The Discrete Logistic Equation

Section 2.2 Problems

2.3 More Population Models

2.3.2 The Discrete Logistic Equation

The most popular discrete-time single-species model is the discrete logistic equation, whose recursion is given by

N_t+1= N_t

1+R

1− N_t K

(2.13) where Rand K are positive constants. R is called thegrowth parameterand K is called thecarrying capacity. The analysis that folllows will explain the terminology.

This model of population growth exhibits very complicated dynamics, described in an influential review paper by Robert May (1976).

Before we illustrate its behavior, we will rewrite the model in what is called thecanonical form. The advantage of this form is that the resulting recursion will be simpler. The algebraic steps presented next are not obvious, but will lead to the canonical form of the discrete logistic equation. We write

N_t+1 =N_t

1+R

1− N_t K

=Nt

1+R− R K Nt

=N_t(1+R)

1− R

K(¹+R)N_t

Now, dividing by 1+Ryields 1

1+RN_t+1 =N_t

1− R

K(¹+R)N_t

Let’s multiply both sides byR/K (you'll see why in a moment):

R

K(¹+R)N_t+1 = R K N_t

1− R

K(¹+R)N_t

(2.14) If we define the new variable as

xt = R

K(¹+R)Nt (2.15)

then

R

K(1+R)N_t+1=x_t+1 and R

K N_t =(¹+R)x_t and (2.14) becomes

x_t+1=(¹+R)x_t(¹−x_t)

At this point, the new parameterr = ¹+ Ris customarily introduced. Note that r >^{1, since} R>0. We thus arrive at the canonical form of the logistic recursion:

x_t+1 =r x_t(¹−x_t) ^(2.16)

The advantage of this form is threefold: (1) The recursion (2.16) looks simpler than the original recursion (2.13); (2) instead of two parameters (RandK), there is just one (r); and (3) the quantityxt isdimensionless. The last point needs some expla- nation. The original variableNt has units (or dimension) of number of individuals;

the parameter K has the same units. Dividing N_t by K in (2.15), we see that the units cancel and we say that the quantityx_tis dimensionless. [The parameterRdoes not have a dimension, so multiplying N_t/K by R/(1+ R)does not introduce any additional units.] A dimensionless variable has the advantage that it has the same numerical value regardless of what the units of measurement are in the original variable. (See Problems 31–34.) The process of making a quantity dimensionless is callednondimensionalization.

Let’s go back to the discrete logistic equation in its canonical form (2.16) and see what its behavior is. The function f(x)=r x(¹−x)is an upside-down parabola, sincer > 1 (Figure 2.16). We see from the figure that ifx is outside of the interval (⁰,¹)^, f(x)is nonpositive. Sincext = _K(1+R)^R Nt [see (2.15)], and we wantNt to be positive (it is a population size, after all), we requirex_tto be positive. This means that we need to ensure thatx_t+1 = f(x_t)stays within the interval(⁰,¹). The maximum value of f(x)^{occurs at}x = ¹/^{2, and} f(¹/²)= r/4, so, in order to make sure that f(xt) ∈ (⁰,¹), we require thatr/⁴ < ^{1, or}r < 4. We already require thatr > ^1, sinceR>0. To summarize, if 1<r <^{4, then}xtstays within the interval(⁰,¹)^{for all} t =¹,²,³, . . ., provided thatx₀ ∈(⁰,¹). In what follows, we will therefore assume that 1<r <^{4 and}x0∈(⁰,¹)^.

2 1.5 1 0.5 0 0.5 1

0 0.5 1 1.5 2 x

rx(1 x)

Figure 2.16 A graph of the discrete logistic equation in its canonical form. (Here,r =2.5.)

We first compute fixed points of (2.16). We need to solve x =r x(¹−x)

Solving immediately yields the solutionx = ^{0. If}x =0, we divide both sides byx and find that

1=r(¹−x), ^or x =¹− ¹ r

(See Figure 2.17.) Provided thatr >1, both fixed points are in[⁰,¹)^.

1 1 1/r

0 x

0.5 0.5

rx(1 x) y f(x)

y x

Figure 2.17 A graphical illustration of the fixed points of the discrete logistic equation in its canonical form. The fixed points are where the parabola and the liney=x

intersect.

We return to the original variableN_t for a moment to see whatx = ^{0 and}x = 1−¹/r mean in terms ofN. Sincex = _K(1+R)^R N, the fixed pointx =0 corresponds to the fixed point N = 0, which is why we callx = 0 a trivial equilibrium. When x =1−1/r, then, usingr =1+R, we obtain

N = K(¹+R)

R x= K(¹+R) R

1− ¹

1+R

= K(¹+R) R

1+R−¹ 1+R = K soN =K is the other fixed point.

The long-term behavior of the discrete logistic equation is very complicated. We will go through the different cases by simply listing them. Later, in Section 5.6, we will be able, at least to some extent, to understand why this equation has such complicated behavior.

When 1<r <3 andx₀∈(0,1),x_t converges to the fixed point 1−1/r(Figure 2.18). Increasingr to a value between 3 and 3.⁴⁴⁹. . ., we learn thatx_t settles into a cycle of period 2 (Figure 2.19). This means that, for large enough times, x_t will oscillate back and forth between a larger and a smaller value. Forrbetween 3.⁴⁴⁹. . . and 3.⁵⁴⁴. . ., the period doubles: A cycle of period 4 appears for large enough times. The population size now oscillates between the same four values (Figure 2.20).

Increasingr continues to double the period: A cycle of period 8 is born whenr = 3.⁵⁴⁴. . ., a cycle of period 16 whenr = ³.⁵⁶⁴. . ., and a cycle of period 32 when r =³.⁵⁶⁷. . .. This doubling of the period continues untilrreaches a value of about 3.57, when the population pattern becomeschaotic(Figure 2.21). The population dynamics seem to be random, although the rules are entirely deterministic! There is no regular pattern we can discern:xtno longer oscillates between the same values; the dynamics areaperiodic. Furthermore, starting from ever so slightly different initial conditions quickly produces very different trajectories (Figure 2.22). This sensitivity to initial conditions is characteristic of chaotic behavior.

0.6 0.5 0.4 0.3 0.2 0.1 0

0 5 10 15

xt

t

Figure 2.18 A graph ofx_tas a function oftwhenr =^2.

0.6 0.5 0.4 0.3 0.2 0.1 0 0.9 0.8 0.7

0 5 10 15

xt

t

Figure 2.19 A graph ofxtas a function oftwhenr =3.2.

0 5 0

0.2 0.4 0.6 0.8 1

10 15

t xt

20 25 30

Figure 2.20 A graph ofx_tas a function oftwhenr =³.^52.

0 5

0 0.2 0.4 0.6 0.8 1

10 15

t xt

20 25 30

Figure 2.21 A graph ofx_tas a function oftwhenr =³.^8.

0 1 0.8 0.6 0.4 0.2 0

5 10

t xt

15 20

x0 0.15 x0 0.2

Figure 2.22 Graphs ofxtas a function oftwhenr =³.8 for two different initial values ofx0.

To obtain biologically sensible results, we needed to restrict bothr andx₀. The reason was that ifx_t >^{1, then}x_t+1is negative. This situation can be easily remedied by changing the dynamics slightly. We discuss such a model in the next subsection.