Generalized Logarithmic and Exponential Functions and Growth Models

Transcription

1 Generalized Logarithmic and Exponential Functions and Growth Models Alexandre Souto Martinez Rodrigo Silva González and Aquino Lauri Espíndola Department of Physics and Mathematics Ribeirão Preto School of Philosophy, Sciences and Literature University of São Paulo, Avenida Bandeirantes, 3900 Ribeirão Preto, São Paulo , Brazil 03/03/2010 Alexandre S. Martinez (USP) Generalized Functions 1 / 20

2 Contents 1 Introduction 2 Generalized Functions q-logarithm q-exponential generalized Euler s number 3 Continous Models Discretization of the Richards model The Generalized θ-ricker model Generalized Skellan Model 4 Conclusion Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 2 / 20

3 Introduction Introduction One-parameter logarithmic and exponential functions have been proposed in the context of: non-extensive statistical mechanics S(A + B) S(A) + S(B); relativistic statistical mechanics; quantum group theory. Two and three-parameter generalization of these functions have also been proposed. One parameter generalization of these functions can be obtained using simple geometrical arguments. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 3 / 20

4 Generalized Functions Non-Symmetrical Hyperboles and Power Laws Consider f q (t) = 1 t 1 q Figure: Behaviors of equation 1/t 1 q as a function of q. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 4 / 20

5 q-logarithm Function Generalized Functions q-logarithm ln q (x) is defined as the value of the area underneath the non-symmetric hyperbole in the interval t [1, x]: ln q (x) = x 1 dt t 1 q x q 1, if q 0, q ln q (x) = ln(x), if q = 0. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 5 / 20

6 Generalized Functions q-logarithm For any q, the area is for: 0 < x < 1, negative x = 1, null x > 1, positive Convergence and divergence: ( q < 0): x 0, ln q (x) x, ln q (x) 1/ q ( q = 0): x 0, ln q (x) x, ln q (x) + ( q > 0): x 0, ln q (x) 1/ q x, ln q (x) + Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 6 / 20

7 Generalized Functions q-exponential Function q-exponential Let x be the area underneath the curve f q (t), for t [0, e q (x)]: e q [ln q (x)] = ln q [e q (x)] = x e q (x) = lim [1 + q x] 1/ q q + q [a] + = max(a, 0) guarantees t > 0 in f q (t) = 1 t 1 q. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 7 / 20

8 Generalized Functions e q (x) > 0 (non-negative function): e q (0) = 1 q ±, e ± (x) = 1 q-exponential Figure: Behaviors of equation lim q q[1 + q x] 1/ q + as a function of q. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 8 / 20

9 Generalized Functions Generalized Euler s number Generalized Euler s Number The Euler s number: lim (1 + n 0 n)1/n = e = The generalized the Euler s number, for x = 1: e q = e q (1) = (1 + q) 1/ q. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 9 / 20

10 Continuous Models Richards Model (arxiv: ) d ln p(t) dt = κ ln q p(t), p(t) = N(t)/N, with: N(t) is the population size at time t N is the carrying capacity κ is the intrinsic growth rate Limiting cases for: ( q = 0) Gompertz model, d ln p/dt = κ ln p ( q = 1) Verhulst model, d ln p/dt = κ ln 1 (p) = κ (1 p) The model solution is the q-generalized logistic equation: p(t) = Notice that: 1/e q (x) = e q ( x). 1 e q [ln q (p 1 0 )e κt ] = e q[ ln q (p 1 0 )e κt ]. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 10 / 20

11 In short Continuous Models Alexandre S. Martinez (USP) Generalized Functions 11 / 20

12 Microscopic Model Continuous Models Alexandre S. Martinez (USP) Generalized Functions 12 / 20

13 Loquistic Map Continuous Models Richard Model To discretize d ln p(t)/dt= κ ln q p(t) dp/dt= κp ln q p: (p i+1 p i ) = κp i (p q [ 1) i 1+κ t, ρ =, x t q q q i =p ρ q 1 i ] q. ρ q This leads to the loquistic map: where ρ q = qρ q. x i+1 = ρ q x i(1 x q i ) = ρ q x i ln q (x i ), If ( q = 1) logistic map obtained from the discretization of Verhulst model. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 13 / 20

14 Continuous Models Loquistic Maps x i+1 = ρ q x i (1 x q ) i Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 14 / 20

15 Normalized Maps Continuous Models Loquistic Maps x i+1 = ρ q x i (1 x q ) i Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 15 / 20

16 θ-ricker Continuous Models The Generalized θ-ricker model θ-ricker model: x i+1 = x i e r[1 (x i /κ) θ]. κ 1 = e r, x = (r 1/θ x)/κ, relevant variable. x i+1 = κ 1 x i e x θ i. Limiting cases, for: θ = 1: standard Ricker model. expanding the exp. to the first order logistic map. θ 1, expanding the exp. to the first order: loquistic map All are scramble models. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 16 / 20

17 Generalized θ-ricker Continuous Models The Generalized θ-ricker model In Eq. x i+1 = κ 1 x i e r(x i /κ) θ : e( x) e q ( x) x i+1 = κ 1 x i e q [ r(x i /κ) θ ] = κ 1 x i [ 1 + qr ( x ) ] i θ 1/ q. (1) κ To obtain standard notation of combined models 1 write: c = 1/ q, κ 2 = r/(κc), κ 1 x i x i+1 = (1 + κ 2 xi θ ) c 1 A. Brannstrom and D. J. T. Sumpter. The role of competition and clustering in population dynamics. Proc. R. Soc. B 272 (2005) Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 17 / 20

18 Continuous Models The Generalized θ-ricker model Limiting Models x i+1 = κ 1 x i e q [ r(x i /κ) θ ] For θ = 1 Hassel model. q = 1, logistic model. q = 0, Ricker model. q = 1, Beverton-Holt model. For θ 1: q = 0, θ-ricker model. q = 1, Maynard-Smith-Slatkin model,. q = 1, loquistic model. For q, the trivial linear model. These models are either scramble or contest. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 18 / 20

19 Continuous Models Generalized Skellan Model Generalized Skellan Model All the models generalized are power-law-like models for q 0. The Skellan (contest) model, x i+1 = κ(1 e rx i ), could not be retrieved. However, if e( x) e q ( x): x i+1 = k [ 1 e q ( rx i ) ]. (2) q =, constant model. q = 1, the trivial linear. q = 0, one retrieves the Skellan model. q = 1 the Beverton-Holt contest model. Alexandre S. Martinez (USP) Generalized Functions asmartinez@usp.br 19 / 20

20 In Short Conclusion Alexandre S. Martinez (USP) Generalized Functions 20 / 20