Physica A. Generalized exponential function and discrete growth models

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1 Physica A 388 (29) Contents lists available at ScienceDirect Physica A journal homepage: Generalized exponential function and discrete growth models Alexandre Souto Martinez a,b, Rodrigo Silva González a, Aquino Lauri Espíndola a, a Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes, 39, 44-9, Ribeirão Preto, São Paulo, Brazil b National Institute of Science and Technology for Complex Systems Avenida Bandeirantes, 39, 44-9, Ribeirão Preto, São Paulo, Brazil a r t i c l e i n f o a b s t r a c t Article history: Received 8 July 28 Received in revised form 5 March 29 Available online 28 March 29 PACS: k n Cc a Keywords: Complex systems Population dynamics (ecology) Nonlinear dynamics Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamic models into a simple formula. A physical interpretation is given to this new introduced parameter in the context of the continuous Richards model, which remains valid for the discrete case. From the discretization of the continuous Richards model (generalization of the Gompertz and Verhulst models), one obtains a generalized logistic map and we briefly study its properties. Notice, however that the physical interpretation for the introduced parameter persists valid for the discrete case. Next, we generalize the (scramble competition) θ-ricker discrete model and analytically calculate the fixed points as well as their stabilities. In contrast to previous generalizations, from the generalized θ-ricker model one is able to retrieve either scramble or contest models. 29 Elsevier B.V. All rights reserved.. Introduction Recently, the generalizations of the logarithmic and exponential functions have attracted the attention of researchers. One-parameter logarithmic and exponential functions have been proposed in the context of non-extensive statistical mechanics [ 5], relativistic statistical mechanics [6,7] and quantum group theory [8]. Two- and three-parameter generalizations of these functions have also been proposed [9 ]. These generalizations are currently in use in a wide range of disciplines since they permit the generalization of special functions: hyperbolic and trigonometric [2], Gaussian/Cauchy probability distribution function [3] etc. Also, they permit the description of several complex systems [4 9], for instance in generalizing the stretched exponential function [2]. As mentioned above, the one-parameter generalizations of the logarithm and exponential functions are not univoquous. The q-logarithm function ln q (x) is defined as the value of the area underneath the non-symmetric hyperbola, f q (t) = /t q, in the interval t [, x] [2]: ln q (x) = x dt x q = lim. t q q q q This function is not the ordinary logarithmic function in the basis q, namely [log q (x)], but a generalization of the natural logarithmic function definition, which is recovered for q =. The area is negative for < x <, it vanishes for x = and it is positive for x >, independently of the q values. () Corresponding address: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Departamento de Fisica e Matematica, Avenida Bandeirantes, 39, 44-9, Ribeirão Preto, São Paulo, Brazil. addresses: asmartinez@ffclrp.usp.br (A. Souto Martinez), caminhos_rsg@yahoo.com.br (R. Silva González), aquinoespindola@usp.br (A. Lauri Espíndola) /$ see front matter 29 Elsevier B.V. All rights reserved. doi:.6/j.physa

2 A. Souto Martinez et al. / Physica A 388 (29) Given the area x underneath the curve f q (t), for t [, y], the upper limit y is the generalized q-exponential function: y = e q (x). This is the inverse function of the q-logarithmic e q [ln q (x)] = x = ln q [e q (x)] and it is given by: { for qx < e q (x) = lim ( + q x) / q for qx. (2) q q This is a non-negative function e q (x), with e q () =, for any q. For q ±, one has that e (x) =, for x and e (x) =, for x. Notice that letting x = one has generalized Euler s number: e q () = ( + q) / q. Instead of using the standard entropic index q in Eqs. () and (3), we have adopted the notation q = q. The latter notation permits us to write simple relations as: ln q (x) = ln q (x) or e q ( x) = /e q (x), bringing the inversion point around q =. These relations lead to simpler expressions in population dynamics problems [22] and the generalized stretched exponential function [2] contexts. Also, they simplify the generalized sum and product operators [2], where a link to the arithmetical and geometrical averages of the generalized functions is established. This logarithm generalization, as shown in Ref. [23, p. 83], is the one of non-extensive statistical mechanics [2]. It turns out to be precisely the form proposed by Montroll and Badger [24] to unify the Verhulst ( q = ) and Gompertz ( q = ) one-species population dynamics model. The q-logarithm leads exactly to Richards growth model [25,22]: d ln p(t) = κ ln q p(t), (4) dt where p(t) = N(t)/N, N(t) is the population size at time t, N is the carrying capacity and κ is the intrinsic growth rate. The solution of Eq. (4) is the q-generalized logistic equation p(t) = /e q [ln q (p )e κt ] = e q [ ln q (p )e κt ] = e q [ln q (p )e κt ]. The competition among cells drive to replicate and inhibitory interactions, that are modeled by long range interaction among these cells. These interactions furnish an interesting microscopic mechanism to obtain Richards model [26,27]. The long range interaction is dependent on the distance r between two cells as a power law r γ. These cells have a fractal structure characterized by a fractal dimension D f. Here we call the attention to Eq. (7) of Ref. [26], namely ṅ(t) = n(t){ G JI[n(t)]}, where I(n(t)) = ω { [D f n(t)/ω] γ /D f } /[D f ( γ /D f )]. Here, ω is a constant related to geometry of the problem, G is the mean intrinsic replication rate of the cells and J is the interaction factor. Using Eq. (), one can rewrite it simply as: d ln n(t)/dt = G /n(t) Jω ln q [D f n(t)/ω]/d f. Calling, p = D f n/ω, κ = Jω/D f and q = γ /D f, this equation is Richard s model [Eq. (4)] with an effort rate G /n(t). In this context the parameter q acquires a physical meaning related to the interaction range γ and fractal dimension of the cellular structure D f. If the interaction does not depend on the distance, γ =, and it implies that q =. This physical interpretation of q has only been possible due to Richards model underlying microscopic description. Introduced by Nicholson in 954 [28], scramble and contest are types of intraspecific competition models that differ between themselves in the way that limited resources are shared among individuals. In scramble competition, the resource is equally shared among the individuals of the population as long as it is available. In this case, there is a critical population size N c, above which, the amount of resource is not enough to assure population survival. In the contest competition, stronger individuals get the amount of resources they need to survive. If there is enough resources to all individuals, population grows, otherwise, only the strongest individuals survive (strong hierarchy), and the population maintains itself stable with size N. From experimental data, it is known that other than the important parameter κ (and sometimes N ), additional parameters in more complex models are needed to adjust the model to the given population. One of the most general discrete model is the θ-ricker model [29,3]. This model describes well scramble competition models but it is unable to put into a unique formulation the contest competition models such as Hassel model [28], Beverton Holt model [3] and Maynard-Smith Slatkin model [32]. Our main purpose is to show that Eq. (2) is suitable to unify most of the known discrete growth models into a simple formula. This is done in the following way. In Section 2, we show that Richards model [Eq. (4)], which has an underlying microscopic model, has a physical interpretation to the parameter q, and its discretization leads to a generalized logistic map. We briefly study the properties of this map and show that some features of it (fixed points, cycles etc.) are given in terms of the q-exponential function. Curiously, the map attractor can be suitably written in terms of q-exponentials, even in the logistic case. In Section 3, using the q-exponential function, we generalize the θ-ricker model and analytically calculate the model fixed points, as well as their stability. In Section 4, we consider the generalized Skellam model. These generalizations allow us to recover most of the well-known scramble/contest competition models. Final remarks are presented in Section Discretization of Richards model To discretize Eq. (4), call (p i+ p i )/ t = kp i (p q i )/ q, ρ q = + k t/ q and x i = p i [(ρ q )/ρ q ] q, which leads to: (3) x i+ = ρ q x i( x q i ) = ρ qx i ln q (x i ), where ρ q = qρ. We notice that q keeps its physical interpretation of the continuous model. q (5)

3 2924 A. Souto Martinez et al. / Physica A 388 (29) In Eq. (5), if q = and ρ = ρ = 4a, with a [, ], one obtains the logistic map, x i+ = 4ax i ( x i ), which is the classical example of a dynamic system obtained from the discretization of the Verhulst model. Although simple, this map presents a extremely rich behavior, universal period duplication, chaos etc. [33]. Let us digress considering Feigenbaum s map [34]: y i+ = µy q+ i, with q >, < µ 2 and y i. Firstly, let us consider the particular case q =. If one writes y i = ỹ i b, with b being a constant, then: ỹ i+ = µb 2 + b + + 2µbỹ i [ ỹ i /(2b)]. Imposing that µb 2 + b + = leads to b ± = ( + 4µ)/(2µ) and calling x i = ỹ i /(2b), one obtains the logistic map with ρ = ρ = 4a = + + 4µ, so that < ρ 4. One can easily relate the control parameter of these two maps, making the maps equivalent. For arbitrary values of q, there is not a general closed analytical form to expand ỹ i b q+ and one cannot simply transform the control parameters of Eq. (5) to Feigenbaum s map. Here, in general, these two maps are not equivalent. It would then be interesting, to study the sensitivity of Eq. (5) with respect to initial conditions as it has been extensively studied in Feigenbaum s map [35 38]. Returning to Eq. (5), in the domain x, f (x i ) = ρ q x i ln q (x i ) (non-negative), for ρ q >. Since e q (x) is real only for qx >, f is real only for q >. The maximum value of the function is f = f ( x) = which occurs at x = e q (), ρ q e ( q)e q (), i. e., the inverse of the generalized Euler s number e q () [Eq. (3)]. For the generalized logistic map, x, so that f, it leads to the following domain for the control parameter ρ q ρ max : ρ max = e q ()e ( q) = ( + q) +/ q. The map fixed points [x = f (x )] are x =, x 2 = e q( /ρ q ). The fixed point x is stable for ρ q < q and x 2 is stable for q ρ q < ρ pd, where ρ pd = q + 2. Notice the presence of the q-exponentials in the description of the attractors, even for the logistic map q =. The generalized logistic map also presents the rich behavior of the logistic map as depicted by the bifurcation diagram of Fig.. The inset of Fig. displays the Lyapunov exponents as function of the central parameter ρ q. In Fig. 2 we have scaled the axis to ρ q [ ( q + 2)/ q]/(ρ max q), where ρ max is given by Eq. (8) and we plotted the bifurcation diagram for q = /, and. We see that the diagrams display the same structure but each one has its own scaling parameters. The role of increasing q is to lift the bifurcation diagram to relatively anticipating the chaotic phase. The period doubling region starts at x 2 ( q) = e q[ /( q + 2)] = [ /( + 2/ q)] / q, so that for x 2 (/) = (2/2).6, x 2 () = 2/3.67 and x 2 () = (/6)/.84. When ρ q = e q ()e ( q), then x i (, ). In Fig. 3 we show the histograms of the distribution of the variable x i. We see that as q increases, the histograms have the same shape as the logistic histogram has, but it is crooked in the counter clock sense around x = /2. 3. The generalized θ-ricker model The θ-ricker model [29,3] is given by: x i+ = x i e r[ (x i/κ) θ ], where θ >. Notice that x = r /θ x/κ is the relevant variable, where κ = e r >. In this way Eq. (2) can be simply written as x i+ = k x i e xθ i. For θ =, one finds the standard Ricker model [39]. For arbitrary θ, expanding the exponential to the first order one obtains the generalized logistic map [Eq. (5)] which becomes the logistic map, for θ =. The θ-ricker, Ricker and quadratic models are all scramble competition models. If one switches the exponential function for the q-generalized exponential in Eq. (2), one gets the generalized θ-ricker model: ( xi ) θ ] κ x i x i+ = κ x i e q [ r = [ κ + ( x qr i ) ] θ / q. (3) κ To obtain standard notation, write c = / q and k 2 = r/(kc), so that x i+ = k x i /( + k 2 x i ) c [4]. (6) (7) (8) (9) () () (2)

4 A. Souto Martinez et al. / Physica A 388 (29) X i ρq ~ Lyapunov Exponent ρq ~ Fig.. Bifurcation diagram of Eq. (5) for q = 2, where we see that the period doubling starts at ρ = ( q + 2)/ q = 2 [Eq. ()] and the chaotic phase finishes at ρ max = e 2 ()e (2)/ q = 27/2 2.6 [Eq. (8)]. Inset: Lyapunov exponents as functions of ρ q for q = 2..9 q ~ = q ~ =/2 q ~ = (ρq ~ ~ ~ (q+2)/q)/ ρmax Fig. 2. Bifurcation diagram of Eq. (5) for q = /2, q = (logistic) and q = 2. The fixed points are given by Eqs. (9) and (). The generalized model with θ =, leads to the Hassel model [28], which can be a scramble or contest competition model. One well-known contest competition model is the Beverton Holt model [3], which is obtained taking q = c =. For q =, one recovers the Ricker model and for q =, one recovers the logistic model. It is interesting to mention that the Beverton Holt model [3] is one of the few models that have the time evolution explicitly written: x i = κ i x /[ + ( κ i ) x /( κ )]. From this equation, one sees that x i( ) =, for κ and x i( ) = κ for κ. Using arbitrary values of θ in Eq. (3), for q = one recovers the θ-ricker model, and for q =, the Maynard- Smith Slatkin model [32] is recovered. The latter is a scramble/contest competition model. For q =, one recovers the generalized logistic map. The trivial linear model is retrieved for q.

5 2926 A. Souto Martinez et al. / Physica A 388 (29) q ~ =. q ~ =. q ~ = frequency X X X Fig. 3. Histograms of x for ρ q = e q ()e ( q) in Eq. (5) and q = /, q = (logistic) and q =. There is no a drastic change with respect to the logistic map. Only at the corners one is able to see the difference as shown by the insets. Table Summary of the parameters to obtain discrete growth models from Eq. (4). In the competition type column, s and c stand for scramble and contest models, respectively. The symbol * stands for arbitrary values. Model q θ Competition type Linear > Logistic s Generalized logistic > s Ricker s θ-ricker > s Hassel * s ( q < /2) or c ( q /2) Maynard Smith Slatkin > s (θ > 2) or c (θ 2) Beverton Holt c In terms of the relevant variable x, Eq. (3) is rewritten as: x i+ = κ x i e q ( x θ i ), where x i and we stress that the important parameters are κ >, q and θ >. Eq. (4) is suitable for data analysis and the most usual known discrete growth models are recovered with the judicious choice of the q and θ parameters as it shown in Table. Some typical bifurcation diagrams of Eq. (4) are displayed in Fig. 4. Now, let us obtain some analytical results for the map of Eq. (4), which we write as x i+ = f gtr ( x i ), with (4) f gtr ( x) = κ xe q ( x θ ) = κ x ( + q x θ ) / q. (5) The x-domain is unbounded ( x ), for q. However, for q <, f gtr ( x) = κ x( q x θ ) / q and the x-domain is bounded to the interval: x x m, with x m = ( q) /θ, so that for q <, x m > ; for q =, with x m = (for q =, it is the generalized logistic case [Eq. (5) in θ instead of q] and for q =, the Maynard-Smith Slatkin model) and for q >, x m <. The derivative of f gtr with respect to x is: f ( x) = κ gtr [ + ( q θ) x θ ]/( + q x θ ) +/ q. Imposing f gtr ( x max) =, one obtains the maximum of Eq. (5), f gtr = f gtr ( x max ) = κ x m e q ( /θ). e θ ( / q) (6) (7)

6 A. Souto Martinez et al. / Physica A 388 (29) a 4 2 q ~ =/2 θ= κ b 5 4 q ~ = θ=3 3 2 c κ q ~ =2 θ= κ at x max = Fig. 4. Typical bifurcation diagrams of Eq. (4). (a) q = /2 and θ = 2, (a) q = and θ = 3 and (a) q = 2 and θ = 5. (θ q) /θ = x m e θ ( / q). The control parameter κ is unbounded (κ > ), for q, but for q <, since fgtr x m, it belongs to the interval (8)

7 2928 A. Souto Martinez et al. / Physica A 388 (29) X 2 * X * e -q ~ (2/θ) κ m κ Fig. 5. Fixed points, given by Eqs. (2) and (2), as functions of k. Instability regions are represented by dashed lines. The value of κ m is given by Eq. (9). < κ κ m, where: κ m = e θ( / q) e q ( /θ) = e θ( / q)e q (/θ). (9) From Eq. (8), one sees that for θ < q, f gtr ( x) does not have a hump, it is simply a monotonically increasing function of x, which characterizes the contest models. At the critical θ = q value, the function f gtr ( x) starts to have a maximum value at infinity. For θ > q, the function f gtr ( x) has a hump, with maximum value at x max [Eq. (8)] such that as θ then x max. The map fixed points [ x = f gtr ( x )] are: x = (2) x 2 = [ln q(κ )] /θ. (2) These fixed points are show as function of κ in Fig. 5. The fixed point x represents the species extinction and is stable, for < κ <. Both fixed points x and x 2 are marginal, for κ =. For < κ < e q (2/θ), x becomes unstable and x 2 > is stable and represents the species survival. For κ = e q (2/θ), x 2 becomes unstable and as κ increases, a stable cycle-2 appears. For q <, as κ increases further, the cycle-2 becomes unstable at some value of κ giving rise to a route to chaos as in the logistic map, via period doubling. Nevertheless, for q, several scenarios may take place. Even though q > and θ > q, if q < θ < 2 q, the map f gtr has a hump, but it is not thin enough to produce periods greater than unity. In this case, f gtr produces only the two fixed points x and x 2, which characterize the context models. Nevertheless, scramble models (θ > 2 q) have maps with a hump thin enough to produce stable cycles with period greater than unity. Thus, in scramble models one has more complex scenarios such as period doubling, as a route to chaos, as κ > e q (2/θ). We have not being able to obtain analytically the behavior of the system q > and θ > 2 q. The θ q diagram is depicted in Fig. 6. For q = θ =, one retrieves the Beverton Holt model, with the fixed points x = and x = ln (κ ) = κ. For q, one retrieves the generalized logistic map. 4. Generalized Skellam model All the contest competition models generalized by Eq. (3) are power-law-like models for q. However, the Skellam contest model cannot be obtained from this approach. It is the complement of an exponential decay x i+ = κ( e rx i ) [4]. Nevertheless, it is interesting to replace the exponential function to the q-exponential in this model: x i+ = k[ e q ( rx i )] and write x = rx and κ = rk, which leads to: x i+ = κ [ e q ( x i ) ]. (22) For q, Eq. (22) leads to the constant model, for q =, the trivial linear growth is found. If q =, one recovers the Skellam model and finally, q = leads to the Beverton Holt contest model (see Table 2).

8 A. Souto Martinez et al. / Physica A 388 (29) θ θ = 2 q ~ Hassel model Generalized Logistic map θ-ricker map Maynard-Smith-Slatkin θ = q ~ Logistic Ricker Beverton-Holt X m = (-q ~ ) -/θ e -θ (-/q ~ ) - X m = q ~ Fig. 6. Diagram distinguishing the several types of behavior of the generalized θ-ricker model [Eq. (4)]. For q <, one has the map x i+ = κ x i ( q x θ i )/ q and x x m, with x m given by Eq. (6). This system represents scramble models and achieves chaos, period doubling. For q, x and the behavior of the system is driving by the θ values. For θ 2 q, we do no have analytical results. Numerical simulation indicate regions with finite order of period doubling. For θ < 2 q, the system represent context models, with only two fixed points given by Eqs. (2) and (2). The map presents a hump, for θ > 2 q, but for θ q, the map is a monotonically increasing function. Table 2 Summary of the parameters to obtain contest competition discrete growth models from Eq. (22). Model Constant Linear Skellam Beverton Holt q 5. Conclusion We have shown that the q-generalization of the exponential function is suitable to describe discrete growth models. The q parameter is related to the range of a repulsive potential and the dimensionality of the fractal underlying structure. From the discretization of Richard s model, we have obtained a generalization for the logistic map and briefly studied its properties. An interesting generalization is the one of θ-ricker model, which allows us to have several scramble or contest competition discrete growth models as particular cases. Eq. (4) allows the use of softwares to fit data to find the most suitable known model throughout the optimum choice of q and θ. Furthermore, one can also generalize the Skellam contest model. Only a few specific models mentioned in Ref. [4] are not retrieved from our generalization. Actually, we propose a general procedure where we do not necessarily need to be tied to a specific model, since one can have arbitrary values of q and θ. Acknowledgments The authors thank C.A.S. Terçariol for the fruitful discussions. ASM acknowledges the Brazilian agency CNPq (3399/27-4 and /27-8) for support. RSG also acknowledges CNPq (442/27-) for support. ALE acknowledges CNPq for the fellowship and FAPESP and MCT/CNPq Fundo Setorial de Infra-Estrutura (26/6333-). References [] C. Tsallis, J. Statist. Phys. 52 (988) [2] C. Tsallis, Química Nova 7 (994) [3] L. Nivanen, A.L.M.A., Q.A. Wang, Rep. Math. Phys. 52 (23) [4] E.P. Borges, Physica A 34 (24) 95. [5] N. Kalogeropoulos, Physica A 356 (25)

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