Theory Biosci. (2014) 133:117–124 DOI 10.1007/s12064-014-0199-6
ORIGINAL PAPER
Generalized Allee effect model Lindomar S. dos Santos • Brenno C. T. Cabella Alexandre S. Martinez
•
Received: 21 August 2013 / Accepted: 19 February 2014 / Published online: 17 March 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The Allee effect consists of a positive correlation between very small population size and fitness. Offering a new view point on the weak and strong demographic Allee effect, we propose to combine them with the Richards growth model. In particular, a peculiar manifestation of the Allee effect is analytically predicted and still not validated by experiments. Model validation with ecological data is presented for some special situations. Keywords Strong and weak Allee effects Growth models Generalized functions Richards model Cooperation–competition transition Unified models
Introduction Growth and population dynamics models are important tools used in the study of several time-dependent processes in different research areas. The main simplification in the population growth study is to explicitly represent only one species. In this case, the environment carrying capacity
L. S. dos Santos (&) B. C. T. Cabella A. S. Martinez Faculdade de Filosofia, Cieˆncias e Letras de Ribeira˜o Preto (FFCLRP), Universidade de Sa˜o Paulo (USP), Avenida dos Bandeirantes, 3900, 14.040-901, Ribeira˜o Preto, SP, Brazil e-mail: [email protected] B. C. T. Cabella e-mail: [email protected] A. S. Martinez e-mail: [email protected] A. S. Martinez Instituto Nacional de Cieˆncia e Tecnologia em Sistemas Complexos, Rio de Janeiro, Brazil
K considers all possible interactions among external individuals, species, and dispute for resources (Blanco 1993). When these resources are unlimited, the population has a maximum instantaneous rate at which it grows, called intrinsic growth rate, j0 : These models quantify the population size (number of individuals) NðtÞ 0; at a certain time t; given its initial size N0 Nð0Þ 0; intrinsic growth rate j0 [ 0; and K ¼ Nð1Þ [ 0: For an environment with unlimited resources (K ! 1), the von Foerster et al. (1960) model considers the per capita growth rate as a power law to somehow take medium inhomogeneities into account d ln N=dt ¼ j0 N h ; where the exponent h [ 0 produces a divergence at a finite time (von Foerster et al. 1960; Strzałka and Grabowski 2008). For h ! 0þ ; one retrieves the Malthus model, so that the population grows exponentially. However, when resources are limited, the population grows monotonically towards K. The growth of individual organisms (Laird et al. 1965), tumors (Bajzer et al. 1996), and other biological systems (Zwietering et al. 1996) are well described by sigmoid curves (Boyce and DiPrima 2001; Murray 1993; Edelstein-Keshet 2005), which are solutions of simple population dynamics models. One of these is the theta-logistic model (Clark et al. 2010) d ln p=dt ¼ j0 ð1 ph Þ; where p ¼ N=K and that for h ¼ 1 retrieves the Verhulst model. For p 1; one sees the effect of h [ 0 by diminishing the medium carrying capacity (d ln p=dt j0 ð1 N=ðK=hÞÞ). According to Clark et al. (2010), when h\1 (h [ ), the the growth rate is unable (able) to recover quickly from extrinsic pertubations. Another one is Richards model (Martinez et al. 2008) d ln p=dt ¼ j0 ð1 pq~Þ=~ q; which generalizes the Verhulst (~ q ¼ 1) and Gompertz (~ q ¼ 0) and characterize intraspecific competition (Cabella et al. 2012). This kind of competition generates a negative correlation between population size and
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Fig. 1 Per capita growth rate of populations that follow the Richards model. We present three different curves referring to Gompertz model (~ q ¼ 0), Verhulst model (~ q ¼ 1) and q~ ¼ 2, where q~ is the generalized logarithmic function parameter, from Richards model (see ‘‘Richards model and Allee effect’’). The negative slope of the curves characterizes the intraspecific competition
the per capita growth rate, often used to measure the socalled total fitness (see Fig. 1). The term ‘‘Allee effect’’ is related to a positive correlation between population size and fitness at very low population size. The demographic Allee effect is defined as a positive correlation between the total fitness of a population and the population size. Specifically about the demographic Allee effect, if there is a critical population size that delimits the transition between population survival and extinction, the Allee threshold, one has the strong Allee effect. In the absence of Allee threshold, the population is not extinguished and one has the strong Allee effect. Although there are simple mathematical models of onespecies population growth dynamics dealing with the weak and strong Allee effects (Saha et al. 2013; Courchamp et al. 1999; Boukal and Berec 2002; Dennis 1989), as far as we are aware, there is not a simple analytical model that links these effects with generalized growth models. We propose to unify both the weak and strong Allee effects and Richards model (Martinez et al. 2008). The proposed model permits a new point of view towards the demographic Allee effect on population dynamics, since there is a broad and well established framework concerning the generalized functions (von Foerster et al. 1960; Tsallis 1988, 1994; Arruda et al. 2008) and models (von Foerster et al. 1960; Strzałka and Grabowski 2008; Martinez et al. 2008; Cabella et al. 2011, 2012). Furthermore, the standard Allee effect models only considers one cooperation–competition transition, making the richness of intraspecific interactions oversimplified. The possibility of emergence of more than one cooperation–competition transition, such as occurs in spacial prisoner dilemma (Pereira and Martinez 2010), reveals the more general aspect of our model and its predictive power.
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The presentation is organized as follows: In Sect. ‘‘Richards model and Allee effect’’, we do a review of existing results in the literature to introduce the Richards model, its solution and particular cases. We present the definition of Allee effect, its interpretation based on the relationship between cooperation and competition. To present the generalized Allee effect model, we consider a population size dependent intrinsic growth rate jðpÞ in the Richards model, where p ¼ N=K: In ‘‘Results and discussion’’, particular cases have full analytical solutions and are discussed and the model is validated with ecological data. We present a solution for the Verhulst-like Allee effect model. Also, we point out a peculiar instance of the Allee effect, stressing the predictive power of our model. In the ‘‘Conclusions’’, we conclude presenting our final remarks.
Richards model and Allee effect The Richards model is d lnðpÞ=ds ¼ lnq~ðpÞ (see Fig. 1), where p ¼ N=K; s ¼ j0 t is the time measured in terms of j0 ; and lnq~ðxÞ is the generalized logarithmic function (~ qlogarithm) of the nonextensive thermostatistics (Tsallis 1988, 1994; Arruda et al. 2008): 8 q~ Zx 0 x 1 dt xq~ 1 < ; if q~ 6¼ 0 ; lnq~ðxÞ ¼ ¼ lim ¼ q~ : t1q~ q~0 !q~ q~0 ln x; if q~ ¼ 0 1 ð1Þ which is geometrically interpreted as the area under the 1=t1q~ curve in the interval ½1; x: We stress this is a generalization of the definition of the natural logarithmic function, which is retrieved as a particular case for q~ ¼ 0: The solution of Richards model is (Cabella et al. 2012) pðsÞ ¼ 1=eq~½lnq~ð1=p0 Þes ; where p0 ¼ pð0Þ and eq~ðxÞ is the generalization of the exponential function (~ q-exponential), the inverse of q~-logarithm function: ( 0 limq~0 !q~½1 þ q~0 x1=q~ ; if q~x 1 eq~ðxÞ ¼ ; ð2Þ 0; otherwise where eq~ðxÞ vanishes if q~x\ 1: This property represents transitions in ecological regimes (Cabella et al. 2011). The q~-exponential function is strictly non-negative (eq~ðxÞ 0) and eq~ð0Þ ¼ 1 for any q~: Also, the solution of the von Foerster et al. model mentioned at in the introduction (Strzałka 2009) is NðsÞ ¼ N0 eh ðN0h sÞ: The Richards model steady-state condition dp=ds ¼ 0 produces two solutions (p ¼ pð1Þ): the unstable one p 1 ¼ 0 and p 2 ¼ 1; the stable one. The parameter q~ ¼ 1 c=Df has a microscopic interpretation in the context of repulsive cells interaction r c ; where r is the distance between two cells and growth in a fractal medium of dimensionality
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Df (Martinez et al. 2008; Mombach et al. 2002a, b). For q~ ¼ 0 and q~ ¼ 1; the Richards model retrieves the Gompertz and Verhulst models, respectively (Martinez et al. 2008), so that the Gompertz model corresponds to a medium where dimensionality matches the potential range Df ¼ c and the Verhulst model corresponds to a medium with high dimensionality (mean field case) weakening the influence of potential (Df c). Here, we understand q~ as an intraspecific competitive intensity. The one-species population growth dynamics is permeated by the relationship between the harms from the intraspecific competition for the limited environmental resources and the benefits from the presence of conspecifics. On the one hand, as a result of population growth, the increasing competition and depletion of resources may lead to decreasing natality and survival. On the other hand, for many species, the benefits from conspecific cooperation may outweigh the harms from intraspecific competition, resulting in a net gain of individual fitness in a specific population size range. The conspecific cooperation importance for some species may be exemplified by the increased likelihood of encountering a reproductive partner and by group protection behavior as the population size increases (Deredec and Courchamp 2007). In some cases, the benefits can greatly outweigh the harms, resulting in a positive correlation between population size and fitness at very low population densities. This correlation is known as the Allee effect. The term ‘‘Allee effect’’ refers to different phenomena (Brassil 2001) and has different definitions. This raises the distinction between component and demographic Allee effects (Stephens et al. 1999). The total fitness of an organism is often measured as the per capita growth rate (Allee et al. 1949) (also known as specific survival rate or specific growth rate (Odum and Allee 1954). This total fitness is a combination of different components, e.g., the survivorship or per capita reproduction, and population size or density (Kramer et al. 2009) (for purposes of simplification, we refer only to population size in this paper). The component Allee effect is a positive relationship between total fitness component and population size (Kramer et al. 2009). When the summation of every component Allee effects is not offset by the negative dependence in other components of fitness, it may cause the demographic Allee effect (Stephens et al. 1999). The demographic Allee effect is defined as a positive correlation between the total fitness and population size. Here, we address only the demographic Allee effect. In this case, there is a distinction between the so-called strong and weak Allee effects. The strong Allee effect has a critical population size, the Allee threshold. This threshold delimits the transition between population survival and
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Fig. 2 Curves for the per capita growth rates. The continuous line represents the logistic growth (no Allee effect), the dashed and the dot-dashed lines represent the weak and the strong Allee effects, respectively. Graphically, the Allee effect occurs in the range where the derivative of per capita growth rate with respect to population size is positive. In the weak Allee effect, the per capita growth rate curve has a positive y-intercept. Otherwise, the strong Allee effect has a negative y-intercept and the Allee threshold, that delimits the transition between population survival and extinction. Inset on the left-hand side of the dashed vertical line, the benefits of the cooperation outweigh the harms from the intraspecific competition (Allee effect). On the right-hand side of the dashed line, the competition outweigh the cooperation. The growth rate is maximum when the effect of the cooperation is balanced by the effect of competition
extinction. For initial population sizes below the Allee threshold, the population cannot persist (Stephens et al. 1999), i.e., it declines till extinction. For initial sizes above the Allee threshold, the population grows towards the environment carrying capacity. The per capita growth rate curve for the strong Allee effect is characterized by a negative y-intercept (see Fig. 2). In contrast, for the weak Allee effect, the per capita growth rate curve has a positive y-intercept. Populations that exhibit the weak Allee effect have their growth slowed down as the population sizes decreases, but it does not become negative (Deredec and Courchamp 2007) (see Fig. 2). The induced saturation function GðNÞ (Arruda et al. 2008) represents the per capita growth rate d ln N=dt ¼ GðNÞ: For GðNÞ ¼ j0 [ 0; where j0 is the constant intrinsic growth rate, the population grows unbounded (Malthus model). The intraspecific competition can be modeled by several means. One of them is to consider the theta-logistic model so that Gh ðpÞ: d lnðpÞ=ds ¼ ð1 ph Þ; where p ¼ N=K; s ¼ j0 t and the power parameter h takes into account medium inhomogeneities by decreasing the medium carrying capacity. Although the Verhulst model is a limit case h ¼ 1; the Gompertz model is not since Gh!0 ðpÞ ¼ h ln p: Another one is the Richards model
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Gq~ðpÞ : d lnðpÞ=ds ¼ lnq~ðpÞ; where lnq~ðpÞ is given by Eq. (1). This model generalizes the Gompertz model (G0 ðpÞ ¼ ln p) and Verhulst model (G1 ðpÞ ¼ 1 p). We point out that the h-logistic model Gh ðpÞ ¼ ðph 1Þ is equivalent to the Richards model only for q~; h [ 0; for negative values q~; h\0; the models have different signs and for h ¼ 0; the logistic model does not retrieve naturally the Gompertz model. For the natural Gompertz limit, we have chosen the deal with the Richards model instead of the h-logistic one. To model the weak and strong Allee effect, consider a population size dependent intrinsic growth rate. For the theta-logistic model, Saha et al. (2013) considered: d ln p=ds ¼ ðpK aÞð1 ph Þ; where a is the Allee parameter. Nevertheless, for vanishing a; one does not retrieve the original theta-logistic model (because the pK factor). To have the Gompertz model as a limiting case and to retrieve the basic population dynamics (growth) model we consider the Richards model and jðpÞ ¼ j0 ½1 AðKp=KA 1Þ; where KA [ 0 and A are constants; therefore, Kp GðpÞ ¼ 1 A 1 lnq~ð pÞ: ð3Þ KA The Allee effect is suppressed for A ¼ 0 and Eq. (3) retrieves the Richards model. Therefore, the parameter A is related to the intensity of intraspecific cooperative effect on the Richards model. As A is varied, the Richards model characteristic regime (predominantly competitive) is gradually modified and, in some cases, it gives rise to a cooperative regime in the form of the demographic Allee effect.
Results and discussion Our Allee effect model is validated considering published ecological data. Figure 3 presents the data for the muskox (Ovibos moschatus), a mammal native to the Arctic regions of Canada, Greenland and Alaska. These data were taken directly from the graph of the Ref. (Gregory et al. 2009), which used a database with 1198 population time series from major taxonomic groups and biomes. These time series were drawn from the Global Population Dynamics Database (GPDD: NERC Centre for Population Biology 1999), peer-reviewed literature, gray literature and online sources. Adjusting the data using the generalized model, the value of Gð0Þ almost vanishes, with A ’ 1; q~ ’ 0:2; KA ’ 2; 362 and K ’ 846; which represents the carrying capacity. The large data dispersion produces a poor fit and leads to a coefficient of determination R2 ’ 0:11; which does not invalidate the weak Allee effect observation.
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Fig. 3 Data for the muskox (Ovibos moschatus). The graph shows a special case between the weak and strong demographic Allee effect, characterized by the maximum cooperation regime for population that reaches the brink of extinction, and the data related to the points were taken directly from the graph of the ref. (Gregory et al. 2009), which used a database with 1198 population time series from major taxonomic groups and biomes. Trend line represents the fit made in QtiPlot version 0.9.8.8-3 using Eq. (3) without the scaling, i.e., replacing pðsÞ by NðsÞ to make explicit the values of K and KA : The fitting parameters are: A ’ 1; q~ ’ 0:2; KA ’ 2; 362 and K ’ 846: It was obtained R2 ’ 0:11
Let us now return to the analysis of the proposed model. For q~ 0; GðpÞ diverges as p ! 0; but for q~ [ 0; Gð0Þ ¼
1þA ; q~
ð4Þ
which is independent of KA and K: If Gð0Þ\ð [ Þ0; one retrieves the strong (weak) Allee effect. When Gð0Þ ¼ 0 (A ¼ 1), which is the case depicted on Fig. 3, the Allee threshold is null and there is the transition between the weak and the strong regimes. In this case, the species reaches the brink of extinction. Cooperation is maximum and, despite the marked decrease in the population growth rate, the species is not extinguished (see inset of Fig. 2). The roots of Eq. (3) are p ¼ 1 and p ¼ TA ¼
KA ð1 þ AÞ ; KA
ð5Þ
which does not depend on q~: Considering K [ 0 and TA [ 0; the smallest of these two values represents the Allee threshold and the largest one represents the environment carrying capacity. Despite representing the carrying capacity on Richards model, K may have another meaning in this generalized model. The transition between cooperation and competition regimes is given by the maximum of the function GðpÞ; obtained from dG=dpjpc ¼ ~ 0 : ½A lnq~ðpc Þ=KA ½jðpc Þpq1 c =j0 K ¼ 0; and can be obtained by positive roots of: pqc~fq~½ AðKA Kpc Þ þ KA KApc g þ KApc ¼ 0;
ð6Þ
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Table 1 Parameters values of Eq. (3) and associated Allee effects regimes K
KA
A
q~
Strong
Positive
Positive
\1
Not null
Weak
Positive (negative)
Negative (positive)
Positive
Positive
with Pc ¼ Gðpc Þ [ 0: To Pc be a maximum, the following inequality must be satisfied: ~ d2 pq1 2A ð1 q~Þjðpc Þ c ð7Þ ½ Gðp Þ ¼ þ \0 : c j0 pc K dp2 K KA Equation (3) has two equilibrium solutions: p 1 ¼ 1 and p 2 ¼ ðKA =KÞ½1 þ ð1=AÞ: From stability analysis, one relates these solutions to the equilibria points in the cases where population growth exhibits Allee effect. Table 1 summarizes the possible regimes according the parameters values. In the following, we consider two particular cases of the model, q~ ¼ 1 (Verhulst-like) and q~ ¼ 0 (Gompertz-like), where we can offer a fulll analysis and point out interesting issues of our model.
is the GðpÞ roots ratio. Replacing p by pc in Eq. (3), one obtains
Verhulst-like Allee effect model First, consider the Verhulst-like Allee effect with q~ ¼ 1: The annual counts of free-living Vancouver Island marmots (Marmota vancouverensis), who are endemic to Vancouver Island, BC, Canada (Brashares et al. 2010) are depicted in Fig. 4. The strong Allee effect is observed, beyond the reduction of colony size, has led to an increase in the inter-colony distances and opened up space for possible mechanisms for Allee effects, including impaired mate-finding, reduced predator detection/avoidance and reduced foraging (Brashares et al. 2010). According to authors, the reduction in colony sizes and proximity between them contributed to declines in survival and reproduction. The curve obtained by the generalized model has q~ ¼ 1 and it intercepts the GðpÞ axis in Gð0Þ\0; which shows a typical strong Allee effect, with A ’ 1:27: It is important to observe that, because K\TA (K ’ 186 and TA ’ 342; since KA ’ 1; 524), K and TA represent the Allee threshold and the environmental carrying capacity, respectively. It was obtained R2 ’ 0:55: The population size that sets the transition between the predominance of cooperation and competition is 1 pc ð1Þ ¼ ð1 þ X Þ; 2
ð8Þ
where X¼
TA K
Fig. 4 Annual counts of free-living Vancouver Island marmots (Marmota vancouverensis). Data related to the points cover the period between 1970 and 2007 and were taken directly from the graph of the ref. (Brashares et al. 2010). The graph shows a range with positive correlation of the per capita growth rate with population size— demographic Allee effect—and a negative y-intercept—strong Allee effect. Trend line represents the fit made using Eq. (3) in QtiPlot version 0.9.8.8-3. Fixing q~ ¼ 1; the fitting parameters are A ’ 1:27; KA ’ 1; 524 and K ’ 186: It was obtained R2 ’ 0:55
ð9Þ
Pc ð1Þ ¼
AðK=KA 1Þ 1 ð X 1Þ; 4
ð10Þ
where Pc ð1Þ ¼ Gðpc ð1ÞÞ: Hence, one has d 2 G=dp2 jpc ¼ 2A=ðKA KÞ and the Allee effect emerges for A\0: Since Eq. (8) is positive, then A\1=½1 þ ðK=KA Þ: The intercept is Gð0Þ ¼ 1 þ A: The condition 1\A\0 leads to Gð0Þ [ 0; that characterizes the weak Allee effect. The strong Allee effect occurs for A\ 1; leading to Gð0Þ\0: Verhulst-like Allee effect model is similar to the one used by Gruntfest et al. (1997) and Courchamp et al. (1999), but, according Boukal et al. (2002), the only possible scenario when using this model is the ‘‘extinctionsurvival’’ (ES), i.e., the strong Allee effect. The ability of Verhulst-like Allee effect model to describe both, strong and weak, Allee effects shows its wider range of descriptive possibilities. Gompertz-like Allee effect model The Gompertz-like Allee effect model, q~ ¼ 0; leads to a different kind of Allee effect, where a new competition– cooperation transition emerges. Contrary to the standard Allee effect models, for very small populations there is a range with a negative correlation between population size and fitness (predominant competition) (see Fig. 5a, b). However, as the population increases, the cooperative effect surpass the competition leading to the Allee effect.
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Fig. 5 Per capita growth rate curves for the Gompertz-like Allee effect model. This model produces a positive correlation between population number and fitness at very low population, characteristic of the demographic Allee effect, but GðpÞ diverges as the population vanishes. The parameters are K ¼ 100; KA ¼ 20 and: a A ¼ 0:65; b A ¼ 0:8; c A ¼ 1:0 and d A ¼ 1:5: In a and b, the curves are similar to the conventional weak Allee effect. In c, there is a transition similar to the one between the weak and strong Allee effect. In d, a case similar to that observed in the strong Allee effect, where there is an Allee threshold
For larger populations, the competition effect takes over again and the Allee effect vanishes. In this case, K [ 0 always represents the carrying capacity and, if A\ 1; TA is the Allee threshold. The extreme points are obtained from the condition ðAflnðpc ÞKpc þ KA ½ðpc K=KA Þ 1g KA Þ=ðKA Kpc Þ ¼ 0; so that pc ð0Þ ¼ eWðeXÞ1 ;
ð11Þ
The series in Eq. (13) is absolutely convergent and it can be rearranged into the ‘‘approximated’’ form (Hassani 2005): WðxÞ ¼ L1 ðxÞ L2 ðxÞ þ
L2 ðxÞ þ : L1 ðxÞ
ð14Þ
Considering Eq. (14), it is possible to obtain some sharper bounds for it (see Ref. Hoorfar and Hassani 2008).
where WðxÞ is the Lambert W function, also called the omega function or the product log function, which satisfies (Corless et al. 1996): WðxÞeWðxÞ ¼ x;
ð12Þ
for x e1 : If x is real, WðxÞ has two branches for e1 x\0 (see Fig. 6). The principal branch satisfies WðxÞ 1 and the other one where W1 ðxÞ 1: Since WðxÞ\ lnðxÞ; for x [ e; and L1 ðxÞ L2 ðxÞ\WðxÞ; where L1 ðxÞ ¼ lnðxÞ and L2 ðxÞ ¼ ln½lnðxÞ; for x [ 41:19 (Hassani 2005). For x 1 (Hassani 2005; Hoorfar and Hassani 2008): WðxÞ ¼ L1 ðxÞ L2 ðxÞ þ
1 X 1 X k¼0 m¼1
ckm
Lm 2 ðxÞ ; kþm L1 ðxÞ
ð13Þ
where ckm ¼ ½ð1Þk =m!S½k þ m; k þ 1; where S½k þ m; k þ 1 is the Stirling cycle number (Corless et al. 1996).
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Fig. 6 The two real branches of Lambert W function WðxÞ; also called the omega function or the product log function, which satisfies WðxÞeWðxÞ ¼ x (graph extracted from ref. Corless et al. 1996). For real x; the principal branch WðxÞ satisfies 1 WðxÞ and otherwise the branch W1 ðxÞ
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sign from positive (concave upward) to negative (concave downward). Thus pinf ¼ X
ð19Þ
and, consequently, Pinf ¼ 2ð1 þ AÞ lnðX Þ :
The existence of two cooperation–competition transitions in this limit case reveals the more general aspect of this model (Fig. 7). The analytical study of this limiting case permitted us to calculate the A parameter ranges for which there is one or two transitions and define the points when the transitions occur.
0
Fig. 7 The Gompetz-like Allee effect GðpÞ curve with 1\A\ ðK þ KA e2 Þ=KA e2 : Unlike the standard Allee effect models, there is a range with negative correlation between population size and fitness (a predominantly competitive regime) that precedes the Allee effect. As the population increases, the cooperative effect surpass the competition and one has the ordinary Allee effect. The competition-cooperation transitions occur at pcmin and pcmax : The population sizes pcmin and pcmax delimit the range where the Allee effect occurs—when the cooperative effect surpass the competition—and are related to the minimum and maximum Pcmin and Pcmax ; respectively. The inflection point pinf indicates concavity switch in the cooperation regime (Allee effect)
The WðeXÞ domain is defined by A ðK þ KA e2 Þ=KA e2 : If A ¼ ðK þ KA e2 Þ=KA e2 or A 1; GðpÞ has only one extreme point, related to the maximum Pc (see Fig. 5c, d). Since the Lambert W has two branches, GðpÞ has two extreme points if 1\A\ðK þ KA e2 Þ=KA e2 (see Fig. 5a, b). Consequently, pc has two values, one related to each branch: pcmin ¼ eW1 ðeXÞ1 ;
ð15Þ
and pcmax ¼ eWðeXÞ1 :
ð16Þ
Replacing p by pc in Eq. (3), we obtain Pcmin ð0; pcmin Þ ¼
ð20Þ
ð1 þ AÞð1 W1 ðeX ÞÞ X ln ; W1 ðeX Þ W1 ðeX Þ
Conclusions We have presented a simple analytical model that unifies the weak and strong demographic Allee effects and the Richards growth model, which has the Verhulst and Gompertz models as particular cases. Our generalized model expands the range of descriptive possibilities of species population dynamics that exhibit Allee effect, since it permits a new view point based on a broad and wellestablished growth models framework. Ecological data, regarding species that provenly exhibit Allee effect, validate our model and are presented as examples of the model fitting possibilities. Ecological interpretations have been assigned to the model parameters, where q~ and A represent the intraspecific competitive and cooperative intensities, respectively. Interpreting the demographic Allee effect as a regime in which the cooperative effect outweighs the competitive one, the Gompertz-like model, q~ ¼ 0; presents a range with negative correlation between population size and fitness that precedes the Allee effect, unlike standard Allee effect models (Verhulst-like, with q~ ¼ 1), which presents only one cooperation–competition transition. As far we know, there are not real data in one-species studies that show explicitly this effect. The analysis of such regimes is difficult in very small population sizes and some of nuances in the cooperation–cooperation are not easily detected.
ð17Þ and ð1 þ AÞð1 W ðeX ÞÞ X ln ; Pcmax ð0; pcmax Þ ¼ W ðeX Þ W ðeX Þ ð18Þ
Acknowledgments The authors thank the fruitful discussions with Fabiano Ribeiro and acknowledge the Brazilian agencies for support. LSS holds grants from Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior (CAPES), ASM and BCTC from Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) (305738/ 2010-0, 127151/2012-5).
where Pcmin ð0; pcmin Þ ¼ Gðpcmin Þ and Pcmax ð0; pcmax Þ ¼ Gðpcmax Þ: In this way, from condition d2 G=dp2 jpinf ¼
References
½AðKA þ Kpinf Þ þ KA =KA ðKpinf Þ2 ¼ 0; we obtain the inflection point, at which the curve concavity changes it
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ORIGINAL PAPER
Generalized Allee effect model Lindomar S. dos Santos • Brenno C. T. Cabella Alexandre S. Martinez
•
Received: 21 August 2013 / Accepted: 19 February 2014 / Published online: 17 March 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The Allee effect consists of a positive correlation between very small population size and fitness. Offering a new view point on the weak and strong demographic Allee effect, we propose to combine them with the Richards growth model. In particular, a peculiar manifestation of the Allee effect is analytically predicted and still not validated by experiments. Model validation with ecological data is presented for some special situations. Keywords Strong and weak Allee effects Growth models Generalized functions Richards model Cooperation–competition transition Unified models
Introduction Growth and population dynamics models are important tools used in the study of several time-dependent processes in different research areas. The main simplification in the population growth study is to explicitly represent only one species. In this case, the environment carrying capacity
L. S. dos Santos (&) B. C. T. Cabella A. S. Martinez Faculdade de Filosofia, Cieˆncias e Letras de Ribeira˜o Preto (FFCLRP), Universidade de Sa˜o Paulo (USP), Avenida dos Bandeirantes, 3900, 14.040-901, Ribeira˜o Preto, SP, Brazil e-mail: [email protected] B. C. T. Cabella e-mail: [email protected] A. S. Martinez e-mail: [email protected] A. S. Martinez Instituto Nacional de Cieˆncia e Tecnologia em Sistemas Complexos, Rio de Janeiro, Brazil
K considers all possible interactions among external individuals, species, and dispute for resources (Blanco 1993). When these resources are unlimited, the population has a maximum instantaneous rate at which it grows, called intrinsic growth rate, j0 : These models quantify the population size (number of individuals) NðtÞ 0; at a certain time t; given its initial size N0 Nð0Þ 0; intrinsic growth rate j0 [ 0; and K ¼ Nð1Þ [ 0: For an environment with unlimited resources (K ! 1), the von Foerster et al. (1960) model considers the per capita growth rate as a power law to somehow take medium inhomogeneities into account d ln N=dt ¼ j0 N h ; where the exponent h [ 0 produces a divergence at a finite time (von Foerster et al. 1960; Strzałka and Grabowski 2008). For h ! 0þ ; one retrieves the Malthus model, so that the population grows exponentially. However, when resources are limited, the population grows monotonically towards K. The growth of individual organisms (Laird et al. 1965), tumors (Bajzer et al. 1996), and other biological systems (Zwietering et al. 1996) are well described by sigmoid curves (Boyce and DiPrima 2001; Murray 1993; Edelstein-Keshet 2005), which are solutions of simple population dynamics models. One of these is the theta-logistic model (Clark et al. 2010) d ln p=dt ¼ j0 ð1 ph Þ; where p ¼ N=K and that for h ¼ 1 retrieves the Verhulst model. For p 1; one sees the effect of h [ 0 by diminishing the medium carrying capacity (d ln p=dt j0 ð1 N=ðK=hÞÞ). According to Clark et al. (2010), when h\1 (h [ ), the the growth rate is unable (able) to recover quickly from extrinsic pertubations. Another one is Richards model (Martinez et al. 2008) d ln p=dt ¼ j0 ð1 pq~Þ=~ q; which generalizes the Verhulst (~ q ¼ 1) and Gompertz (~ q ¼ 0) and characterize intraspecific competition (Cabella et al. 2012). This kind of competition generates a negative correlation between population size and
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Fig. 1 Per capita growth rate of populations that follow the Richards model. We present three different curves referring to Gompertz model (~ q ¼ 0), Verhulst model (~ q ¼ 1) and q~ ¼ 2, where q~ is the generalized logarithmic function parameter, from Richards model (see ‘‘Richards model and Allee effect’’). The negative slope of the curves characterizes the intraspecific competition
the per capita growth rate, often used to measure the socalled total fitness (see Fig. 1). The term ‘‘Allee effect’’ is related to a positive correlation between population size and fitness at very low population size. The demographic Allee effect is defined as a positive correlation between the total fitness of a population and the population size. Specifically about the demographic Allee effect, if there is a critical population size that delimits the transition between population survival and extinction, the Allee threshold, one has the strong Allee effect. In the absence of Allee threshold, the population is not extinguished and one has the strong Allee effect. Although there are simple mathematical models of onespecies population growth dynamics dealing with the weak and strong Allee effects (Saha et al. 2013; Courchamp et al. 1999; Boukal and Berec 2002; Dennis 1989), as far as we are aware, there is not a simple analytical model that links these effects with generalized growth models. We propose to unify both the weak and strong Allee effects and Richards model (Martinez et al. 2008). The proposed model permits a new point of view towards the demographic Allee effect on population dynamics, since there is a broad and well established framework concerning the generalized functions (von Foerster et al. 1960; Tsallis 1988, 1994; Arruda et al. 2008) and models (von Foerster et al. 1960; Strzałka and Grabowski 2008; Martinez et al. 2008; Cabella et al. 2011, 2012). Furthermore, the standard Allee effect models only considers one cooperation–competition transition, making the richness of intraspecific interactions oversimplified. The possibility of emergence of more than one cooperation–competition transition, such as occurs in spacial prisoner dilemma (Pereira and Martinez 2010), reveals the more general aspect of our model and its predictive power.
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The presentation is organized as follows: In Sect. ‘‘Richards model and Allee effect’’, we do a review of existing results in the literature to introduce the Richards model, its solution and particular cases. We present the definition of Allee effect, its interpretation based on the relationship between cooperation and competition. To present the generalized Allee effect model, we consider a population size dependent intrinsic growth rate jðpÞ in the Richards model, where p ¼ N=K: In ‘‘Results and discussion’’, particular cases have full analytical solutions and are discussed and the model is validated with ecological data. We present a solution for the Verhulst-like Allee effect model. Also, we point out a peculiar instance of the Allee effect, stressing the predictive power of our model. In the ‘‘Conclusions’’, we conclude presenting our final remarks.
Richards model and Allee effect The Richards model is d lnðpÞ=ds ¼ lnq~ðpÞ (see Fig. 1), where p ¼ N=K; s ¼ j0 t is the time measured in terms of j0 ; and lnq~ðxÞ is the generalized logarithmic function (~ qlogarithm) of the nonextensive thermostatistics (Tsallis 1988, 1994; Arruda et al. 2008): 8 q~ Zx 0 x 1 dt xq~ 1 < ; if q~ 6¼ 0 ; lnq~ðxÞ ¼ ¼ lim ¼ q~ : t1q~ q~0 !q~ q~0 ln x; if q~ ¼ 0 1 ð1Þ which is geometrically interpreted as the area under the 1=t1q~ curve in the interval ½1; x: We stress this is a generalization of the definition of the natural logarithmic function, which is retrieved as a particular case for q~ ¼ 0: The solution of Richards model is (Cabella et al. 2012) pðsÞ ¼ 1=eq~½lnq~ð1=p0 Þes ; where p0 ¼ pð0Þ and eq~ðxÞ is the generalization of the exponential function (~ q-exponential), the inverse of q~-logarithm function: ( 0 limq~0 !q~½1 þ q~0 x1=q~ ; if q~x 1 eq~ðxÞ ¼ ; ð2Þ 0; otherwise where eq~ðxÞ vanishes if q~x\ 1: This property represents transitions in ecological regimes (Cabella et al. 2011). The q~-exponential function is strictly non-negative (eq~ðxÞ 0) and eq~ð0Þ ¼ 1 for any q~: Also, the solution of the von Foerster et al. model mentioned at in the introduction (Strzałka 2009) is NðsÞ ¼ N0 eh ðN0h sÞ: The Richards model steady-state condition dp=ds ¼ 0 produces two solutions (p ¼ pð1Þ): the unstable one p 1 ¼ 0 and p 2 ¼ 1; the stable one. The parameter q~ ¼ 1 c=Df has a microscopic interpretation in the context of repulsive cells interaction r c ; where r is the distance between two cells and growth in a fractal medium of dimensionality
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Df (Martinez et al. 2008; Mombach et al. 2002a, b). For q~ ¼ 0 and q~ ¼ 1; the Richards model retrieves the Gompertz and Verhulst models, respectively (Martinez et al. 2008), so that the Gompertz model corresponds to a medium where dimensionality matches the potential range Df ¼ c and the Verhulst model corresponds to a medium with high dimensionality (mean field case) weakening the influence of potential (Df c). Here, we understand q~ as an intraspecific competitive intensity. The one-species population growth dynamics is permeated by the relationship between the harms from the intraspecific competition for the limited environmental resources and the benefits from the presence of conspecifics. On the one hand, as a result of population growth, the increasing competition and depletion of resources may lead to decreasing natality and survival. On the other hand, for many species, the benefits from conspecific cooperation may outweigh the harms from intraspecific competition, resulting in a net gain of individual fitness in a specific population size range. The conspecific cooperation importance for some species may be exemplified by the increased likelihood of encountering a reproductive partner and by group protection behavior as the population size increases (Deredec and Courchamp 2007). In some cases, the benefits can greatly outweigh the harms, resulting in a positive correlation between population size and fitness at very low population densities. This correlation is known as the Allee effect. The term ‘‘Allee effect’’ refers to different phenomena (Brassil 2001) and has different definitions. This raises the distinction between component and demographic Allee effects (Stephens et al. 1999). The total fitness of an organism is often measured as the per capita growth rate (Allee et al. 1949) (also known as specific survival rate or specific growth rate (Odum and Allee 1954). This total fitness is a combination of different components, e.g., the survivorship or per capita reproduction, and population size or density (Kramer et al. 2009) (for purposes of simplification, we refer only to population size in this paper). The component Allee effect is a positive relationship between total fitness component and population size (Kramer et al. 2009). When the summation of every component Allee effects is not offset by the negative dependence in other components of fitness, it may cause the demographic Allee effect (Stephens et al. 1999). The demographic Allee effect is defined as a positive correlation between the total fitness and population size. Here, we address only the demographic Allee effect. In this case, there is a distinction between the so-called strong and weak Allee effects. The strong Allee effect has a critical population size, the Allee threshold. This threshold delimits the transition between population survival and
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Fig. 2 Curves for the per capita growth rates. The continuous line represents the logistic growth (no Allee effect), the dashed and the dot-dashed lines represent the weak and the strong Allee effects, respectively. Graphically, the Allee effect occurs in the range where the derivative of per capita growth rate with respect to population size is positive. In the weak Allee effect, the per capita growth rate curve has a positive y-intercept. Otherwise, the strong Allee effect has a negative y-intercept and the Allee threshold, that delimits the transition between population survival and extinction. Inset on the left-hand side of the dashed vertical line, the benefits of the cooperation outweigh the harms from the intraspecific competition (Allee effect). On the right-hand side of the dashed line, the competition outweigh the cooperation. The growth rate is maximum when the effect of the cooperation is balanced by the effect of competition
extinction. For initial population sizes below the Allee threshold, the population cannot persist (Stephens et al. 1999), i.e., it declines till extinction. For initial sizes above the Allee threshold, the population grows towards the environment carrying capacity. The per capita growth rate curve for the strong Allee effect is characterized by a negative y-intercept (see Fig. 2). In contrast, for the weak Allee effect, the per capita growth rate curve has a positive y-intercept. Populations that exhibit the weak Allee effect have their growth slowed down as the population sizes decreases, but it does not become negative (Deredec and Courchamp 2007) (see Fig. 2). The induced saturation function GðNÞ (Arruda et al. 2008) represents the per capita growth rate d ln N=dt ¼ GðNÞ: For GðNÞ ¼ j0 [ 0; where j0 is the constant intrinsic growth rate, the population grows unbounded (Malthus model). The intraspecific competition can be modeled by several means. One of them is to consider the theta-logistic model so that Gh ðpÞ: d lnðpÞ=ds ¼ ð1 ph Þ; where p ¼ N=K; s ¼ j0 t and the power parameter h takes into account medium inhomogeneities by decreasing the medium carrying capacity. Although the Verhulst model is a limit case h ¼ 1; the Gompertz model is not since Gh!0 ðpÞ ¼ h ln p: Another one is the Richards model
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Gq~ðpÞ : d lnðpÞ=ds ¼ lnq~ðpÞ; where lnq~ðpÞ is given by Eq. (1). This model generalizes the Gompertz model (G0 ðpÞ ¼ ln p) and Verhulst model (G1 ðpÞ ¼ 1 p). We point out that the h-logistic model Gh ðpÞ ¼ ðph 1Þ is equivalent to the Richards model only for q~; h [ 0; for negative values q~; h\0; the models have different signs and for h ¼ 0; the logistic model does not retrieve naturally the Gompertz model. For the natural Gompertz limit, we have chosen the deal with the Richards model instead of the h-logistic one. To model the weak and strong Allee effect, consider a population size dependent intrinsic growth rate. For the theta-logistic model, Saha et al. (2013) considered: d ln p=ds ¼ ðpK aÞð1 ph Þ; where a is the Allee parameter. Nevertheless, for vanishing a; one does not retrieve the original theta-logistic model (because the pK factor). To have the Gompertz model as a limiting case and to retrieve the basic population dynamics (growth) model we consider the Richards model and jðpÞ ¼ j0 ½1 AðKp=KA 1Þ; where KA [ 0 and A are constants; therefore, Kp GðpÞ ¼ 1 A 1 lnq~ð pÞ: ð3Þ KA The Allee effect is suppressed for A ¼ 0 and Eq. (3) retrieves the Richards model. Therefore, the parameter A is related to the intensity of intraspecific cooperative effect on the Richards model. As A is varied, the Richards model characteristic regime (predominantly competitive) is gradually modified and, in some cases, it gives rise to a cooperative regime in the form of the demographic Allee effect.
Results and discussion Our Allee effect model is validated considering published ecological data. Figure 3 presents the data for the muskox (Ovibos moschatus), a mammal native to the Arctic regions of Canada, Greenland and Alaska. These data were taken directly from the graph of the Ref. (Gregory et al. 2009), which used a database with 1198 population time series from major taxonomic groups and biomes. These time series were drawn from the Global Population Dynamics Database (GPDD: NERC Centre for Population Biology 1999), peer-reviewed literature, gray literature and online sources. Adjusting the data using the generalized model, the value of Gð0Þ almost vanishes, with A ’ 1; q~ ’ 0:2; KA ’ 2; 362 and K ’ 846; which represents the carrying capacity. The large data dispersion produces a poor fit and leads to a coefficient of determination R2 ’ 0:11; which does not invalidate the weak Allee effect observation.
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Fig. 3 Data for the muskox (Ovibos moschatus). The graph shows a special case between the weak and strong demographic Allee effect, characterized by the maximum cooperation regime for population that reaches the brink of extinction, and the data related to the points were taken directly from the graph of the ref. (Gregory et al. 2009), which used a database with 1198 population time series from major taxonomic groups and biomes. Trend line represents the fit made in QtiPlot version 0.9.8.8-3 using Eq. (3) without the scaling, i.e., replacing pðsÞ by NðsÞ to make explicit the values of K and KA : The fitting parameters are: A ’ 1; q~ ’ 0:2; KA ’ 2; 362 and K ’ 846: It was obtained R2 ’ 0:11
Let us now return to the analysis of the proposed model. For q~ 0; GðpÞ diverges as p ! 0; but for q~ [ 0; Gð0Þ ¼
1þA ; q~
ð4Þ
which is independent of KA and K: If Gð0Þ\ð [ Þ0; one retrieves the strong (weak) Allee effect. When Gð0Þ ¼ 0 (A ¼ 1), which is the case depicted on Fig. 3, the Allee threshold is null and there is the transition between the weak and the strong regimes. In this case, the species reaches the brink of extinction. Cooperation is maximum and, despite the marked decrease in the population growth rate, the species is not extinguished (see inset of Fig. 2). The roots of Eq. (3) are p ¼ 1 and p ¼ TA ¼
KA ð1 þ AÞ ; KA
ð5Þ
which does not depend on q~: Considering K [ 0 and TA [ 0; the smallest of these two values represents the Allee threshold and the largest one represents the environment carrying capacity. Despite representing the carrying capacity on Richards model, K may have another meaning in this generalized model. The transition between cooperation and competition regimes is given by the maximum of the function GðpÞ; obtained from dG=dpjpc ¼ ~ 0 : ½A lnq~ðpc Þ=KA ½jðpc Þpq1 c =j0 K ¼ 0; and can be obtained by positive roots of: pqc~fq~½ AðKA Kpc Þ þ KA KApc g þ KApc ¼ 0;
ð6Þ
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Table 1 Parameters values of Eq. (3) and associated Allee effects regimes K
KA
A
q~
Strong
Positive
Positive
\1
Not null
Weak
Positive (negative)
Negative (positive)
Positive
Positive
with Pc ¼ Gðpc Þ [ 0: To Pc be a maximum, the following inequality must be satisfied: ~ d2 pq1 2A ð1 q~Þjðpc Þ c ð7Þ ½ Gðp Þ ¼ þ \0 : c j0 pc K dp2 K KA Equation (3) has two equilibrium solutions: p 1 ¼ 1 and p 2 ¼ ðKA =KÞ½1 þ ð1=AÞ: From stability analysis, one relates these solutions to the equilibria points in the cases where population growth exhibits Allee effect. Table 1 summarizes the possible regimes according the parameters values. In the following, we consider two particular cases of the model, q~ ¼ 1 (Verhulst-like) and q~ ¼ 0 (Gompertz-like), where we can offer a fulll analysis and point out interesting issues of our model.
is the GðpÞ roots ratio. Replacing p by pc in Eq. (3), one obtains
Verhulst-like Allee effect model First, consider the Verhulst-like Allee effect with q~ ¼ 1: The annual counts of free-living Vancouver Island marmots (Marmota vancouverensis), who are endemic to Vancouver Island, BC, Canada (Brashares et al. 2010) are depicted in Fig. 4. The strong Allee effect is observed, beyond the reduction of colony size, has led to an increase in the inter-colony distances and opened up space for possible mechanisms for Allee effects, including impaired mate-finding, reduced predator detection/avoidance and reduced foraging (Brashares et al. 2010). According to authors, the reduction in colony sizes and proximity between them contributed to declines in survival and reproduction. The curve obtained by the generalized model has q~ ¼ 1 and it intercepts the GðpÞ axis in Gð0Þ\0; which shows a typical strong Allee effect, with A ’ 1:27: It is important to observe that, because K\TA (K ’ 186 and TA ’ 342; since KA ’ 1; 524), K and TA represent the Allee threshold and the environmental carrying capacity, respectively. It was obtained R2 ’ 0:55: The population size that sets the transition between the predominance of cooperation and competition is 1 pc ð1Þ ¼ ð1 þ X Þ; 2
ð8Þ
where X¼
TA K
Fig. 4 Annual counts of free-living Vancouver Island marmots (Marmota vancouverensis). Data related to the points cover the period between 1970 and 2007 and were taken directly from the graph of the ref. (Brashares et al. 2010). The graph shows a range with positive correlation of the per capita growth rate with population size— demographic Allee effect—and a negative y-intercept—strong Allee effect. Trend line represents the fit made using Eq. (3) in QtiPlot version 0.9.8.8-3. Fixing q~ ¼ 1; the fitting parameters are A ’ 1:27; KA ’ 1; 524 and K ’ 186: It was obtained R2 ’ 0:55
ð9Þ
Pc ð1Þ ¼
AðK=KA 1Þ 1 ð X 1Þ; 4
ð10Þ
where Pc ð1Þ ¼ Gðpc ð1ÞÞ: Hence, one has d 2 G=dp2 jpc ¼ 2A=ðKA KÞ and the Allee effect emerges for A\0: Since Eq. (8) is positive, then A\1=½1 þ ðK=KA Þ: The intercept is Gð0Þ ¼ 1 þ A: The condition 1\A\0 leads to Gð0Þ [ 0; that characterizes the weak Allee effect. The strong Allee effect occurs for A\ 1; leading to Gð0Þ\0: Verhulst-like Allee effect model is similar to the one used by Gruntfest et al. (1997) and Courchamp et al. (1999), but, according Boukal et al. (2002), the only possible scenario when using this model is the ‘‘extinctionsurvival’’ (ES), i.e., the strong Allee effect. The ability of Verhulst-like Allee effect model to describe both, strong and weak, Allee effects shows its wider range of descriptive possibilities. Gompertz-like Allee effect model The Gompertz-like Allee effect model, q~ ¼ 0; leads to a different kind of Allee effect, where a new competition– cooperation transition emerges. Contrary to the standard Allee effect models, for very small populations there is a range with a negative correlation between population size and fitness (predominant competition) (see Fig. 5a, b). However, as the population increases, the cooperative effect surpass the competition leading to the Allee effect.
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Fig. 5 Per capita growth rate curves for the Gompertz-like Allee effect model. This model produces a positive correlation between population number and fitness at very low population, characteristic of the demographic Allee effect, but GðpÞ diverges as the population vanishes. The parameters are K ¼ 100; KA ¼ 20 and: a A ¼ 0:65; b A ¼ 0:8; c A ¼ 1:0 and d A ¼ 1:5: In a and b, the curves are similar to the conventional weak Allee effect. In c, there is a transition similar to the one between the weak and strong Allee effect. In d, a case similar to that observed in the strong Allee effect, where there is an Allee threshold
For larger populations, the competition effect takes over again and the Allee effect vanishes. In this case, K [ 0 always represents the carrying capacity and, if A\ 1; TA is the Allee threshold. The extreme points are obtained from the condition ðAflnðpc ÞKpc þ KA ½ðpc K=KA Þ 1g KA Þ=ðKA Kpc Þ ¼ 0; so that pc ð0Þ ¼ eWðeXÞ1 ;
ð11Þ
The series in Eq. (13) is absolutely convergent and it can be rearranged into the ‘‘approximated’’ form (Hassani 2005): WðxÞ ¼ L1 ðxÞ L2 ðxÞ þ
L2 ðxÞ þ : L1 ðxÞ
ð14Þ
Considering Eq. (14), it is possible to obtain some sharper bounds for it (see Ref. Hoorfar and Hassani 2008).
where WðxÞ is the Lambert W function, also called the omega function or the product log function, which satisfies (Corless et al. 1996): WðxÞeWðxÞ ¼ x;
ð12Þ
for x e1 : If x is real, WðxÞ has two branches for e1 x\0 (see Fig. 6). The principal branch satisfies WðxÞ 1 and the other one where W1 ðxÞ 1: Since WðxÞ\ lnðxÞ; for x [ e; and L1 ðxÞ L2 ðxÞ\WðxÞ; where L1 ðxÞ ¼ lnðxÞ and L2 ðxÞ ¼ ln½lnðxÞ; for x [ 41:19 (Hassani 2005). For x 1 (Hassani 2005; Hoorfar and Hassani 2008): WðxÞ ¼ L1 ðxÞ L2 ðxÞ þ
1 X 1 X k¼0 m¼1
ckm
Lm 2 ðxÞ ; kþm L1 ðxÞ
ð13Þ
where ckm ¼ ½ð1Þk =m!S½k þ m; k þ 1; where S½k þ m; k þ 1 is the Stirling cycle number (Corless et al. 1996).
123
Fig. 6 The two real branches of Lambert W function WðxÞ; also called the omega function or the product log function, which satisfies WðxÞeWðxÞ ¼ x (graph extracted from ref. Corless et al. 1996). For real x; the principal branch WðxÞ satisfies 1 WðxÞ and otherwise the branch W1 ðxÞ
Theory Biosci. (2014) 133:117–124
123
sign from positive (concave upward) to negative (concave downward). Thus pinf ¼ X
ð19Þ
and, consequently, Pinf ¼ 2ð1 þ AÞ lnðX Þ :
The existence of two cooperation–competition transitions in this limit case reveals the more general aspect of this model (Fig. 7). The analytical study of this limiting case permitted us to calculate the A parameter ranges for which there is one or two transitions and define the points when the transitions occur.
0
Fig. 7 The Gompetz-like Allee effect GðpÞ curve with 1\A\ ðK þ KA e2 Þ=KA e2 : Unlike the standard Allee effect models, there is a range with negative correlation between population size and fitness (a predominantly competitive regime) that precedes the Allee effect. As the population increases, the cooperative effect surpass the competition and one has the ordinary Allee effect. The competition-cooperation transitions occur at pcmin and pcmax : The population sizes pcmin and pcmax delimit the range where the Allee effect occurs—when the cooperative effect surpass the competition—and are related to the minimum and maximum Pcmin and Pcmax ; respectively. The inflection point pinf indicates concavity switch in the cooperation regime (Allee effect)
The WðeXÞ domain is defined by A ðK þ KA e2 Þ=KA e2 : If A ¼ ðK þ KA e2 Þ=KA e2 or A 1; GðpÞ has only one extreme point, related to the maximum Pc (see Fig. 5c, d). Since the Lambert W has two branches, GðpÞ has two extreme points if 1\A\ðK þ KA e2 Þ=KA e2 (see Fig. 5a, b). Consequently, pc has two values, one related to each branch: pcmin ¼ eW1 ðeXÞ1 ;
ð15Þ
and pcmax ¼ eWðeXÞ1 :
ð16Þ
Replacing p by pc in Eq. (3), we obtain Pcmin ð0; pcmin Þ ¼
ð20Þ
ð1 þ AÞð1 W1 ðeX ÞÞ X ln ; W1 ðeX Þ W1 ðeX Þ
Conclusions We have presented a simple analytical model that unifies the weak and strong demographic Allee effects and the Richards growth model, which has the Verhulst and Gompertz models as particular cases. Our generalized model expands the range of descriptive possibilities of species population dynamics that exhibit Allee effect, since it permits a new view point based on a broad and wellestablished growth models framework. Ecological data, regarding species that provenly exhibit Allee effect, validate our model and are presented as examples of the model fitting possibilities. Ecological interpretations have been assigned to the model parameters, where q~ and A represent the intraspecific competitive and cooperative intensities, respectively. Interpreting the demographic Allee effect as a regime in which the cooperative effect outweighs the competitive one, the Gompertz-like model, q~ ¼ 0; presents a range with negative correlation between population size and fitness that precedes the Allee effect, unlike standard Allee effect models (Verhulst-like, with q~ ¼ 1), which presents only one cooperation–competition transition. As far we know, there are not real data in one-species studies that show explicitly this effect. The analysis of such regimes is difficult in very small population sizes and some of nuances in the cooperation–cooperation are not easily detected.
ð17Þ and ð1 þ AÞð1 W ðeX ÞÞ X ln ; Pcmax ð0; pcmax Þ ¼ W ðeX Þ W ðeX Þ ð18Þ
Acknowledgments The authors thank the fruitful discussions with Fabiano Ribeiro and acknowledge the Brazilian agencies for support. LSS holds grants from Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior (CAPES), ASM and BCTC from Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) (305738/ 2010-0, 127151/2012-5).
where Pcmin ð0; pcmin Þ ¼ Gðpcmin Þ and Pcmax ð0; pcmax Þ ¼ Gðpcmax Þ: In this way, from condition d2 G=dp2 jpinf ¼
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