A Discrete Equivalent of the Logistic Equation
© Eugenia N. Petropoulou. 2010
Received: 29 September 2010
Accepted: 10 November 2010
Published: 29 November 2010
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.
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.
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) , epidemiology [2, 3], atmospheric pollution, flow in a river , 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 .
has been known as the discrete logistic equation and it serves as an analogue to the initial value problem (1.2) (see, e.g., ).
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]
- (b)by discretizing (1.1) using a forward difference scheme for the derivative, which gives rise to the difference equation
where , being the step size of the scheme , 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:
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]).
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  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.
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  under the framework of a more general difference equation.
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
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 .
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 .
is an isomorphism from onto , that is, it is a 1 − 1 mapping from onto which preserves the norm .
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)
2.3. Existence and Uniqueness Theorems
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.
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] ).
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 ) .
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.
If conditions (2.16) and (2.21) hold, then the discrete equivalent logistic equation (2.11), has a unique solution in bounded by .
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)).
Following the same technique as the one applied for the proof of Theorems 2.6 and 2.7, conditions were given in , so that the difference equation (1.4) is to have a unique solution in or , . Indeed, it was proved that
It is obvious that conditions (2.16) and (2.21) are very similar to the conditions derived in .
3. Solution of the Discrete Equivalent Logistic Equation
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 .
which is a Bernoulli differential equation with respect to . (Remember that the original differential equation (1.2) was also of Bernoulli type!)
which is the solution of (3.1).
are of course satisfied.
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.
will be given.
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 ). However, when restricted to and is not necessary a sequence in the following more general result holds [7, pages 43–45]:
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."
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.
This work is dedicated to the memory of the author's Professor, P. D. Siafarikas, who left so early at the age of 57.
- Dykstra KH, Wang HY: Changes in the protein profile of streptomyces griseus during a cycloheximide fermentation. Annals of the New York Academy of Sciences 1987, 506: 511-522. 10.1111/j.1749-6632.1987.tb23846.xView ArticleGoogle Scholar
- Brauer F, Castillo-Chávez C: Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics. Volume 40. Springer-Verlag, New York, NY, USA; 2001:xxiv+416.View ArticleGoogle Scholar
- Hethcote HW: Three basic epidemiological models. In Applied Mathematical Ecology (Trieste, 1986), Biomathematics. Volume 18. Springer, Berlin, Germany; 1989:119-144.View ArticleGoogle Scholar
- Stoker JJ: Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics. Volume 4. Interscience Publishers, New York, NY, USA; 1957:xxviii+567.Google Scholar
- Gabriel JP, Saucy F, Bersier L-F: Paradoxes in the logistic equation? Ecological Modelling 2005, 185: 147-151. 10.1016/j.ecolmodel.2004.10.009View ArticleGoogle Scholar
- Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 155. Marcel Dekker, New York, NY, USA; 1992:xiv+777. Theory, methods, and applicationGoogle Scholar
- Elaydi S: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 3rd edition. Springer, New York, NY, USA; 2005:xxii+539.Google Scholar
- Mickens RE: Difference Equations. 2nd edition. Van Nostrand Reinhold, New York, NY, USA; 1990:xii+448. Theory and applicationMATHGoogle Scholar
- Mickens RE: Nonstandard Finite Difference Models of Differential Equations. World Scientific Publishing, River Edge, NJ, USA; 1994:xii+249.MATHGoogle Scholar
- Mickens RE: Discretizations of nonlinear differential equations using explicit nonstandard methods. Journal of Computational and Applied Mathematics 1999,110(1):181-185. 10.1016/S0377-0427(99)00233-2MathSciNetView ArticleMATHGoogle Scholar
- Patidar KC: On the use of nonstandard finite difference methods. Journal of Difference Equations and Applications 2005,11(8):735-758. 10.1080/10236190500127471MathSciNetView ArticleMATHGoogle Scholar
- Petropoulou EN, Siafarikas PD, Tzirtzilakis EE: A "discretization" technique for the solution of ODEs. Journal of Mathematical Analysis and Applications 2007,331(1):279-296. 10.1016/j.jmaa.2006.08.084MathSciNetView ArticleMATHGoogle Scholar
- Petropoulou EN, Siafarikas PD, Tzirtzilakis EE: A "discretization" technique for the solution of ODEs. II. Numerical Functional Analysis and Optimization 2009,30(5-6):613-631. 10.1080/01630560902987576MathSciNetView ArticleMATHGoogle Scholar
- Elaydi S: Stability and asymptoticity of Volterra difference equations: a progress report. Journal of Computational and Applied Mathematics 2009,228(2):504-513. 10.1016/j.cam.2008.03.023MathSciNetView ArticleMATHGoogle Scholar
- Kolmanovskii VB, Castellanos-Velasco E, Torres-Muñoz JA: A survey: stability and boundedness of Volterra difference equations. Nonlinear Analysis. Theory, Methods & Applications 2003,53(7-8):861-928. 10.1016/S0362-546X(03)00021-XView ArticleMATHGoogle Scholar
- Applelby JAD, Győri I, Reynolds DW: On exact convergence rates for solutions of linear systems of Volterra difference equations. Journal of Difference Equations and Applications 2006,12(12):1257-1275. 10.1080/10236190600986594MathSciNetView ArticleMATHGoogle Scholar
- Baker CTH, Song Y: Periodic solutions of discrete Volterra equations. Mathematics and Computers in Simulation 2004,64(5):521-542. 10.1016/j.matcom.2003.10.002MathSciNetView ArticleMATHGoogle Scholar
- Elaydi S, Murakami S: Uniform asymptotic stability in linear Volterra difference equations. Journal of Difference Equations and Applications 1998,3(3-4):203-218. 10.1080/10236199808808097MathSciNetView ArticleMATHGoogle Scholar
- Gil' MI, Medina R:Nonlinear Volterra difference equations in space . Discrete Dynamics in Nature and Society 2004,2004(2):301-306. 10.1155/S1026022604312021MathSciNetView ArticleMATHGoogle Scholar
- Győri I, Horváth L: Asymptotic representation of the solutions of linear Volterra difference equations. Advances in Difference Equations 2008, -22.Google Scholar
- Győri I, Reynolds DavidW: Sharp conditions for boundedness in linear discrete Volterra equations. Journal of Difference Equations and Applications 2009,15(11-12):1151-1164. 10.1080/10236190902932726MathSciNetView ArticleGoogle Scholar
- Kolmanovskii VB: Asymptotic properties of the solutions for discrete Volterra equations. International Journal of Systems Science 2003,34(8-9):505-511. 10.1080/00207720310001609048MathSciNetView ArticleMATHGoogle Scholar
- Kolmanovskii V: Boundedness in average for Volterra nonlinear difference equations. Functional Differential Equations 2005,12(3-4):295-301.MathSciNetMATHGoogle Scholar
- Kolmanovskii V, Shaikhet L: Some conditions for boundedness of solutions of difference Volterra equations. Applied Mathematics Letters 2003,16(6):857-862. 10.1016/S0893-9659(03)90008-5MathSciNetView ArticleMATHGoogle Scholar
- Medina R, Gil' M: Solution estimates for nonlinear Volterra difference equations. Functional Differential Equations 2004,11(1-2):111-119.MathSciNetMATHGoogle Scholar
- Messina E, Muroya Y, Russo E, Vecchio A: Asymptotic behavior of solutions for nonlinear Volterra discrete equations. Discrete Dynamics in Nature and Society 2008, -18.Google Scholar
- Murakami S, Nagabuchi Y: Stability properties and asymptotic almost periodicity for linear Volterra difference equations in a Banach space. Japanese Journal of Mathematics 2005,31(2):193-223.MathSciNetMATHGoogle Scholar
- Song Y: Almost periodic solutions of discrete Volterra equations. Journal of Mathematical Analysis and Applications 2006,314(1):174-194. 10.1016/j.jmaa.2005.03.073MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Baker CTH: Perturbation theory for discrete Volterra equations. Journal of Difference Equations and Applications 2003,9(10):969-987. 10.1080/1023619031000080844MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Baker CTH: Linearized stability analysis of discrete Volterra equations. Journal of Mathematical Analysis and Applications 2004,294(1):310-333. 10.1016/j.jmaa.2004.02.019MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Baker CTH: Admissibility for discrete Volterra equations. Journal of Difference Equations and Applications 2006,12(5):433-457. 10.1080/10236190600563260MathSciNetView ArticleMATHGoogle Scholar
- Xu Daoyi: Invariant and attracting sets of Volterra difference equations with delays. Computers & Mathematics with Applications 2003,45(6–9):1311-1317. 10.1016/S0898-1221(03)00104-4MathSciNetView ArticleMATHGoogle Scholar
- Elaydi S: Stability of Volterra difference equations of convolution type. In Dynamical Systems (Tianjin, 1990/1991), Nankai Series in Pure, Applied Mathematics and Theoretical Physics. Volume 4. World Scientific Publishing, River Edge, NJ, USA; 1993:66-72.Google Scholar
- Elaydi S, Messina E, Vecchio A: On the asymptotic stability of linear Volterra difference equations of convolution type. Journal of Difference Equations and Applications 2007,13(12):1079-1084. 10.1080/10236190701264529MathSciNetView ArticleMATHGoogle Scholar
- Elaydi S, Murakami S: Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type. Journal of Difference Equations and Applications 1996,2(4):401-410. 10.1080/10236199608808074MathSciNetView ArticleMATHGoogle Scholar
- Tang XH, Jiang Z: Asymptotic behavior of Volterra difference equation. Journal of Difference Equations and Applications 2007,13(1):25-40. 10.1080/10236190601008810MathSciNetView ArticleMATHGoogle Scholar
- Ifantis EK: On the convergence of power series whose coefficients satisfy a Poincaré-type linear and nonlinear difference equation. Complex Variables. Theory and Application 1987,9(1):63-80. 10.1080/17476938708814250MathSciNetView ArticleMATHGoogle Scholar
- Ifantis EK: An existence theory for functional-differential equations and functional-differential systems. Journal of Differential Equations 1978,29(1):86-104. 10.1016/0022-0396(78)90042-6MathSciNetView ArticleMATHGoogle Scholar
- Ifantis EK: Analytic solutions for nonlinear differential equations. Journal of Mathematical Analysis and Applications 1987,124(2):339-380. 10.1016/0022-247X(87)90004-7MathSciNetView ArticleMATHGoogle Scholar
- Ifantis EK: Global analytic solutions of the radial nonlinear wave equation. Journal of Mathematical Analysis and Applications 1987,124(2):381-410. 10.1016/0022-247X(87)90005-9MathSciNetView ArticleMATHGoogle Scholar
- Gohberg I, Goldberg S: Basic Operator Theory. Birkhäuser Boston, Boston, Mass, USA; 2001:xiv+285.MATHGoogle Scholar
- Earle CJ, Hamilton RdS: A fixed point theorem for holomorphic mappings. In Global Analysis Proceedings Symposium Pure Mathematics, Vol. XVI, Berkeley, Calif., (1968). American Mathematical Society, Providence, RI, USA; 1970:61-65.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.