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The dynamics of a discrete population model with threshold
The Dynamics of a Discrete Population Model with Threshold FREDERICK R. MAROTTO Division 01 Science und Mrrrhemrrtics. Fordham New’ York 10023
Unioersl& ut Lincoln Center,
New
York,
Reserved 29 June 1981
ABSTRACT The dynamics gated. In addition equation previous
of the discrete, scalar to density dependence,
population model xI + , =u.x~(1 -.xk) are investiwhich has been studied previously by many, this
models the threshold phenomenon. models are observed. In particular,
Some similarities to and differences for large (1 values this model exhibits
from chaos
which is restricted to a nowhere dense Cantor set of measure 0. In order to explain this, a piecewise linear simplification of the model is considered. Other models exhibiting similar dynamics
I.
are also mentioned.
INTRODUCTION
The dynamics of density-dependent population models have been extensively investigated lately by many authors. This is particularly true of discrete scalar models of the type xk + , =f(x,) where f: Iw-+ R is continuous, and xI. ER represents the population at discrete time intervals. Density dependence implies that small initial populations will tend to grow, but large populations with limited life-sustaining resources will decrease. The simplest mapping f which expresses this relationship is the logistic mapping f(x) = ax( 1 -x), which has been widely studied by researchers in many fields. (See May [3, 41, for example.) What this mapping and others of this type fail to model, however, is another interesting property of biological populations, namely the threshold phenomenon. At extremely low levels many populations tend to lead to extinction rather than to growth. However, beyond some threshold the population will tend to grow in a manner similar to that predicted by the logistic equation. We are thus led to consider the model xk+, =a$( The reason for this choice is evident-it MATHEMATICAL
BIOSCIENCES
1_Xk).
(1)
is perhaps the simplest such model 123
58: 123- 128 (1982)
6Elsevier North Holland. Inc., 1982 52 Vanderbilt Ave., New York, NY I00 I7
0025-5564/82/O
I23 +06$02.75
124
FREDERICK
R. MAROTTO
which exhibits the threshold phenomenon. This can be easily seen by investigating the fixed point x = 0 of the mapping f(x) = ax ‘( 1 -x). For all values of the parameter u, f’(0) =O and so 0 is a stable fixed point. Therefore, for any growth rate a there will be a region near 0 in which any initial value x0 will lead to xk + 0 as k -+cc under (1). We shall investigate (1) for different choices of the parameter a. As we shall see, for some values of a this model can exhibit rather surprising dynamics. II.
AN ANALYSIS
OF THE MODEL
Let us restrict the analysis of (1) to OGuG6.75 and OGX,,< 1, which will guarantee that 04xkG 1 for all k. As previously mentioned, 0 is a fixed point of j(x)=ux*(l -x). It can be easily checked that for O~u<4,f(x)4 there are two other fixed points: x, =(u*~~)/~u. (See Figure 1.) Here, x_ is unstable for all u>4, but for some values of u exceeding 4, x+ will be a locally stable equilibrium. Let us remark on the threshold nature of the model for any value of u>4. Since x_ is an unstable equilibrium withf’(x_)> 1, then for any x,E[O, x-),
1
*+
X
0
FIG. I.
The mappingf(x)=ux2(l
-x)
for n>4.
DYNAMICS
OF A DISCRETE
POPULATION TABLE
I
The Dynamics of xk + , = a~( for O~uG6.75 0.00~u~4.00 4.oo
125
MODEL
I - xL)
Extinction Stable I -cycles Stable 2-cycles Stable 4-cycles Stable X-cycles Stable 2”-cycles, Chaos Nowhere
na4
dense chaos
X~-) 0 as k- cc under (1) but for any x,, exceeding (but close to) x_ growth will occur. Thus the fixed point x- is precisely the threshold value. As the parameter a is increased beyond 4, the dynamics of this model begin to resemble those found for the previously investigated logistic equation. A numerical study reveals that there are cascades of bifurcations from stable points of period 2” to stable points of period 2”+’ as a is increased. Table I indicates the approximate values of c1 at which these bifurcations occur. The 2” cycles appear to be stable for any initial value x0)x_. The parameter values corresponding to these bifurcations approach a limiting value of 5.89, after which the equation exhibits chaos. Chaos is a phenomenon which has been studied extensively in recent years. Analytic conditions for the existence of chaos are well known. One such condition is: if a scalar function f is continuous and has a point of period 3, then f is chaotic, i.e., there are an infinite number of periodic points, as well as other randomlike behavior (Li and Yorke [l], Sarkovskii [5]). Although this result has been useful in explaining the chaotic behavior of one-dimensional mappings, it is not extendable to multidimensional models. Another condition that does apply for any differentiable f:R n-+IF4 nis the following. If there exists an unstable fixed point x of f and a point x0 “close” to x with f O’( x0)= x and Det[ Of”( x0)] #O, then x is called a snapback repeller. (By “close” is meant that the negative limit set of x0 under f is x). If f has a snapback repeller, then f is chaotic in the previous sense (Marotto [2]). Ether of these two conditions can help explain the chaotic behavior of (l), since the equation has both a point of period 3 and a snapback repeller for large enough values of a. Up to this point (1) has exhibited behavior similar to previous models (except for the threshold), but as Table 1 indicates, for a>6.54 a new phenomenon emerges. Although chaos is still present theoretically for a > 6.54, i.e., there still exist both points of period 3 and snapback repellers, this chaos is confined to a nowhere dense set of measure 0. That is, for almost any
126
FREDERICK
R. MAROTTO
choice of xg in [0, I], except for a Cantor set, xk -0 as k- 00. Further, it appears that this Cantor set has measure 0. Thus the chaos must reside in a nowhere dense Cantor set of measure 0, and the point x=0 is for all practical purposes a globally stable fixed point. This behavior has not been observed in previous models, and in fact if such chaos exists, then “chaos” is not an appropriate name for it. The following discussion is an attempt to explain these unusual dynamics. III.
A POSSIBLE
EXPLANATION
In order to explain nowhere dense chaos, let us instead consider a piecewise linear simplification of f(x)=ax*( I -x) exhibiting the same general characteristics:
g(x)=
0
for
OGx
b(x-f)
for
$
-b(x-I)
for
$GxGl.
1
For any b satisfying O~b<2 it can be easily checked that all solutions of x~,, =g(x,) approach 0, but for b>2 there are again two nontrivial fixed points, the lesser of which is again the threshold value. This threshold value is given by x*=b/(3b-3). Let y* be the point in [2, I] satisfying g(y*)=x*, i.e., r*=(3b-4)/(3b-3). It can be easily seen that for any x,E[O,x*) or x,E(y*,l], x,-O as k-m. Also, for any b>2, g(j)=b/3>y*. (See Figure 2.) CLAIM
I
For all b>2,
g is chaotic.
Proof. Chaos can be proven by exhibiting either a point of period 3 or a snapback repeller. We shall do the latter. (Finding a point of period 3 for b>2 is also not very difficult.) Note that since g(i)=b/3>y*, then ])>[x*,y*]. Thus there must exist x,E(x*,$) with g(x,)=y* and g(]x*, i so g2(xo)=g(g(x0))=x*. Also dg2(x0)/dx#0. But for any point in [x*,3] a sequence of inverse images can be found inside [ x *, $1 such that the limit of this sequence is x*. Thus g2(x0)=x*, Det[dg2(xo)/dx]#0 and the negative limit set of x0 is x*, proving that x* is a snapback repeller. CLAIM
2
For ali b>2, aN solutions of x,+,=g(x,) dense Cantor set of measure 0.
approach 0 except for u nowhere
Proof. Note first that the inverse image in [x*, y*] of any set of measure m inside [x*, b/3] has measure 2m/b. This is true because the slopes of the
DYNAMICS
OF A DISCRETE
FIG. 2.
POPULATION
The piecewise
127
MODEL
linear mapping
g(x)
for h>2
line segments of the multivalued inverse are 2 l/b and there are two distinct inverse image sets. Let I,,=(y*,b/3) and r=b/3-~*=(b-2)~/(3b-3). The inverse image I, C [ x*, y *] of Z,, has measure 2r/b. The inverse image I2 c [ x*, y *] of I, therefore has measure r(2/b)‘. Continuing in this manner, we can find a sequence of inverse images Ik C [ x *, y *] of I, ~, having measure r(2/b)k. Now since all Ik are disjoint, then the set S= UT= ,{lk} C[x*, y*] has measure equal to
2r =---zz b-2
2b-4 3b-3’
It can be easily checked that y*-x* also equals (26-4)/(36-3), and so the measure of the complement of S in [x*, y*] is 0. Note that from the way the Ik were constructed the complement of S is a Cantor set. Also, for any x0 ES some iterate xk of x0 under g must he in (y*, b/3]. Thus for any x0 inside [0,x*), (y*, l] or S we have xk+ 0 as k -+ co. Hence all solutions approach 0
128
FREDERICK
R. MAROTTO
except for the complement of S, which is a nowhere dense Cantor set of measure 0, proving the claim. These two claims together prove that nowhere dense chaos can exist. What appears to be its cause in this case is: (a) the existence of the threshold region [0, x*1, and (b) a large enough value of the maximum g(i)=b/3 so that an interval around S is mapped into [_v*, l] and thereafter into the threshold region [0, x*]. If the maximum of g were less, then it would be impossible for points inside [x*, y *] to escape and approach 0. If this escape were not possible, then it is likely that the usual form of chaos would prevail. This appears to explain what occurs for the mapping f(x) = ax’( I- x). For 5.89~ a ~6.54 points near the maximum point x = i cannot escape into the threshold region [0, x- 1, and thus there is truely chaotic chaos. However, for u >6.54 the maximum satisfiesf( f((:))< x- , i.e., less than the threshold value, and almost all solutions tend to 0. IV.
CONCLUSION
The analysis of this model has revealed some similarities differences from nonthreshold models. Preliminary numerical of other threshold models such as xk+,=ax;(l-xxh)‘%ndx
r+,=ax;exp
(-x,)
to and some investigations
forn,m>l
indicate similar dynamics: a threshold region for all values of a, cascades of bifurcations leading to chaos, and for large values of a nowhere dense chaos. This latter phenomenon is of particular interest, since it implies that chaos does not have to be chaotic. That is, chaos, as it has been commonly defined, is not in general equivalent to ergodicity or randomness. Finding general conditions under which chaos and randomness are equivalent is an interesting problem and remains to be investigated. REFERENCES I 2 3 4 5
T.-Y. Li and J. A. Yorke. Period three implies chaos, Anw. Moth. Month!) X2:9X5-992 (1975). F. R. Marotto. Snap-back repellers imply chaos in Iw”. J. /W&h. Awl. Appl. 63: 199-223 (197X). R. M. May, Biological populations with nonoverlapping generations: stable points. stable cycles and chaos, Scrence I X6:645-647 (1974). R. M. May, Simple mathematical models with complicated dynamics, Nature 261:459467 (1976). A. N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, UXrcri’n. Mur. i. 16:61-71 (1964).
Unioersl& ut Lincoln Center,
New
York,
Reserved 29 June 1981
ABSTRACT The dynamics gated. In addition equation previous
of the discrete, scalar to density dependence,
population model xI + , =u.x~(1 -.xk) are investiwhich has been studied previously by many, this
models the threshold phenomenon. models are observed. In particular,
Some similarities to and differences for large (1 values this model exhibits
from chaos
which is restricted to a nowhere dense Cantor set of measure 0. In order to explain this, a piecewise linear simplification of the model is considered. Other models exhibiting similar dynamics
I.
are also mentioned.
INTRODUCTION
The dynamics of density-dependent population models have been extensively investigated lately by many authors. This is particularly true of discrete scalar models of the type xk + , =f(x,) where f: Iw-+ R is continuous, and xI. ER represents the population at discrete time intervals. Density dependence implies that small initial populations will tend to grow, but large populations with limited life-sustaining resources will decrease. The simplest mapping f which expresses this relationship is the logistic mapping f(x) = ax( 1 -x), which has been widely studied by researchers in many fields. (See May [3, 41, for example.) What this mapping and others of this type fail to model, however, is another interesting property of biological populations, namely the threshold phenomenon. At extremely low levels many populations tend to lead to extinction rather than to growth. However, beyond some threshold the population will tend to grow in a manner similar to that predicted by the logistic equation. We are thus led to consider the model xk+, =a$( The reason for this choice is evident-it MATHEMATICAL
BIOSCIENCES
1_Xk).
(1)
is perhaps the simplest such model 123
58: 123- 128 (1982)
6Elsevier North Holland. Inc., 1982 52 Vanderbilt Ave., New York, NY I00 I7
0025-5564/82/O
I23 +06$02.75
124
FREDERICK
R. MAROTTO
which exhibits the threshold phenomenon. This can be easily seen by investigating the fixed point x = 0 of the mapping f(x) = ax ‘( 1 -x). For all values of the parameter u, f’(0) =O and so 0 is a stable fixed point. Therefore, for any growth rate a there will be a region near 0 in which any initial value x0 will lead to xk + 0 as k -+cc under (1). We shall investigate (1) for different choices of the parameter a. As we shall see, for some values of a this model can exhibit rather surprising dynamics. II.
AN ANALYSIS
OF THE MODEL
Let us restrict the analysis of (1) to OGuG6.75 and OGX,,< 1, which will guarantee that 04xkG 1 for all k. As previously mentioned, 0 is a fixed point of j(x)=ux*(l -x). It can be easily checked that for O~u<4,f(x)
1
*+
X
0
FIG. I.
The mappingf(x)=ux2(l
-x)
for n>4.
DYNAMICS
OF A DISCRETE
POPULATION TABLE
I
The Dynamics of xk + , = a~( for O~uG6.75 0.00~u~4.00 4.oo
125
MODEL
I - xL)
Extinction Stable I -cycles Stable 2-cycles Stable 4-cycles Stable X-cycles Stable 2”-cycles, Chaos Nowhere
na4
dense chaos
X~-) 0 as k- cc under (1) but for any x,, exceeding (but close to) x_ growth will occur. Thus the fixed point x- is precisely the threshold value. As the parameter a is increased beyond 4, the dynamics of this model begin to resemble those found for the previously investigated logistic equation. A numerical study reveals that there are cascades of bifurcations from stable points of period 2” to stable points of period 2”+’ as a is increased. Table I indicates the approximate values of c1 at which these bifurcations occur. The 2” cycles appear to be stable for any initial value x0)x_. The parameter values corresponding to these bifurcations approach a limiting value of 5.89, after which the equation exhibits chaos. Chaos is a phenomenon which has been studied extensively in recent years. Analytic conditions for the existence of chaos are well known. One such condition is: if a scalar function f is continuous and has a point of period 3, then f is chaotic, i.e., there are an infinite number of periodic points, as well as other randomlike behavior (Li and Yorke [l], Sarkovskii [5]). Although this result has been useful in explaining the chaotic behavior of one-dimensional mappings, it is not extendable to multidimensional models. Another condition that does apply for any differentiable f:R n-+IF4 nis the following. If there exists an unstable fixed point x of f and a point x0 “close” to x with f O’( x0)= x and Det[ Of”( x0)] #O, then x is called a snapback repeller. (By “close” is meant that the negative limit set of x0 under f is x). If f has a snapback repeller, then f is chaotic in the previous sense (Marotto [2]). Ether of these two conditions can help explain the chaotic behavior of (l), since the equation has both a point of period 3 and a snapback repeller for large enough values of a. Up to this point (1) has exhibited behavior similar to previous models (except for the threshold), but as Table 1 indicates, for a>6.54 a new phenomenon emerges. Although chaos is still present theoretically for a > 6.54, i.e., there still exist both points of period 3 and snapback repellers, this chaos is confined to a nowhere dense set of measure 0. That is, for almost any
126
FREDERICK
R. MAROTTO
choice of xg in [0, I], except for a Cantor set, xk -0 as k- 00. Further, it appears that this Cantor set has measure 0. Thus the chaos must reside in a nowhere dense Cantor set of measure 0, and the point x=0 is for all practical purposes a globally stable fixed point. This behavior has not been observed in previous models, and in fact if such chaos exists, then “chaos” is not an appropriate name for it. The following discussion is an attempt to explain these unusual dynamics. III.
A POSSIBLE
EXPLANATION
In order to explain nowhere dense chaos, let us instead consider a piecewise linear simplification of f(x)=ax*( I -x) exhibiting the same general characteristics:
g(x)=
0
for
OGx
b(x-f)
for
$
-b(x-I)
for
$GxGl.
1
For any b satisfying O~b<2 it can be easily checked that all solutions of x~,, =g(x,) approach 0, but for b>2 there are again two nontrivial fixed points, the lesser of which is again the threshold value. This threshold value is given by x*=b/(3b-3). Let y* be the point in [2, I] satisfying g(y*)=x*, i.e., r*=(3b-4)/(3b-3). It can be easily seen that for any x,E[O,x*) or x,E(y*,l], x,-O as k-m. Also, for any b>2, g(j)=b/3>y*. (See Figure 2.) CLAIM
I
For all b>2,
g is chaotic.
Proof. Chaos can be proven by exhibiting either a point of period 3 or a snapback repeller. We shall do the latter. (Finding a point of period 3 for b>2 is also not very difficult.) Note that since g(i)=b/3>y*, then ])>[x*,y*]. Thus there must exist x,E(x*,$) with g(x,)=y* and g(]x*, i so g2(xo)=g(g(x0))=x*. Also dg2(x0)/dx#0. But for any point in [x*,3] a sequence of inverse images can be found inside [ x *, $1 such that the limit of this sequence is x*. Thus g2(x0)=x*, Det[dg2(xo)/dx]#0 and the negative limit set of x0 is x*, proving that x* is a snapback repeller. CLAIM
2
For ali b>2, aN solutions of x,+,=g(x,) dense Cantor set of measure 0.
approach 0 except for u nowhere
Proof. Note first that the inverse image in [x*, y*] of any set of measure m inside [x*, b/3] has measure 2m/b. This is true because the slopes of the
DYNAMICS
OF A DISCRETE
FIG. 2.
POPULATION
The piecewise
127
MODEL
linear mapping
g(x)
for h>2
line segments of the multivalued inverse are 2 l/b and there are two distinct inverse image sets. Let I,,=(y*,b/3) and r=b/3-~*=(b-2)~/(3b-3). The inverse image I, C [ x*, y *] of Z,, has measure 2r/b. The inverse image I2 c [ x*, y *] of I, therefore has measure r(2/b)‘. Continuing in this manner, we can find a sequence of inverse images Ik C [ x *, y *] of I, ~, having measure r(2/b)k. Now since all Ik are disjoint, then the set S= UT= ,{lk} C[x*, y*] has measure equal to
2r =---zz b-2
2b-4 3b-3’
It can be easily checked that y*-x* also equals (26-4)/(36-3), and so the measure of the complement of S in [x*, y*] is 0. Note that from the way the Ik were constructed the complement of S is a Cantor set. Also, for any x0 ES some iterate xk of x0 under g must he in (y*, b/3]. Thus for any x0 inside [0,x*), (y*, l] or S we have xk+ 0 as k -+ co. Hence all solutions approach 0
128
FREDERICK
R. MAROTTO
except for the complement of S, which is a nowhere dense Cantor set of measure 0, proving the claim. These two claims together prove that nowhere dense chaos can exist. What appears to be its cause in this case is: (a) the existence of the threshold region [0, x*1, and (b) a large enough value of the maximum g(i)=b/3 so that an interval around S is mapped into [_v*, l] and thereafter into the threshold region [0, x*]. If the maximum of g were less, then it would be impossible for points inside [x*, y *] to escape and approach 0. If this escape were not possible, then it is likely that the usual form of chaos would prevail. This appears to explain what occurs for the mapping f(x) = ax’( I- x). For 5.89~ a ~6.54 points near the maximum point x = i cannot escape into the threshold region [0, x- 1, and thus there is truely chaotic chaos. However, for u >6.54 the maximum satisfiesf( f((:))< x- , i.e., less than the threshold value, and almost all solutions tend to 0. IV.
CONCLUSION
The analysis of this model has revealed some similarities differences from nonthreshold models. Preliminary numerical of other threshold models such as xk+,=ax;(l-xxh)‘%ndx
r+,=ax;exp
(-x,)
to and some investigations
forn,m>l
indicate similar dynamics: a threshold region for all values of a, cascades of bifurcations leading to chaos, and for large values of a nowhere dense chaos. This latter phenomenon is of particular interest, since it implies that chaos does not have to be chaotic. That is, chaos, as it has been commonly defined, is not in general equivalent to ergodicity or randomness. Finding general conditions under which chaos and randomness are equivalent is an interesting problem and remains to be investigated. REFERENCES I 2 3 4 5
T.-Y. Li and J. A. Yorke. Period three implies chaos, Anw. Moth. Month!) X2:9X5-992 (1975). F. R. Marotto. Snap-back repellers imply chaos in Iw”. J. /W&h. Awl. Appl. 63: 199-223 (197X). R. M. May, Biological populations with nonoverlapping generations: stable points. stable cycles and chaos, Scrence I X6:645-647 (1974). R. M. May, Simple mathematical models with complicated dynamics, Nature 261:459467 (1976). A. N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, UXrcri’n. Mur. i. 16:61-71 (1964).