1.2.
Outside of the contour is the basin of the fixed point (0, 0).
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Fig. 5. Division of regions in the parameter plane for the
prey-predator relation.
√
√
points can not exist (a > ln 3, b > ln 3), the point r 5
becomes stable, and all species are exterminated.
We can test the stability of the fixed points with the
Jacobian matrices. For the survival fixed point,
!
∂xn+1 ∂xn+1
0 a
∂xn
∂yn
=
.
(4)
∂yn+1 ∂yn+1
b 0
∂xn
∂yn
(xn ,yn )=(0.5,0.5)
The simple form of the Jacobian matrix is the consequence of the symmetry of the map, and we can derive
the stability condition of r 1 as |ab| < 1.
With the relating function µ(y), which is an increasing
function of y (a > 0), the species whose population variable is y plays beneficial role for the other one. On the
other hand, a decreasing relating function corresponds
to a harmful relation. Therefore, interspecific relations
Fig. 6. Bifurcation diagram (a, x) along the line of b =
−1.0 in the parameter plane of Fig. 5.
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Journal of the Korean Physical Society, Vol. 30, No. 1, February 1997
Fig. 7. Bifurcation diagram (a, y) along the line of b =
−1.0 in the parameter plane of Fig. 5.
can be classified with the sets of signs of (a, b) in Eq. (2).
(+, +) corresponds to mutualism (a mutually beneficial
relation), (−, −) is competition type (a mutually harmful
relation), and (+, −) or (−, +) is a prey-predator relation. Formulated with the different sets of signs of the
parameters in the same equations, the actual dynamics
of the interspecific relations are very different from each
other. Hence, we will discuss the three types separately.
III. TYPES OF INTERSPECIFIC RELATIONS
1. Mutualism
The dynamics of the bilaterally beneficial interaction
in the map in Eq. (2) is so simple that there are no periodic cycles, no chaos, but only attractors of fixed points.
Fig. 8. Division of regions in the parameter plane for a
competition relation. The initial point is (0.49,0.51).
Fig. 9. Bifurcation diagram (a, x) along the line a = b
from a = 0 to a = −2.5 in the parameter plane of Fig. 8.
This result may not be general. For other forms of µ(y)
and ν(x), periodic cycles and chaos may appear. For
ab > 1, the survival fixed point r 1 is unstable, and the
only fixed point which can be stable is the extinct one.
Therefore, at least one of the two species is exterminated.
For ab < 1, r 1 is stable. However, its basin coexists with
that of the extinct fixed point. The basins are depicted in
Figs. 2-4 for three values of (a, b). Inside of the contours
are the basins of the survival fixed point and outside of
them are those of extinction. We can see that the basin
of the survival fixed point r 1 shrinks as ab approaches
1. The interspecific relation of Fig. 4 can be said to be
so fragile that a small external perturbation may lead to
extermination easily. Considering the survival possibility
of one species in mutualism, the degree of dependence on
the other one (it is a or b in this map) plays the key role.
As shown in √
Table ??, species 1 will be exterminated
when a > ln 3 in the basin of extinction. When the
degree of dependence becomes higher, not only the possibilty of breakdown of the interspecific relation but also
that of extermination of the species becomes greater.
Fig. 10. Bifurcation diagram (a, y) along the line a = b
from a = 0 to a = −2.5 in the parameter plane of Fig. 8.
Coupled Logistic Map for Symbiotic Relations - Gyu-Seung Shin
2. Prey-Predator Relation
In the prey-predator relation, r 1 is the focus of a helix and is stable for |ab| < 1. Just out of this region,
regions of period-4 cycle, period-8, period-16,... appear
successively and eventually the chaotic region is arrived
at as shown in Fig. 5, which is the phase diagram of the
prey-predator relation. The boundaries of the regions
are obtained numerically by plotting points at which periodic doublings or phase changes appear. As seen in the
upper right region, a quasi-periodic region is put into the
periodic regions. Also in the lower left region, the boundaries of the regions seems to have complicated structures
which are not understood yet. The bifurcation diagrams
(a, x) and (a, y) are depicted in Fig. 6 and Fig. 7 along
the line b = −1.0 from a = 0 to a = 2.5. With this
relation, the whole phase plane is composed of survival
basins of attraction.
3. Competition
Competitive interspecific interactions show more complicated dynamics than the other types. From top to
bottom in Fig. 8, there appear the regions of the stable
fixed point and the period-2 cycle, the period-4 cycle, the
period-8 cycle,... successively and the chaotic region. As
with the predation relation, the bifurcation diagrams are
depicted, in Fig. 9 and Fig. 10 along a line in the parameter plane. There are two types of basins in the phase
plane, the survival ones and the extinct ones. In Fig. 8,
the boundaries can be largely divided into two classes:
one being parallel lines and the other being curves. The
independence from one parameter (parallel lines) indicates the extinction of one species. Species 1 (species
2) is exterminated in the regions around the horizontal
lines (vertical lines). The lines are connected with curves
continuously in the upper region of Fig. 8, but a tip of
the survival region penetrates the region of extinction in
the lower right region. It must be noted that this diagram depends on the initial point and that conversely
the basins also depends on the parameters. Fig. 8 is for
the initial point (0.49, 0.51).
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IV. CONCLUSION
We have dealt with the three types of interspecific relations with a simple and symmetric form of a coupled
logistic map. The results can be summarized as follows:
(i) For mutualism and competition, two distinct types
of basins occur, extinct and survival. However for the
prey-predator relation, there are only survival- type
basins.
(ii) For mutualism, the stability of the relation depends on the degree of mutualism. The degree of dependence on the other species is crucial for the survival of a
species; a higher degree of dependence means a greater
possibility of extinction.
(iii) Mutualism shows simple dynamics while the parameter plane of other two types can be divided into
regions for which the relation shows different dynamics
(stable fixed points, limit cycles and chaos).
We presented the coupled logistic map as a formulation for symbiotic relations and have dealt with a simple
example. With more general forms of µ(y) and ν(x),
richer dynamics of the relations can be manifested.
ACKNOWLEDGMENTS
Helpful discussions with Professor Y. I. Cha are gratefully acknowledged. This work was supported by a grant
(1994-1995) from Kyung Hee University.
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