A Discrete Equivalent of the Logistic Equation

Abstract

A discrete equivalent and not analogue of the well-known logistic differential equation is proposed. This discrete equivalent logistic equation is of the Volterra convolution type, is obtained by use of a functional-analytic method, and is explicitly solved using the -transform method. The connection of the solution of the discrete equivalent logistic equation with the solution of the logistic differential equation is discussed. Also, some differences of the discrete equivalent logistic equation and the well-known discrete analogue of the logistic equation are mentioned. It is hoped that this discrete equivalent of the logistic equation could be a better choice for the modelling of various problems, where different versions of known discrete logistic equations are used until nowadays.

1. Introduction

The well-known logistic differential equation was originally proposed by the Belgian mathematician Pierre-François Verhulst (1804–1849) in 1838, in order to describe the growth of a population under the assumptions that the rate of growth of the population was proportional to

the existing population and

the amount of available resources.

When this problem is "translated" into mathematics, results to the differential equation

(1.1)

where denotes time, is the initial population, and , are constants associated with the growth rate and the carrying capacity of the population. A more general form of (1.1), which will be used in this paper, is

(1.2)

where and , , are real constants with (in order to exclude trivial cases).

Equation (1.2) can be regarded as a Bernoulli differential equation or it can be solved by applying the simplest method of separation of variables. In any case, the solution of the initial value problem (1.2) is given by

(1.3)

Although, (1.2) can be considered as a simple differential equation, in the sense that it is completely solvable by use of elementary techniques of the theory of differential equations, it has tremendous and numerous applications in various fields. The first application of (1.2) was already mentioned, and it is connected with population problems, and more generally, problems in ecology. Other applications of (1.2) appear in problems of chemistry, medicine (especially in modelling the growth of tumors), pharmacology (especially in the production of antibiotic medicines) [1], epidemiology [2, 3], atmospheric pollution, flow in a river [4], and so forth.

Nowadays, the logistic differential equation can be found in many biology textbooks and can be considered as a cornerstone of ecology. However, it has also received much criticism by several ecologists. One may find the basis of these criticisms and several paradoxes in [5].

However, as it often happens in applications, when modelling a realistic problem, one may decide to describe the problem in terms of differential equations or in terms of difference equations. Thus, the initial value problem (1.2) which describes the population problem studied by Verhulst, could be formulated instead as an initial value problem of a difference equation. Also, there is a great literature on topics regarding discrete analogues of the differential calculus. In this context, the general difference equation

(1.4)

has been known as the discrete logistic equation and it serves as an analogue to the initial value problem (1.2) (see, e.g., [6]).

There are several ways to "end up" with (1.4) starting from (1.1) or (1.2) as:

• (a) by iterating the function , , which gives rise to the difference equation [7, page 43]

  

  (1.5)

• (b) by discretizing (1.1) using a forward difference scheme for the derivative, which gives rise to the difference equation

  

  (1.6)

  where , being the step size of the scheme [8], or

• (c) by "translating" the population problem studied by Verhulst in terms of differences: if is the population under study at time , its growth is indicated by . Thus, according to the assumptions and , the following initial value problem appears:

  

  (1.7)

  Notice of course that all three equations (1.5)–(1.7) are special cases of (1.4).

The similarities between (1.2) and (1.4) are obvious even at a first glance. However, these similarities are only superficial, since there are many qualitative differences between their solutions. Perhaps the most important difference between (1.2) and (1.4) is that in contrast to (1.2), (the solution of which is given explicitly in (1.3)) (1.4) (or even its simplest form (1.5)) cannot be solved explicitly so as to obtain its solution in closed form (except for certain values of the parameters) (see, e.g., [6, page 120] and [7, page 14]).

Also, (1.4) is one of the simplest examples of discrete autonomous equations leading to chaos, whereas the solution (1.3) of (1.2) guarantees the regularity of (1.2). Finally, it worths mentioning that the numerical scheme (1.6) or other nonlinear difference equations approximations of (1.2) given for example in [6, page 120] or in [8, pages 297–303] gives rise to approximate solutions of (1.2), which are qualitatively different from the true solution (1.3). These solutions are many times referred to as spurious solutions. These spurious solutions "disappear" when better approximations are used, for example, by applying nonstandard difference schemes (see, e.g., [9–11]).

Recently, in [12, 13] a nonstandard way was proposed for solving "numerically" an ordinary differential equation accompanied with initial or boundary conditions in the real or complex plane. This method was successfully applied to the Duffing equation, the Lorenz system, and the Blasius equation. The technique used is based on the equivalent transformation of the ordinary differential equation under consideration to an ordinary difference equation through an operator equation utilizing a specific isomorphism in specific Banach spaces. One of the aims of the present paper is to apply this technique to (1.2) so as to obtain the following equation:

(1.8)

where , are constants, which in the rest of the paper will be calleddiscrete equivalent logistic equation. It should be mentioned at this point that although the application of the technique in [12] to (1.2) is interesting on each own, its side effect, that is, the derivation of (1.8) is more important, since it is proposed as the discrete equivalent of (1.2). It is also emphasized that (1.8) is the discrete equivalent logistic equation derived by straightforward analytical means unlike the known versions of discrete logistic equation such as (1.4). Thus, the solutions of (1.8) are expected to have similar behavior with those of the differential logistic equation and not the peculiar characteristics appearing in the solutions of (1.4) discussed above. Conclusively it is the main aim of the present paper to convince the reader, that (1.8) deserves to be called discrete equivalent logistic equation. It is also hoped that (1.8) could be a better choice for the modelling of various problems, where different versions of known discrete logistic equations are used until nowadays.

Equation (1.8) is a nonlinear Volterra difference equation of convolution type. The Volterra difference equations have been thoroughly studied, and there exists an enormous literature for them. For example, there are several results concerning the boundedness, asymptotic behavior, admissibility, and periodicity of the solution of a Volterra difference equation. Although the list of papers cited in the present work is by no means exhaustive, the review papers [14, 15] on the boundedness, stability, and asymptoticity of Volterra difference equations should be mentioned (see also the references in these two papers). Indicatively, one could also mention the papers [16–32], the general results of which can also be applied to convolution-type Volterra difference equations. Also, in [33–36], linear Volterra difference equations of convolution type are exclusively studied.

In Section 2, (1.8) is fully derived. Moreover, in the same section conditions are given for the existence of a unique solution of (1.2) in the Banach space

(1.9)

where and of (1.8) in the Banach space

(1.10)

It should be mentioned at this point that the issue of the existence of a unique solution in of the discrete analogue logistic equation (1.4) has been studied in [37] under the framework of a more general difference equation.

In Section 3, (1.8) is explicitly solved by applying the -transform method. Finally, in Section 4, several differences between (1.4) and (1.8) are discussed. These differences concern their solutions (see Figure 1), their bifurcation diagrams, and their stability.

Figure 1

Solutions of (4. 1) and (4.2).

2. Derivation of the Discrete Equivalent Logistic Equation

In this section, the method proposed in [12, 13] will be applied to (1.2). As already mentioned in the introduction, the main idea is to transform (1.2) into an equivalent operator equation in an abstract Banach space and from this to deduce the equivalent difference equation (1.8). This method can be applied only when the ordinary differential equation under consideration is studied in the Banach space defined by (1.9). Moreover, the solution of (1.8), which will eventually give the solution of (1.2), belongs to the Banach space of absolutely summable sequences defined by (1.10).

2.1. Basic Definitions and Propositions

First of all, define the Hilbert space by

(2.1)

where . Denote now by an abstract separable Hilbert space over the real field, with the orthonormal base , Denote by and the inner product and the norm in , respectively. Define also in the shift operator and its adjoint

(2.2)

as well as the diagonal operator

(2.3)

Proposition 2.1.

The representation

(2.4)

is a one-by-one mapping from onto which preserves the norm, where , , is the complete system in of eigenvectors of and an element of [38].

The unique element appearing in (2.4) is called theabstract form of in . In general, if is a function from to and is the unique element in for which

(2.5)

then is called the abstract form of in .

Consider now the linear manifold of all which satisfy the condition . Define the norm . Then, this manifold becomes the Banach space defined by (1.9). Denote also by the corresponding by the representation (2.4), abstract Banach space of the elements for which .

The following properties hold [38–40]:

• (1) is invariant under the operators , , as well as under every bounded diagonal operator;

• (2) the abstract form of is the element , that is, ;

• (3) the abstract form of is the element , that is,

  

  (2.6)

• (4) the operator is the Frechét differentiable in .

Proposition 2.2.

The linear function

(2.7)

is an isomorphism from onto , that is, it is a 1 − 1 mapping from onto which preserves the norm [37].

Remark 2.3.

The basic Propositions 2.1 and 2.2 were originally proved for complex valued sequences and functions ( also in ), as well as for , defined over the complex field. However, in the present paper a restriction to the real plane is made due to the physical applications of the logistic equation.

2.2. Derivation of (1.8)

In order to apply the method of [12, 13] to the logistic differential equation (1.2), it is considered that , finite and (1.2) is restricted to by using the simple transformation , . Then, (1.2) becomes

(2.8)

Using Proposition 2.1 and what mentioned in Section 2.1, (2.8) is rewritten as

(2.9)

which holds for all , . But is the complete system in of eigenvectors of , which gives the following equivalent operator equation:

(2.10)

By taking the inner product of both parts of (2.10) with and taking into consideration Proposition 2.2 one obtains

(2.11)

where , , which is (1.8), the discrete equivalent logistic equation. It is obvious that in (2.11), it is and that , since and .

Of course, for all the above to hold, one has to assure that and . This is guaranteed by the theorems presented in the next section.

2.3. Existence and Uniqueness Theorems

As mentioned in Section 2.2, conditions must be found so that and . In order to do so, it is helpful to work with the operator equation (2.10), which is equivalent to both (2.8) and (2.11). Equation (2.10) can be rewritten as

(2.12)

where is the bounded operator , or as

(2.13)

due to the definition of , where is a constant which can be defined by taking the inner product of both parts of (2.13) with the element . Indeed, this gives

(2.14)

since . Thus (2.13) becomes

(2.15)

In order to assure the existence of a unique solution of the nonlinear operator equation (2.15) in , some conditions must be imposed on the parameters appearing in the equation. Moreover, since it is a non linear equation, a fixed-point theorem would be useful. Indeed, the following well-known theorems concerning the inversion of linear operators and the existence of a unique fixed point of an equation will be used.

Theorem 2.4.

If is a linear bounded operator of a Hilbert space or a Banach space , with , then is invertible with and is defined on all or (see, e.g., [41, pages 70-71] ).

Theorem 2.5.

If is holomorphic, that is, its Fréchet derivative exists, and lies strictly inside , then has a unique fixed point in , where is a bounded, connected, and open subset of a Banach space . (By saying that a subset of lies strictly inside , it is meant that there exists an such that for all and ) [42].

If it is assumed that

(2.16)

then and due to Theorem 2.4, the operator is defined on all and is bounded by . Thus, (2.15) takes the form

(2.17)

from which one finds that

(2.18)

Suppose that . Then, from (2.18) it is obvious that

(2.19)

Define the function , which attains its maximum at the point . Then, for , , it follows that if

(2.20)

or if

(2.21)

then (2.19) gives , which means that Theorem 2.5 is applied to (2.17). Thus, the following has just been proved.

Theorem 2.6.

If conditions (2.16) and (2.21) hold, then the abstract operator equation (2.10) has a unique solution in bounded by .

Equivalently, this theorem can be "translated" to the following two.

Theorem 2.7.

If conditions (2.16) and (2.21) hold, then the discrete equivalent logistic equation (2.11), has a unique solution in bounded by .

Theorem 2.8.

If conditions (2.16) and (2.21) hold, then the logistic differential equation (1.2) has a unique analytic solution of the form bounded by , which together with its first derivative converges absolutely for . (The coefficients are defined of course by (2.11)).

Remark 2.9.

Following the same technique as the one applied for the proof of Theorems 2.6 and 2.7, conditions were given in [37], so that the difference equation (1.4) is to have a unique solution in or , . Indeed, it was proved that

• (a) if and , then (1.4) has a unique solution in and

• (b) if and then (1.4) has a unique solution in , .

It is obvious that conditions (2.16) and (2.21) are very similar to the conditions derived in [37].

3. Solution of the Discrete Equivalent Logistic Equation

In this section, the discrete equivalent logistic equation (2.11), that is, equation

(3.1)

will be solved by applying the well-known -transform method (see, e.g., [6, pages 77–82], [7, Chapter 6], and [8, pages 159–172]). Suppose is the -transform of the unknown sequence . It is obvious that is required. However, since starts from 1, an "overstepping" should be made, by defining arbitrarily in such a way so that (3.1) is consistent. Indeed, by setting to (3.1), one obtains .

Equation (3.1) is of convolution type, and it can be rewritten as

(3.2)

Taking the -transform of both sides of (3.2), one obtains

(3.3)

which is a Bernoulli differential equation with respect to . (Remember that the original differential equation (1.2) was also of Bernoulli type!)

The solution of (3.3) is

(3.4)

where is the arbitrary constant of integration. This constant can be determined by using the following property of this -transform (since ):

(3.5)

from which it is easily obtained that . Thus, (3.4) becomes

(3.6)

It should be mentioned at this point that since according to Theorem 2.7, the function defined by (3.6) is analytic for (see [7, Theorem  6.14, page 292] ). By expanding , it is found that

(3.7)

which is the solution of (3.1).

Remark 3.1.

The well-known properties of the -transform

(3.8)

are of course satisfied.

Remark 3.2.

The solution (3.7) of (3.1) is expected, since the analytic solution (1.3) of (1.2) is known. However, the same technique can be applied to other nonlinear ordinary differential equations of interest for which their analytic solutions are not available. In this sense, a connection between specific ordinary differential and difference equations can be established.

4. Discussion

In this section, a comparison between the solutions of (1.4) and (1.8) will be given. However, since the most commonly used form of the discrete logistic equation is (1.5)

(4.1)

a comparison between the preceding equation and (1.8) for , that is,

(4.2)

will be given.

First of all, something should be mentioned regarding the stability of the equilibrium points of (4.1) and (4.2). At Remark 2.9, it was mentioned that if

(4.3)

equation (4.1) has a unique solution in , whereas if

(4.4)

equation (4.1) has a unique solution in .

This means that 0 and are locally asymptotically stable equilibrium points of (4.1) with regions of attraction given by (4.3) and (4.4), respectively. Actually, these results hold for , , and complex and not only real as regarded in the present paper (see [37]). However, when restricted to and is not necessary a sequence in the following more general result holds [7, pages 43–45]:

"The equilibrium point 0 of (4.1) is asymptotically stable for and unstable for , and the equilibrium point of (4.1) is asymptotically stable for and unstable for ."

In a similar way, due to Theorem 2.7, if

(4.5)

equation (4.2) has a unique solution in , and, thus, 0 is a locally asymptotically stable equilibrium point of the nonautonomous equation (4.2) with region of attraction given by (4.5).

In Figure 1, the solutions of (4.1) and (4.2) are graphically represented for some representative values of the parameters. More precisely in Figure 1(a), the solutions of (4.1) and (4.2) are given for and initial conditions . For these values, both conditions (4.3) and (4.5) are satisfied and thus these solutions of (4.1) and (4.2) both belong in . Moreover, it is obvious from Figure 1(a) that both and exhibit a very similar behavior and of course 0 is an asymptotically stable equilibrium point.

In Figures 1(b) and 1(c), the solutions of (4.1) and (4.2) are given for and initial conditions and , respectively. For these value of , the first condition of both (4.3) and (4.5) is violated. However, it is known that for this value of , the point is an asymptotically stable equilibrium point of (4.1) [7, pages 43–45]. In these cases, and do not exhibit a similar behavior, but both of them tend to a specific point as tends to infinity, to the equilibrium point 0.5 and to 0. This observation is a quite promising fact that the equilibrium point 0 of (4.2) may remain asymptotically stable even for values of the parameters that do not satisfy (4.5) (since this condition is not necessary and sufficient). However, this needs further study.

Finally, in Figure 1(d), the solutions of (4.1) and (4.2) are given for and initial conditions . For this value of , the first condition of both (4.3) and (4.5) is again violated. In this case, however, and exhibit again a very similar behavior, as they both seem to oscillate with continuously growing amplitude to infinity.

Last, but not least, it should be mentioned that as is well known, (4.1) exhibits chaotic behavior and this can be deduced from its period doubling bifurcation diagram [7, page 47]. However, numerical results for (4.2) indicate no chaotic behavior. Actually, its bifurcation diagram is a straight line at 0. For (which is a region of values for which condition (4.5) is violated), starts for some initial conditions to "blow up."

5. Conclusions

In this paper, a discrete equivalent to the well-known logistic differential equation is proposed. This discrete equivalent equation is of the Volterra convolution type and is obtained using a functional-analytic technique. From what mentioned in Sections 3 and 4, it seems that this discrete equivalent logistic equation better resembles the behaviour of the corresponding logistic differential equation in the sense that (a) it can be solved explicitly and (b) it does not seem to present chaotic behaviour. This author believes that although the discrete equivalent logistic equation was derived in a very specific way under the conditions of Theorem 2.6, it would be interesting to further study this equation on its own regardless of conditions with respect to the appearing parameters. In other words, the study of the discrete equivalent logistic equation, not only in the space but also in several other spaces of sequences, could give rise to interesting results. It would also be interesting to investigate the possibility, (1.8) being useful in the study of biology or physics problems.