The Verhulst Equation

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The Verhulst Equation

Let

denote the population of a given species at time

, then

is the total growth rate at time

, which is assumed to be constant, i.e.

$\displaystyle \frac{u'(t)}{u(t)}= \gamma$ with $\displaystyle \gamma \in {\mathbb {R}}^+.$

If $u\neq 0$ , hence $\gamma = \frac{u'(t)}{u(t)} = \frac{d}{dt}\ln(u)$ integrating it is obtained the formula for unlimited growth

$u(t)= e^{\gamma(t- t_0)}u_0$ for all $t\in {\mathbb {R}}$ .

In reality, a population cannot increase without limit. It is more realistic to assume that, when the population level exceeds a limiting population value $\eta$ , the growth rate is negative (reasons for a negative growth rate could be insanity, smog, overcrowding...). Let us now suppose the growth rate is proportional to the difference between this limiting value and the actual population:

$\gamma = \beta(\eta - u)$ , $\quad \beta>0$ a constant.

Thus the equation of limited growth is obtained:

(2.1)

$\displaystyle u'=\beta(\eta - u)u=(\alpha - \beta u)u$

where $\alpha \equiv \beta \eta > 0$ . The solution is

$\displaystyle u(t)= \frac{u_0 \alpha}{(\alpha - \beta u_0)e^{-\alpha(t- t_0)} + \beta u_0}$ with $\displaystyle \ u_0=u(t_0).$

This is the logistic equation, or Verhulst equation, called after the $19^{th}$ century Belgian mathematician Piérre François Verhulst (1804-1849).

The equilibria of (2.1) occur at

and $u^1=\frac{\alpha}{\beta}$ . The equilibrium at

is asymptotically stable since the derivative of

at

is negative (i.e. $u'(u^1)= -\alpha$ ).

will increase to

as a limit if

, and decrease to

as a limit if

. The fixed point

is unstable, or a repellor, since $u'(u^0)=\alpha > 0$ .
The point of inflection marks the turning point where the second derivative becomes negative (

), and hence the point beyond which the yearly population growth rate begins to decrease.

**Figure 2.1:** Vector Field and Graphs of Solutions
$\begin{figure} \epsfxsize =6cm \centerline {\fbox {\epsffile{logcont.eps}} } \vspace*{-1mm} \end{figure}$

Let us consider the discrete logistic equation $u_{t+1}=u_t(\alpha-\beta u_t)$ , which has the typical equivalent version

(2.2)

$\displaystyle x_{n+1} \ = \ \alpha x_n(1- x_n)$

(redefining $\eta= \frac{\beta}{\alpha} ,\ x_n= \eta u_t$ ).
The equation (2.2) represents a one-parameter family of discrete dynamical systems defined on

, provided the initial value

is also in this range and $\alpha$ enough small.
Fixed points are computed by setting $\ \overline{x}= \alpha \overline{x}(1 - \overline{x})$ , so $\ x^0=0$ and $x^1= 1- 1/\alpha$ are obtained. The following analysis is found:

$0<\alpha< 1$: : is the only fixed point and it is asymptotically stable;
$\alpha =1$: : is a neutral fixed point;
$1<\alpha \leq 3$: : is a repelling point, while the fixed point is asymptotically stable with domain of attraction ;
$\alpha> 3$: : is a repelling fixed point. On increasing $\alpha$ from 3 to 4, periodic orbit arise with at each step a doubling of the period: 2,4,8... At each value of where such a new periodic point with double period arises, a bifurcation occurs and there are an infinite number of such period-doubling bifurcation until the value $\alpha_\infty = 3,569946...$ . The values of $\alpha$ with $\alpha_\infty< \alpha< 4$ produce mapping which are indicated by chaotic. The site http://www.lboro.ac.uk/departments/ma/gallery/doubling/ provides a graphical simulation, based on the Cobwedding method).

**Figure 2.2:** Discrete Logistic Equation
$\begin{figure} \epsfxsize =8cm \centerline {\fbox {\epsffile{bifurcLog.eps}} } \vspace{-1mm} \end{figure}$

The parameter $\alpha$ is usually restricted to the interval

to ensure that $u_t\geq 0$ for all

, since otherwise the population becomes extinct.
The complicated behaviour exhibited by the logistic map is typical for a whole class of families of one-dimensional maps of a finite interval that have a single smooth maximum, known as unimodal maps.

Remarks 2.1

For the differential equations the dimension of the problem plays an important role in determining the dynamics. In one dimension the only possible limit set for bounded solutions are steady states and in two dimension only steady states and periodic solutions can be found as the limit set of bounded solutions. One shall require a dimension of at least three if we are to see complicated dynamical behaviour.
One of the crucial differences between discrete system and continuous systems, in one- and two-dimension, is the fact that it is plainly impossible for the dynamics of the differential equation to be chaotic!
For the difference equations the situation is different: for maps it is possible even in one dimension to obtain chaotic (or seemingly random) orbits.
With the setting of $c=(2\alpha-\alpha^2)/4$ and $y_n=\alpha /2-\alpha x_n \$ the discrete logistic law (2.2) may be rewritten as the quadratic law $y_{n+1} = y_n^2 + c$ (Euler's model). There are various important aspects of behaviour which can be introduced through the quadratic map and, however, there are many different problems to be solved with it. Let we fix the parameter c and start with : whenever there is an inaccuracy (as decimal representation of real number) in the wild iteration process, this error is dramatically amplified, due to the quadratic character of the expression. Unfortunately, an estimate of this error is usually not available without knowledge of the actual solution.
In order to draw a discrete analogy with the logistic equation (2.1) (i.e. which reproduces the same behaviour), one should consider the difference equation produced by a numerical method, and not the direct counterpart. The earliest numerical algorithm was the Euler's method. Let us apply the Euler's method to the equation (2.1), then it produces

    $\displaystyle u_{n+1}$ $\displaystyle =u_n + h u_n(\alpha - \beta u_n).$

Let us set h=1 and , thus

    $\displaystyle u_{n+1}$ $\displaystyle = u_n(\alpha ' - \beta u_n)$

bring again to the sequence (2.2), which dramatically has shown that, passing to discrete approximation, complex behaviour results as $\alpha$ is varied, while in the continuous equation the parameter $\alpha$ only influences scaling of the qualitative behaviour. This is not also a honest-to-goodness approximate model.(Other methods may be graphically tested through the applets regarding the logistic equation, see chapter 4 section The Applets.)
In contrast, the difference equation

(2.3) $\displaystyle y_{n+1}= \frac{(1+\beta) y_n}{1+ \frac{\beta}{\eta} y_n}$

is analogous to the logistic equation (2.1). Precisely the fixed points of (2.3) are and $u^1=\frac{\alpha}{\beta}$ . The stability is now discussed:

    $\displaystyle \frac{d}{d u} f (u^0)$ $\displaystyle = 1+ \beta > 1$

then is unstable,

    $\displaystyle \frac{d}{d u}f (u^1)$ $$\displaystyle = \frac{1}{1+ \beta}><br></TD> </TR> <TR> <TD COLSPAN=3 ALIGN=$
then is asymptotically stable for any , as desired.

Thus a choice of an appropriate difference equation (i.e. a numerical method which produces the difference) is a very important step in studying the qualitative behaviour of continuous systems.

Next: The Lotka-Volterra Equation Up: Classical Problems Previous: Classical Problems

Nardin Patrizia
2001-03-30