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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  @xcite , relativistic statistical mechanics  @xcite and quantum group theory  @xcite . two and three - parameter generalization of these functions have also been proposed  @xcite . these generalizations are in current use in a wide range of disciplines since they permit the generalization of special functions : hyperbolic and trigonometric  @xcite , gaussian / cauchy probability distribution function  @xcite etc . also , they permit the description of several complex systems  @xcite , for instance in generalizing the stretched exponential function  @xcite . as mentioned above , the one - parameter generalizations of the logarithm and exponential functions are not univoquous . the @xmath1-logarithm function @xmath2 is defined as the value of the area underneath the non - symmetric hyperbole , @xmath3 , in the interval @xmath4 $ ] @xcite : @xmath5 this function is _ not _ the ordinary logarithmic function in the basis @xmath1 , namely @xmath6 $ ] , but a generalization of the natural logarithmic function definition , which is recovered for @xmath7 . the area is negative for @xmath8 , it vanishes for @xmath9 and it is positive for @xmath10 , independently of the @xmath1 values .    given the area @xmath11 underneath the curve @xmath12 , for @xmath13 $ ] , the upper limit @xmath14 is the generalized @xmath1-exponential function : @xmath15 . this is the inverse function of the @xmath1-logarithmic @xmath16=x=\ln_{\tilde q}[e_{\tilde q}(x)]$ ] and it is given by : @xmath17 this is a non - negative function @xmath18 , with @xmath19 , for any @xmath1 . for @xmath20 , one has that @xmath21 , for @xmath22 and @xmath23 , for @xmath24 . notice that letting @xmath9 one has generalized the euler s number : @xmath25    instead of using the standard entropic index @xmath26 in eqs . ( [ eq : gen_log ] ) and ( [ eq : eqtilde ] ) , we have adopted the notation @xmath27 . the latter notation permits us to write simple relations as : @xmath28 or @xmath29 , bringing the inversion point around @xmath30 . these relations lead to simpler expressions in population dynamics problems  @xcite and the generalized stretched exponential function  @xcite contexts . also , they simplify the generalized sum and product operators  @xcite , where a link to the aritmethical and geometrical averages of the generalized functions is established .    this logarithm generalization , as shown in ref . @xcite , is the one of non - extensive statistical mechanics  @xcite . it turns out to be precisely the form proposed by montroll and badger  @xcite to unify the verhulst ( @xmath31 ) and gompertz ( @xmath32 ) one - species population dynamics model . the @xmath33-logarithm leads exactly to the richards growth model  @xcite : @xmath34 where @xmath35 , @xmath36 is the population size at time @xmath37 , @xmath38 is the carrying capacity and @xmath39 is the intrinsic growth rate . the solution of eq . ( [ eq : richard_model ] ) is the _ @xmath1-generalized logistic _ equation @xmath40 }   = e_{-{\tilde q}}[-\ln_{\tilde q}(p_0^{-1})e^{-\kappa t } ] =   e_{-{\tilde q}}[\ln_{-\tilde q}(p_0)e^{-\kappa t}]$ ] . 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  @xcite . the long range interaction is dependent on the distance @xmath41 between two cells as a power law @xmath42 . these cells have a fractal structure characterized by a fractal dimension @xmath43 .    here we call the attention to eq .  ( 7 ) of ref .  @xcite , namely @xmath44\}$ ] , where @xmath45^{1-\gamma / d_f}-1\right\}/[d_f(1-\gamma / d_f)]$ ] . here , @xmath46 is a constant related to geometry of the problem , @xmath47 is the mean intrinsic replication rate of the cells and @xmath48 is the interaction factor . using eq . ( [ eq : gen_log ] ) , one can rewrite it simply as : @xmath49/{d_f}$ ] . calling , @xmath50 , @xmath51 and @xmath52 , this equation is the richard s model [ eq . ( [ eq : richard_model ] ) ] with an effort rate @xmath53 . in this context the parameter @xmath33 acquires a physical meaning related to the interaction range @xmath54 and fractal dimension of the cellular structure @xmath43 . if the interaction does not depend on the distance , @xmath55 , and it implies that @xmath56 . this physical interpretation of @xmath33 has only been possible due to richards model underlying microscopic description . introduced by nicholson in 1954  @xcite , 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 @xmath57 , 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 @xmath38 .    from experimental data , it is known that other than the important parameter @xmath39 ( and sometimes @xmath38 ) , 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 @xmath0-ricker model  @xcite . 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  @xcite , beverton - holt model @xcite and maynard - smith - slatkin model  @xcite .    our main purpose is to show that eq . ( [ eq : limite ] ) is suitable to unify most of the known discrete growth models into a simple formula . this is done in the following way . in sec . [ sec : loquistic ] , we show that the richards model [ eq . ( [ eq : richard_model ] ) ] , which has an underlying microscopic model , has a physical interpretation to the parameter @xmath33 , 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 @xmath1-exponential function . curiously , the map attractor can be suitably written in terms of @xmath33-exponentials , even in the logistic case . in sec . [ sec : generalized_theta_ricker ] , using the @xmath1-exponential function , we generalize the @xmath0-ricker model and analytically calculate the model fixed points , as well as their stability . in sec . [ sec : generalizedskellam ] , 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 sec  [ sec : conclusion ] . to discretize eq .  ( [ eq : richard_model ] ) , call @xmath58 and @xmath59^{\tilde q}$ ] , which leads to : @xmath60 where @xmath61 . we notice that @xmath1 keeps its physical interpretation of the continuous model .    in eq . ( [ eq : loquistic ] ) , if @xmath62 and @xmath63 , with @xmath64 $ ] , one obtains the _ logistic map _ , @xmath65 , 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 . @xcite . let us digress considering the feigenbaum s map  @xcite : @xmath66 , with @xmath67 , @xmath68 and @xmath69 . firstly , let us consider the particular case @xmath62 . if one writes @xmath70 , with @xmath71 being a constant , then : @xmath72 $ ] . imposing that @xmath73 leads to @xmath74 and calling @xmath75 , one obtains the logistic map with @xmath76 , so that @xmath77 . one can easily relate the control parameter of these two maps , making the maps equivalent . for arbitrary values of @xmath33 , there is not a general closed analytical form to expand @xmath78 and one can not simply transform the control parameters of eq . ( [ eq : loquistic ] ) to the feigenbaum s map . here , in general , these two maps are not equivalent . it would be then interesting , to study the sensitivity of eq . ( [ eq : loquistic ] ) with respect to initial conditions as it has been extensively studied in the feigenbaum s map  @xcite .    returning to eq . ( [ eq : loquistic ] ) , in the domain @xmath79 , @xmath80 ( non - negative ) , for @xmath81 . since @xmath82 is real only for @xmath83 , @xmath84 is real only for @xmath85 . the maximum value of the function is @xmath86 which occurs at @xmath87 i. e. , the inverse of the generalized euler s number @xmath88 [ eq . ( [ eq : eqtilde ] ) ] . for the generalized logistic map , @xmath89 , so that @xmath90 , it leads to the following domain for the control parameter @xmath91 : @xmath92    the map fixed points @xmath93 $ ] are @xmath94 the fixed point @xmath95 is stable for @xmath96 and @xmath97 is stable for @xmath98 , where @xmath99 notice the presence of the @xmath33-exponentials in the description of the attractors , even for the logistic map @xmath62 . 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 . [ fig1 ] displays the lyapunov exponents as function of the central parameter @xmath100 .    in fig . [ fig2 ] we have scaled the axis to @xmath101/ ( \rho_{max}\tilde{q})$ ] , where @xmath102 is given by eq . ( [ eq : rho_max ] ) and we plotted the bifurcation diagram for @xmath103 and @xmath104 . we see that the diagrams display the same structure but each one has its own scaling parameters . the role of increasing @xmath33 is to lift the bifurcation diagram to relatively anticipating the chaotic phase . the period doubling region start at @xmath105^{1/\tilde{q}}$ ] , so that for @xmath106 , @xmath107 and @xmath108 .    when @xmath109 , then @xmath110 . in fig . [ fig3 ] we show the histograms of the distribution of the variable @xmath111 . we see that as @xmath33 increases , the histograms have the same shape as the logistic histogram has , but it is crooked in the counter clock sense around @xmath112 . the _ @xmath0-ricker _ model  @xcite is given by : @xmath113 } ,    \label{eq : theta_ricker}\ ] ] where @xmath114 .    notice that @xmath115 is the relevant variable , where @xmath116 . in this way eq . ( [ eq : theta_ricker ] ) can be simply written as @xmath117 . for @xmath118 , one finds the standard _ model  @xcite . for arbitrary @xmath0 , expanding the exponential to the first order one obtains the generalized logistic map [ eq . ( [ eq : loquistic ] ) ] which becomes the logistic map , for @xmath118 . the @xmath0-ricker , ricker and quadratic models are all scramble competion models .    if one switches the exponential function for the @xmath119-generalized exponential in eq . ( [ eq : theta_ricker ] ) , one gets the _ generalized @xmath0-ricker model _ : @xmath120           = \frac{\kappa_1 x_i}{\left[1 + \tilde{q } r \left ( \frac{x_i}{\kappa}\right)^{\theta } \right]^{1/\tilde{q } } } \ ; . \label{eq : generalized_theta_ricker_model}\ ] ] to obtain standard notation , write @xmath121 and @xmath122 , so that @xmath123 @xcite . the generalized model with @xmath118 , leads to the _ hassel _ model  @xcite , which can be a scramble or contest competition model . one well - known contest competition model is the _ beverton - holt _ model  @xcite , which is obtained taking @xmath124 . for @xmath125 , one recovers the ricker model and for @xmath126 , one recovers the logistic model . it is interesting to mention that the beverton - holt model  @xcite is one of the few models that have the time evolution explicitly written : @xmath127 $ ] . from this equation , one sees that @xmath128 , for @xmath129 and @xmath130 for @xmath131 .    using arbitrary values of @xmath0 in eq . ( [ eq : generalized_theta_ricker_model ] ) , for @xmath125 one recovers the @xmath0-ricker model , and for @xmath132 , the _ maynard - smith - slatkin _ model  @xcite is recovered . the latter is a scramble / contest competition model . for @xmath126 , one recovers the generalized logistic map . the trivial linear model is retrieved for @xmath133 .    in terms of the relevant variable @xmath134 , eq . ( [ eq : generalized_theta_ricker_model ] ) is rewritten as : @xmath135 where @xmath136 and we stress that the important parameters are @xmath137 , @xmath33 and @xmath114 . ( [ eq : final ] ) is suitable for data analysis and the most usual known discrete growth models are recovered with the judicious choice of the @xmath1 and @xmath0 parameters as it shown in table  [ tabela ] . some typical bifurcation diagrams of eq . ( [ eq : final ] ) are displayed in fig . [ figbdtrm ] . .summary of the parameters to obtain discrete growth models from eq . ( [ eq : final ] ) . in the competition type column , _ s _ and _ c _ stand for scramble and contest models , respectively . the symbol @xmath138 stands for arbitrary values . [ cols="<,^,^ , < " , ] we have shown that the @xmath33-generalization of the exponential function is suitable to describe discrete growth models . the @xmath1 parameter is related to the range of a repulsive potential and the dimensionality of the fractal underlying structure . from the discretization of the richard s model , we have obtained a generalization for the logistic map and briefly studied its properties . an interesting generalization is the one of @xmath0-ricker model , which allows to have several scramble or contest competition discrete growth models as particular cases . equation ( [ eq : final ] ) allows the use of softwares to fit data to find the most suitable known model throughout the optimum choice of @xmath1 and @xmath0 . furthermore , one can also generalize the skellam contest model . only a few specific models mentioned in ref . @xcite 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 @xmath1 and @xmath0 . the authors thank c. a. s. terariol for fruitful discussions . asm acknowledges the brazilian agency cnpq ( 303990/2007 - 4 and 476862/2007 - 8 ) or support . rsg also acknowledges cnpq ( 140420/2007 - 0 ) for support . ale acknowledges cnpq for the fellowship and fapesp and mct / cnpq fundo setorial de infra - estrutura ( 2006/60333 - 0 ) .
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 dynamics 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 verhuslt 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 ) @xmath0-ricker discrete model and analytically calculate the fixed points as well as their stability . 
 in contrast to previous generalizations , from the generalized @xmath0-ricker model one is able to retrieve either scramble or contest models . 
 complex systems , population dynamics ( ecology ) , nonlinear dynamics 89.75.-k , 87.23.-n , 87.23.cc , 05.45.-a