The Logistic Model of Growth

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1 AU J.T. 11(2: (Oct. 27 The Logistic Model of Growth Graham Kenneth Winley Faculty of Science and Technology Assumption University Bangkok Thailand Abstract This article examines the popular logistic model of growth from three perspectives: its sensitivity to initial conditions; its relationship to analogous difference equation models; and the formulation of stochastic models with mean logistic growth. The results indicate that although the logistic model is appealing in terms of its simplicity its appeal is questionable in terms of its realism. Keywords: Real-valued function S-shaped data difference equations stochastic models. Introduction It is common in many fields of study to model the values of a variable of interest over time t using an autonomous differential equation of the form d ( = F( (1 where is a continuous real-valued function of time. In the case where realistically is an integer-valued function of time Eq. (1 is a model of the mean value of the integer-valued process. Assuming that the coefficient of variation of that process is small the continuous model in Eq. (1 is used to 1 d represent. If is constant then the growth in is said to be density independent and otherwise it is density dependent. In many situations experimental values of exhibit an S-shaped graphical representation and although many possible functions F( may be used in Eq. (1 to produce models with sigmoid growth curves the Verhulst (1838 logistic model in Eq. (2 below is certainly one of the most popular (Hazen 1975 d ( = λ [K ] (2 where usually = < K λ > K is referred to as the carrying capacity (i.e. an upper bound on the value of and the product λk is the intrinsic rate of increase. The model in Eq. (2 has been studied and used extensively over a long period of time by researchers in demography the biological sciences ecology genetics applied statistics (logistic regression software metrics and many other fields of study (Feller 194; Bertalanffy 1941; Lotka ; Keyfitz 1968; Pielou 1969; Hoppenstea 1975; Turner et al. 1976; Walpole et al. 22. However despite the popularity of the logistic model which as it will be seen is probably based more on its simplicity than its realism some of its features and its relationship to analogous models using difference equations and stochastic growth models are often not well understood by researchers who use the model simply on the basis of the plausibility of its sigmoid growth curve. In particular Feller (194 warned against blind faith in the use of the logistic. He considered S-shaped data from an experiment and then selected several S-shaped functions at random. Applying the usual criteria for best fit he ranked the various functions. The results were that the logistic fitted the data worse than the other selected functions. His conclusion was that the recorded agreement between the Review Article 99

2 AU J.T. 11(2: (Oct. 27 logistic and actually observed phenomena of growth does not produce any significant new evidence in support of the logistic beyond the plausibility of its deduction. The purpose of this article is to: (a examine the features of the graphical representation of the logistic model in Eq. (2 for various initial conditions; (b examine the formulation and features of models using difference equations which are considered to be analogous to Eq. (2; and (c investigate stochastic birth and death processes which have mean behavior represented by Eq. (2. Throughout the article an attempt has been made to select references which direct the reader to original sources and consequently provide the interested reader with an historical perspective on the topic. It is hoped that the article is of interest to researchers and students from various fields that use the very popular logistic model. In Fig.1 the point of inflection is in: the first quadrant provided K > 2 ; the second quadrant if < K < 2 ; and at ( if K = 2. The slope of the tangent at the point of inflection is λk 2 /4. The curve approaches the t- axis asymptotically as t - and it approaches = K asymptotically as t. The curve is approximately exponential for 1 K values of t < ln( 1 when the growth is λk in a transient stage. However if the initial conditions are = > K > and λ > then as shown in Fig. 2 the solution curve is no longer S-shaped. The Graphical Representation of the Logistic Model The solution to the differential equation (2 if = and λ > is (Verhulst 1838: K = + ( K exp( λk (3 and the graphical representation of for K > > is the usual S-shaped curve as shown in Fig λk K ln( 1 Fig.1. S-shaped logistic curve. 1 ln λ K K Fig. 2. on-s-shaped logistic curve. Difference Equations and the Logistic The rigorous usage of difference equations while defined as known relations between the values of an unknown function at a discrete pattern of values of the argument such as t τ t - 2τ t - 3τ where τ is a fixed known number allow the argument t to vary continuously. Recurrence relations on the other hand while defined in the same way do not allow t to vary continuously and in fact t will take on only equally spaced discrete values which are multiples of τ. However in accordance with common usage the same terms difference equation and recurrence relation will be used interchangeably. There is a considerable literature on the relationship between differential equations and difference equations (May 1974 May et al. 1974; Van Der Vaart Two fundamental questions arise: (a Starting with a differential Review Article 1

3 AU J.T. 11(2: (Oct. 27 equation how one can find the difference equation with the same solutions as the differential equation? and (b In what sense one could want the solutions to be the same? In what follows these questions in relation to Eq. (2 will be examined with different methods that are commonly used to develop a difference equation analogous to the solution of the logistic differential equation (2. Method 1 For the logistic differential equation (2 one can seek the difference equation whose solution at each time t has the same value as the solution in Eq. (3 to the said differential equation. Thus from Eq. (3 with t = t + τ K t + τ = + ( K exp[ λk( t + τ ] and ( ( K ( exp λ Kt =. ( ( K So the required difference equation is K exp( λτk t + τ = K + [ exp( λτk 1] which has the solution K exp( nλτk nτ = (4 K + [ exp( nλτk 1] for n = ote that if making the substitution n = t/τ in Eq. (4 and taking the limit of nτ as τ then one can reproduce the solution to the differential equation given in Eq. (3. Method 2 A different approach to the development of a difference equation analogous to the logistic is proposed by May (1974 and uses d t + τ = Lim τ τ which when applied to Eq. (2 gives the difference equation t + τ = [1 + λτk λτ]. (5 One can further analyze the behavior of Eq. (5 in order to compare it with the solution in Eq. (3 to the logistic differential equation. The following results are presented and the proof of R1(a is provided in the Appendix to illustrate the manner in which the interested reader may construct proofs for these results. R1. If λτk < 1 then: (a < t + τ < K for t when < < K < 1/(λτ. (b > t + τ > K for t when < K < < 1/(λτ. (c > K > t + τ for t when < K < 1/(λτ <. R2. If λτk >1 then: (a t + τ > K > for t when < 1/(λτ < < K. (b t + τ < K < for t when < 1/(λτ < K <. (c < (λτk < t + τ < K for t when < < 1/(λτ < K. R3. From R2(c it is seen that if < t + τ < 1/(λτ < K then K > t + 2τ > (λτkt + τ > (λτk 2 and if this pattern continues then t + nτ > (λτk n > and (λτk n > 1/(λτ for an integer ln( λτ n f. Under these conditions K > ln( λτk t + nτ > 1/(λτ and the subsequent behavior of is described by R2(a and R2(b. From R1(a and R1(c one can see that τ < 2τ < 3τ <... < K which resembles the behavior of in Eq. (3. However under the conditions in R1(b one can see that > τ > 2τ > 3τ > > K and from R2(a R2(b R2(c and R3 one can see that regardless of the positive value of the values of eventually oscillate asymptotically around the value of K. Consequently the difference equation (5 exhibits very different behavior to the difference equation (4 which exhibits exact agreement with the solution to the logistic differential equation. In fact there is no differential equation of the form in Eq. (1 which has a solution that exhibits exact agreement with the difference equation (5. This follows from the result that t + τ in Eq. (5 does not satisfy the group property (Van Der Vaart 1973 which requires that for t 2 > t 1 > [1 + t 2 λk - t 2 λ ] = [1 + (t 2 t 1 λk - (t 2 t 1 λt 1 ]t 1 where t 1 = [1 + t 1 λk t 1 λ ]. Review Article 11

4 AU J.T. 11(2: (Oct. 27 Stochastic Models and the Logistic To examine the relationship between stochastic models and the logistic one can consider stochastic models of birth and death processes encountered in the context of population growth where the population size at time t is represented by a random variable. One can further study such processes which have mean logistic behavior such that the expected value of E[] = M( satisfies the logistic differential equation (2 and so dm ( = λ [K M(] M(. (6 Suppose that: P k ( = P[ = k = ]; U k Δt + o(δ is the probability at time t that in a population of size k there will be a single birth in the next time interval Δt; D k Δt + o(δ is the probability at time t that in a population of size k there will be a single death in the next time interval Δt; and o(y is any function such that o(y/y as y. If the infinitesimal birth and death rates (U k D k are functions of t then they are said to be time inhomogeneous and otherwise they are time homogeneous. It follows that P k (t + Δ = U k-1 P k-1 (Δt + D k+1 P k+1 (Δt + [1 (U k + D k Δt]P k ( + o(δ and subtracting P k ( from both sides dividing through by Δt and then taking the limit as Δt approaches gives dpk ( = D 1 P 1 ( for k = ; and U k-1 P k-1 ( + D k+1 P k+1 ( - (U k + D k P k ( for k = (7 ext the following assumptions are made concerning a single individual in the population: u(δt + o(δ is the probability of a single individual producing a single birth in the time interval (t t + Δ; d(δt + o(δ is the probability of a single individual dying in the time interval (t t + Δ; and births and deaths of individuals are independent. This means that in Eq. (7 U k = ku( and D k = kd( and so Eq. (7 becomes dpk ( = d(p 1 ( for k = and (k 1u(P k-1 ( + (k + 1d(P k+1 ( k[u( + d(]p k ( for k = (8 Multiplying both sides of Eq. (8 by k and summing over the values of k gives dm ( dpk ( = k = [ u( d( ] M ( k where M ( = kpk( = E[ ( t ]. (9 k Also if K is the saturation level for this birth and death process then it is assumed that u( and d( decrease and increase respectively to the same limiting value. ext one can consider choices for u( and d( which result in M( satisfying the logistic differential equation (2. Choice 1 Consider the following choices for u( and d( u( = u γ 1 λm( d( = d + γ 2 λm( where u d = λk. (1 Then if γ 1 + γ 2 =1 one can have from Eq. (9 and Eq. (1 dm ( = [ u( d ( ] M ( = λ [ K M ( ] M ( and M( satisfies the logistic differential equation. Also provided 1 γ 1 u( and d( have the desired behavior and decrease and increase respectively to the same limiting value u(1 - γ 1 + γ 1 d. ote that the u( and d( in (1 are not the only choices that result in M( having logistic behavior but in all such cases u( and d( will depend on M( and this illustrates a problem which is discussed following the next subsection. Choice 2 The stochastic model usually regarded as the analogue of the logistic is described by Pielou (1969 and uses u( = u γ 1 λk d( = d + (1 - γ 1 λk where u d = λk (11 and if 1 γ 1 then u( and d( have same desired behavior as in (1. This means that the probabilities of birth and death for a single individual in the time interval t to t + Δt in a population of size k Review Article 12

5 AU J.T. 11(2: (Oct. 27 depend on the actual population size k rather than the mean population size M( as in (1. Realistically this is a more plausible assumption. However substituting Eq. (11 in Eq. (8 multiplying by k and summing over k using Eq. (9 gives dm ( S( = λ K M ( M ( where S( = E[ 2 (] > {E[]} 2 = M 2 ( is the second moment. Consequently dm ( p λ[ K M ( ] M ( and M( does not satisfy the logistic differential equation (2. Conclusions Regarding Stochastic Models of the Logistic Stochastic models have been described in Eq. (1 that have mean behavior corresponding to the logistic. Superficially these appear to be reasonable models since: they require that the individual birth rate is initially greater than the death rate; and that the birth rate decreases while the death rate increases with time to the same limiting value. At that stage the process becomes a symmetric random walk. However there is a serious defect in these models which makes their use for simulation a suspect. The models assume that the density dependent effect is based on the mean population size M( rather than the actual population size k. Unfortunately any attempt to take account of the actual population size in determining the transition probabilities results in the introduction of higher moments and the loss of the logistic relationship. This means that one cannot simulate processes on a computer relying only on information contained in the simulated population level; one must also evaluate independently the expression for the mean value. Consider the following experiment to illustrate this. Let 1 people be placed in a closed environment with some food. After 2 years an arbitrary person is singled out and one can try to guess the probability of this person dying during the next small time interval. Assuming the ability to observe the number of people at this time surely the most demographically plausible assumption would be that this probability should be proportional to the actual number of people. Thus if there were 1 people present there would be a higher probability of any individual dying than if there were only three people. Unfortunately insisting that the mean behavior must be logistic casts out this plausible assumption and denies that the actual number of people is significant. In its place it assumes that the mean number in the population determines the probability. In other words the logistic behavior implies that instead of counting the number of people present one must ignore this information and look at a clock to see how long it is since the beginning of the experiment this time then it allows one to compute the mean numbers from the logistic equation and only on this basis to calculate the resultant probability. So whether there are 3 or 1 people present logistic behavior implies that the probability of one of these dying is exactly the same and the same implication applies for deaths. The conclusion is that the logistic equation requires individual behavior that is not demographically acceptable. Conclusion This article has examined three aspects concerning the popular logistic model of growth: (a features of its graphical representation under various initial conditions; (b methods for the formulation of difference equations which are considered to be analogous to the logistic model; and (c the development of stochastic models which have mean behavior represented by the logistic equation. It is shown that the appealing S-shaped features of the logistic model are sensitive to the initial conditions. In particular the S- shaped graphical representation of the logistic is lost if the initial population size exceeds the carrying capacity. Two different methods for constructing difference equations analogous to the logistic were considered. The first method produced a difference equation model which agreed exactly with the solution to the logistic differential Review Article 13

6 AU J.T. 11(2: (Oct. 27 equation. However the second method despite the plausibility of its formulation produced a difference equation with very different behavior to the solution of the logistic differential equation. Consequently a clear warning is sounded for those interested in formulating difference equations with logistic behavior and such models are often a more realistic representation of growth phenomena. It is shown that it is possible to construct stochastic models with mean logistic behavior. This was done in the context of stochastic models of birth and death processes and it was demonstrated that in order to construct models with mean logistic behavior it is necessary to have the birth and death rates for individuals in the population dependent on the mean size of the population and this requirement is demographically unacceptable and poses problems for simulating the process. On the other hand if these rates are dependent on the actual population size which is more realistic then the mean size of the population does not satisfy the logistic equation. Thus the plausibility of the process having mean logistic behavior disappears when realistic assumptions are used for the birth and death rates of individuals in the population. These results indicate that although subject to appropriate initial conditions the logistic model is appealing. Its appeal is based more on its simplicity than its plausibility. It is hoped that these results will guide researchers and students to a deeper understanding of the logistic model of growth. References von Bertalanffy L Quantitative laws in metabolism and growth. Quart. Rev. Biol. 32: Van Der Vaart H.R A comparative investigation of certain difference equations and related differential equations: implications for model-building. Bul. Math. Biol. 35: Feller W On the logistic law of growth and its empirical verification. Acta Biotheoretica 5: Hoppenstea F Mathematical theories of population: demographics genetics and epidemics. Series 2. Society for Industrial and Applied Mathematics. Philadelphia PA. USA. Keyfitz Introduction to the mathematics of population. Addison- Wesley Reading MS USA. Lotka A Mode of growth of material aggregates. Amer. J. Sci. 24: Lotka A Elements of Mathematical Biology. Dover ew York Y USA. May R.M Stability and Complexity in Model Ecosystems. Princeton University Press Princeton J USA. May R.M.; Conway G.R.; Hassell M.P.; and Southwood T.R.E Time delays density-dependence and single-species oscillations. J. Anim. Ecol. 16: Pielou E.C An Introduction to Mathematical Ecology. Wiley ew York Y USA. Turner M.E.; Bradley E.L.; Kirk K.A.; and Pruitt K.M A theory of growth. Math. Biosci. 29: Verhulst P.F otice sur la loi que la population pursuit dans son accroissement. Correspondance Mathématique et Physique 1: Walpole R.E.; Myers R.H.; Myers S.L.; and Ye Keying. 22. Probability and Statistics for Engineers and Scientists. Prentice-Hall Englewood Cliff J USA. Appendix A proof for R1(a is provided to illustrate the manner in which the proofs for R1 R2 and R3 may be constructed. R1(a states that if λτk < 1 then < t + τ < K for t when < < K < 1/(λτ. Proof: If < < K < 1/(λτ then < λτ < λτk < 1 = [ K ] [ K ] and so < λτ[k ] < K and adding throughout gives < t + τ < K as stated. Review Article 14