Population Dynamics: Extensions of Verhulst Model and Extremal Asymptotic Behaviour

Transcription

1 Population Dynamics: Extensions of Verhulst Moel an Extremal Asymptotic Behaviour Maria e Fátima Brilhante, Maria Ivette Gomes 2, an Dinis Pestana 3 Universiae os Açores an CEAUL, Ponta Delgaa, Açores, Portugal ( fbrilhante@uac.pt ) 2 DEIO, FCUL an CEAUL, Universiae e Lisboa, Portugal ( ivette.gomes@fc.ul.pt) 3 CEAUL Centro e Estatística e Aplicações a Universiae e Lisboa, Portugal ( inis.pestana@fc.ul.pt) Abstract. Starting from the Beta(2,2) moel, connecte to the Verhulst logistic parabola, several extensions are iscusse, an connections to extremal moels are reveale. Asie from the classical General Extreme Value Moel from the inepenent, ientically istribute case, extreme value moels in ranomly stoppe extremes schemes are iscusse. In this context, the classical logistic Verhulst moel is a max-geo-stable moel, i.e. the geometric thinning of the observations curbs own growth to sustainable patterns. The general ifferential moels presente are a unifie approach to population ynamics growth, with factors of the form [ ln( N(t))] P an the linearization [N(t)] p moeling two very ifferent growth patterns, an factors of the form [ ln N(t)] Q an the linearization [ N(t)] q leaing to very ifferent ambiental resources control of the growth behavior. Keywors: Verhulst logistic moel, Beta an BeTaBoOp moels, population ynamics, extreme value moels, geometric thinning, ranomly stoppe maxima with geometric suborinator. Introuction Let N(t) enote the size of some population at time t. Verhulst ([22], [23], [24]) impose some natural regularity conitions on N(t), namely that t N(t) = A k [N(t)] k, k=0 with A 0 = 0 since nothing can stem out from an extinct population, A > 0 a growing parameter, A 2 < 0 a retroaction parameter controlling sustainable growth tie to available resources. See also Lotka, [4]. The secon orer approximation t N(t) = A N(t) + A 2 [N(t)] 2 can be rewritten as N(t) = r N(t) t [ N(t) K ], ()

2 where r > 0 is frequently interprete as a Malthusian instantaneous growth rate parameter, whenever moeling natural breeing populations, an K > 0 as the equilibrium limit size of the population. The general form of the solution of the ifferential equation approximation, in (), is the family of logistic functions N(t) = K N 0 N 0 + (K N 0 ) e rt, where N 0 is the population size at time t = 0. This is the reason why in the context of population ynamics r x ( x) is frequently referre to as the logistic parabola. Due to the seasonal reprouction an time life of many natural populations, the ifferential equation in () is often iscretise, first taking r such that N(t + ) N(t) = r N(t) [ N(t)/K] an then α = r +, x(t) = r N(t) r +, to obtain x(t + ) = α x(t)[ x(t)], an then the associate ifference equation x n+ = α x n ( x n ), (2) where it is convenient to eal with the assumption x n [0, ], n =, 2,... The equilibrium x n+ = x n leas to a simple secon orer algebraic equation with positive root /α, an to a certain extent it is surprising that anyone woul care to investigate its numerical solution using the fixe point metho, which inee brings in many pathologies when a steep curve i.e., for some values of the iterates α ( 2 x n ) > is approximate by an horizontal straight line. This numerical investigation, apparently evoi of interest, has however been at the root of many theoretical avances (namely Feigenbaum bifurcations an ultimate chaotic behavior), an a posteriori le to many interesting breakthroughs in the unerstaning of population ynamics. Observe also that (2) can be rewritten as x n+ = α 6 6 x n [ x n ], an that f(x) = 6 x ( x) I (0,) (x) is the Beta(2, 2) probability ensity function (pf). Extensions of the Verhulst moel using ifference equations similar to (2), but where the right han sie is tie to a more general Beta(p, q) pf, where as usual B(p, q) = f p,q (x) = xp ( x) q B(p, q) 0 x p ( x) q x = I (0,) (x), (3) Γ (p) Γ (q) Γ (p + q) is Euler s beta function, have been investigate in Aleixo et al., [], an in Rocha et al., [9]. Herein we consier further extensions of population ynamics first iscusse in Pestana et al. [5], Brilhante et al. [5] an Brilhante et al. [3],

3 whose inspiration has been to remark that x is the linear truncation of the series expansion of ln x, as well as x is the linear truncation of the series expansion of ln( x). In Section 2, we escribe the BeT aboop(p, q, P, Q), p, q, P, Q > 0 family of pfs, with special focus on subfamilies for which one at least of those shape parameters is. In Section 3, we iscuss generalise Verhulst ifferential equations an connect them to extreme value theory (EVT). In Section 4, some further points tying population ynamics an statistical extreme value moels are iscusse, namely the connection of the instantaneous growing factors x p an [ ln( x)] P to moels for minima, an of the retroaction control factors ( x) q an [ ln x] Q to moeling population growth using maxima extreme value moels either in the classical extreme value setting or in the geo-stable setting, where the geometric thinning curbs own growth to sustainable patterns. Section 5 iscusses what shoul be expecte from some specially remarkable ifferential escription of growth in terms of proucts of inepenent uniform ranom variables, an proucts of maxima an minima of two inepenent uniforms. 2 The X p,q,p,q BeT aboop(p, q, P, Q) moels, p, q, P, Q > 0 Let {U, U 2,..., U Q } be inepenent an ientically istribute (ii) stanar uniform ranom variables, V = Q k= U p k, p > 0, the prouct of ii Beta(p, ) ranom variables. As ln V Gamma(Q, p ), the pf of V is f V (x) = pq Γ (Q) xp ( ln x) Q I (0,) (x). While for the interpretation of V as a prouct of powers of inepenent uniform ranom variables the parameter Q must be an integer, the above expression makes sense for all Q > 0. This le Brilhante et al. [5] to introuce the so-calle Betinha(p, Q) family of ranom variables {X p,q }, p, Q > 0, with pf f Xp,Q (x) = pq Γ (Q) xp ( ln x) Q I (0,) (x), p, Q > 0, (4) to erive population growth moels that o not comply with the sustainable equilibrium exhibite by the Verhulst logistic growth moel. Observe that the Beta(p, q), p, q > 0 family, in (3), can be viewe as a truncation approximation of this more flexible Betinha(p, Q), in (4), since x is the linear term of the MacLaurin expansion ln x = k= ( x)k /k.

4 On the other han, if X q,p Betinha(q, P ), the pf of X q,p is f Xq,P (x) = qp Γ (P ) ( x)q ( ln( x)) P I (0,) (x), q, P > 0, an the family of such ranom variables also extens the Beta(p, q) family in the sense that x is the linearization of ln( x). Having in min Höler s inequality, it follows that x p ( x) q [ ln( x)] P ( ln x) Q L (0,), p, q, P, Q > 0, an hence f p,q,p,q (x) = xp ( x) q [ ln( x)] P ( ln x) Q I (0,) (x) 0 x p ( x) q [ ln( x)] P ( ln x) Q x (5) is a pf of a ranom variable X p,q,p,q for all p, q, P, Q > 0. Obviously, X p,q,p,q = X q,p,q,p. Brilhante et al. [3] use the notation X p,q,p,q BeT aboop(p, q, P, Q) for the ranom variable with pf (5) obviously the Beta(p, q), p, q > 0 family of ranom variables, in (3), is the subfamily BeT aboop(p, q,, ), an the formerly introuce Betinha(p, Q), p, Q > 0, in (4), is in this more general setting the BeT aboop(p,,, Q) family. The cases for which some of the shape parameters are an the other parameters are 2 are particularly relevant in population ynamics. In the present paper, we shall iscuss in more epth X p,,,q an X,q,P,, an in particular X 2,,,2 an X,2,2,. Some of the 5 subfamilies when one or more of the 4 shape parameters p, q, P, Q are have important applications in moeling. Below we enumerate the most relevant cases, giving interpretations in terms of proucts of powers of inepenent U k Uniform(0, ) ranom variables, for integer parameters an whenever feasible.. X,,, = U Uniform(0, ), f,,, (x) = I (0,) (x). 2. X p,,, = U p Beta(p, ), f p,,, (x) = p x p I (0,) (x). 3. X,q,, = U q Beta(, q), f,q,, (x) = q ( x) q I (0,) (x).

5 4. X,,P,, that for P N is minus the prouct of P ii stanar uniform ranom variables, P X,,P, = U k, k= U k Uniform(0, ), inepenent. More generally, for all P > 0, f,,p, (x) = ( ln( x))p I (0,) (x), Γ (P ) where Γ (P ) = 0 x P e x x is Euler s gamma function. 5. X,,,Q, that for Q N is the prouct of P ii stanar uniform ranom variables, Q X,,,Q = U k, k= U k Uniform(0, ), inepenent. Alternatively, X,,,Q can be escribe in the following hierarchical construction: Let Y = X,,, Uniform(0, ), Y 2 Uniform(0, Y ), Y 3 Uniform(0, Y 2 ),..., Y Q Uniform(0, Y Q ). Then Y Q = X,,,Q BeT aboop(,,, Q). More generally, for all Q > 0, f,,,q (x) = ( ln x)q Γ (Q) I (0,) (x). 6. X p,q,, Beta(p, q), with pf f p,q,, (x) f p,q (x), alreay given in (3). Observe that if p, q N, we have an interesting interpretation in terms of orer statistics of an uniform ranom sample: X p,q,, is then the p-th ascening orer statistic from an uniform ranom sample of size p+q, usually enote U q:p+q. As alreay observe, the pf f 2,2,, (x) = 6 x ( x) I (0,) (x) of X 2,2,, is proportional to the logistic parabola, a lanmark in the evelopment of applications of ynamic systems an chaos to analyze biological phenomena, an namely in population ynamics. Observe also that X 2,2,, is U 2:3, the meian of an uniform ranom sample of size X p,,p,, with pf f p,,p, (x) = C p,,p, x p [ ln( x)] P I (0,) (x), where C p,,p, = / 0 xp [ ln( x)] P x. Observe that for p N, C p,,p, = / p ( k= ( )k+ p k ) Γ (P ). k P

6 8. X p,,,q, with f p,,,q (x) = pq Γ (Q) xp ( ln x) Q I (0,) (x), that for Q N is the prouct of Q ii Beta(p, ), i.e. stanar uniform ranom variables raise to the power /p, cf. also Arnol et al. [2]. As x can be viewe as the linear truncation of ln x, the traitional Beta(p, q) family, with pf given in (3), can be viewe as an approximation, in what concerns the retroactive curbing own factor, of this X p,,,q family. Such a family is thus suite to moel more complex growth control patterns. Observe that ( ln x) ν > ( x) ν for each ν >, while the reverse inequality hols for ν (0, ). 9. X,q,P,, with pf f,q,p, (x) = qp Γ (P ) ( x)q [ ln( x)] P I (0,) (x). Similarly to what happens in the previous case, for some fixe value ν >, the growing factors x ν < [ ln( x)] ν, while for ν (0, ) x ν > [ ln( x)] ν. As alreay unerline, x can be viewe as the linear truncation of ln( x), an henceforth the traitional Beta(p, q) family can be viewe as an approximation, in what concerns the growing factor, of this X,q,P, family, that exhibits more complex growth patterns. 0. X,q,,Q, with pf f,q,,q (x) = C,q,,Q ( x) q [ ln x] Q I (0,) (x), where C,q,,Q = / 0 ( x)q [ ln x] Q x. More generally, the notation C p,q,p,q = / 0 xp ( x) q [ ln( x)] P [ ln x] Q x is use in the sequel.. X,,P,Q, with pf 2. X p,q,p,, with pf f,q,,q (x) = C,,P,Q [ ln( x)] P [ ln x] Q I (0,) (x). f p,q,p, (x) = C p,q,p, x p ( x) q [ ln ( x)] P I (0,) (x). 3. X p,q,,q, with pf 4. X p,,p,q, with pf f p,q,,q (x) = C p,q,,q x p ( x) q [ ln x] Q I (0,) (x). f p,,p,q (x) = C p,,p,q x p [ ln ( x)] P [ ln x] Q I (0,) (x).

7 5. X,q,P,Q, with pf f,q,p,q (x) = C,q,P,Q ( x) q [ ln ( x)] P [ ln x] Q I (0,) (x). Observe that the enominator of the norming constants C p,q,p,q = / 0 x p ( x) q [ ln( x)] P [ ln x] Q x can be viewe as moments of functions of BeT aboop ranom variables with aitional shape parameters with value. For instance, C p,q,,q = E Xp,q,, [( ln X) Q ] = E X,q,, [X p ( ln X) Q ]. In what concerns the applicability of some of the above moels (an namely 5), we have to recognize that computations are unfeasible, even if we ecie to use only lower moments an approximations instea of more powerful methos using the exact moel. However, computational algorithms can, at least partially, resolve this question when ealing with precise practical applications. 3 Generalise Verhulst ifferential equations Looking to the Verhulst equation, in (), an observing that in it N(t) f 2,2,,, with f p,q,, f p,q, given in (3), it seems worth consiering similar ifferential equations with N(t) f p,q,p,q, in (5), t N(t) = r [N(t)]p [ N(t)] q [ ln( N(t))] P [ ln(n(t))] Q, (6) p, q, P, Q > 0, namely when one at least of the parameters is. The situation p + q + P + Q = 6, with p, q, P, Q { 2,, 3 2, 2} seems also worth exploring. The solution is straightforwar for very simple cases, such as p = q = P = Q = linear growth; p = 2, q = P = Q = exponential growth; q = 2, p = P = Q = exponential ecay; p = q = 2, P = Q = logistic growth. For some combinations of the parameters, Mathematica s proceure DSolve prouces explicit (but in general very cumbersome) solutions, for instance: q = P = Q = = N(t) = [ (p 2) (c + rt)] 2 p ; p = Q = 2, q = P = = N(t) = exp(e rt+c ) Gompertzian (or Gumbel) growth;

8 ( ) p = 2, q = P =, Q = + γ = N(t) = exp [ γ(rt c)] γ Fréchet growth if γ > 0, Weibull growth if γ (, 0) (when γ 0, the limiting growth is of Gumbel type). Looking back at the biological interpretations of (), it seems reasonable to consier that in (6) [N(t)] p an [ ln( N(t))] P are growing factors; [ N(t)] q an [ ln(n(t))] Q are retroaction factors whose role in the moel is to take into account bouns impose by finite environmental resources. Therefore, some sort of equilibrium is to be expecte when p + P = q + Q (although slight eviance from such equilibrium may match some forms of extreme growth or of extinction, as iscusse later on). Looking at some plots gives some visual insight on the balance of the expaning an contracting factors: f 22 (x) f 22 (x) f 22 (x) f 22 (x) More rigorous algebraic comparisons can be mae. For instance, as shown in [5], 4 f 22 (x) = k(k + )(k + 2) f 2,k+,,(x), k=

9 an as on the other han f 2,k+,, (x) = k j=0 ( ) ( ) j k j (j + 2) B(2, k + ) f j+2,,,, it follows that X 2,,,2 is a pseuo-convex mixture (a term we use to characterise mixtures where negative weights are allowe, provie that the sum of all weights is ) of power laws, each positive even component forcing population growth, followe by a negative o component counteracting this growth impetus. The cases p + P q + Q an p + P q + Q obviously lea to explosive growth N(t) or to ultimate population extinction N(t) 0, respectively. Once again, visual insight can be gaine from some simple plots: f 2 (x) f 2 (x) f 2 (x) 0.0 f 2 (x) In what follows we shall consier only those cases for which we have some explicit solutions connecte to EVT, namely. N(t) = r [N(t)][ N(t)], t whose normalise solution is the Logistic istribution function, an its extension t N(t) = r ([N(t)])+γ [ N(t)] γ,

10 2. whose normalise solutions are the log-logistic or the symmetrise loglogistic istribution functions. N(t) = r [N(t)][ ln(n(t))], t whose normalise solution is the Gumbel istribution function (for maxima), an its extension t N(t) = r [N(t)][ ln(n(t))]+γ, whose normalise solutions are the Fréchet istribution function (for maxima) when γ > 0, an the max-weibull istribution function when γ < N(t) = r [ N(t)][ ln( N(t))], t whose normalise solution is the min-gumbel istribution function, an its extension t N(t) = r [ N(t)][ ln( N(t))]+γ, whose normalise solutions are the min-fréchet istribution function when γ > 0, an the Weibull istribution function (for minima) when γ < 0. 4 Geo-stable laws for the maxima of ii ranom variables Rachev an Resnick [6] evelope a theory of stable limits of ranomly stoppe maxima with geometric suborinator (also calle max-geo stability) similar to what ha been inepenently achieve by Rényi [7], Kovalenko [2] an in all generality by Kozubowski [3]. For a panorama cf. also Gneenko an Korolev [0]. A ranom variable is max-geo-stable if an only if geometric ranomly stoppe maxima of inepenent replicas is of the same Khinchine type. More precisely, if X, X 2,..., X n,... are inepenent replicas of X, with istribution function F, an Y Geometric(θ) inepenent of the X k s, the istribution function of max{x,..., X Y } is F k (x)θ( θ) k θf (x) = ( θ)f (x). (7) k=0 We then say that X is a max-geo-stable ranom variable (or that F is a max-geo-stable istribution function) if for all θ (0, ) there exist a θ > 0

11 an b θ R such that F (a θ x + b θ ) = θ F (x) ( θ) F (x). (8) Let us efine G(x) = e F (x), x > αf, where α F enotes the left-enpoint of F, i. e. α F = inf{x : F (x) > 0. Then (8) is equivalent to G(a θ x + b θ ) = G θ (x), (9) i.e. G is a max-stable istribution. If there is no nee of the shift parameter b θ, we say that X (or F ) is strictly max-geo-stable, an we get the maxstability equation G(a θ x) = G θ (x), first investigate by Lévy in the context of stability of sums in the ii context (an so for characteristic functions, instea of istribution functions), an aapte by Fréchet, [8], to establish the max-stability of the type G(x) = exp ( x ) γ I[0, ) (x), γ > 0. In fact, Fisher an Tippet, [7], have shown that istribution functions G of the type [ ] G(x) G γ (x) = exp ( + γx) γ I {x: +γx>0} (x), γ R, (0) with the Gumbel limiting form G 0 (x) = exp e x, x R, when γ 0, satisfy the functional equation (9), an Gneenko, [9], has shown that this general extreme value (GEV) istribution (sometimes presente in three separate branches, for γ > 0 (Fréchet), γ = 0 (Gumbel) an γ < 0 (max-weibull), while the general expression (0) is known as von Mises-Jenkinson GEV family of istributions). This, together with the characterisation of the omains of attraction of the Fréchet an Weibull types by Gneenko, [9], an of the Gumbel type by e Haan, [], form the core of classical EVT. Hence, the max-geo-stable istribution functions, F (x) = ln G γ(x), x > α F, are given by F (x) F γ (x) = ln G γ (x) =, + γx > 0, () + ( + γx) /γ The max-geo-stable istribution functions, in (), can thus be written as one of the following types:. F (x) = + x /γ I [0, ), γ > 0, a log-logistic istribution (i.e., the istribution of a ranom variable whose natural logarithm follows the logistic istribution) tie to the classical max-stable Fréchet-γ istribution,

12 2. 3. F (x) = + e x I R, the logistic istribution tie to the classical max-stable Gumbel extreme value istribution, F (x) = + ( x) /γ I (,0), γ < 0, symmetric to the log-logistic, an tie to the classical max-stable Weibullγ extreme value istribution, as first establishe by Rachev an Resnick [6]. From tail equivalence results obtaine by Resnick, [8], an by Cline, [6], it follows that the characterisation of the omains of attraction of max-geostable laws are similar to the characterisation of the omains of attraction of the classical maxima extreme value laws. It is obvious that for the same parent population, the maximum of a geometrically thinne sequence is necessarily stochastically smaller than the maximum of the full sequence, an hence max-geo-stable laws are stochastically smaller than the corresponing classical extreme value laws, as can be seen in Fig.. 0. g (x) f (x) g!0.5 (x)!(x) 0 g!0.5 (x) g 0 (x) f 0 (x) !(x) f g 0 (x) 0 (x) f (x) g (x) x Fig.. Pf s g γ(x) = G γ(x)/x, for γ = 0.5, γ = 0 an γ =, together with the normal pf, ϕ(x), an the max-geo-stable pf s f 0(x) an f (x). From the fact that min{x,..., X n } = max{ X,..., X n }, similar results follow in what concerns min-geo-stability. In what regars stochastic orering, min-geo-stable laws are stochastically greater than the corresponing classical minimum extreme value laws.

13 5 Population Dynamics, BeT aboop(p, q, P, Q) an extreme value moels Brilhante et al., [3], use ifferential equations t N(t) = r N(t) [ ln[n(t)]]+γ, (2) obtaining as solutions the three extreme value moels for maxima, max- Weibull when γ < 0, Gumbel when γ = 0 an Fréchet when γ > 0. The result for γ = 0 has also been presente in Tsoularis [20] an in Waliszewski an Konarski [25], where as usual in population growth context the Gumbel istribution is calle Gompertz function. Brilhante et al., [3], have also shown that the associate ifference equations x n+ = α x n [ ln x n ] +γ exhibit bifurcation an ultimate chaos, when numerical root fining using the fixe point metho, when α = α(γ) increases beyon values maintaining the absolute value of the erivative limite by. On the other han, if instea of the right han sie N(t) [ ln[n(t)]] +γ associate to the BeT aboop(2,,, 2 + γ) we use as right han sie [ ln[ N(t)]] +γ [ N(t)], associate to the BeT aboop(, 2 + γ, 2, ), t N(t) = r [ ln[ N(t)]]+γ [ N(t)] the solutions obtaine are the corresponing extreme value moels for minima (an bifurcation an chaos appear when solving the associate ifference equations using the fixe point metho). In view of the uality of extreme orer statistics for maxima an for minima, in the sequel we shall restrict our observation to the case (2) an the associate BeT aboop(2,,, 2 + γ) moel. As [ N(t)] k ln N(t) = > N(t), k k= for the same value of the malthusian instantaneous growth parameter r we have r N(t) [ N(t)] < r N(t) ( ln[n(t))], an hence while Verhulst ifferential equation () moels sustainable growth in view of the available resources, extreme value ifferential equations (2) moel extreme, arguably estructive unsustaine growth for instance cell growth in tumours. The connection to EVT suggests further observations: Assume that U, U 2, U 3, U 4 are ii stanar uniform ranom variables.

14 . The pf of min(u, U 2 ) is f min(u,u 2)(x) = 2 ( x) I (0,) (x) an the pf of max(u, U 2 ) is f max(u,u 2)(x) = 2 x I (0,) (x). Hence the Beta(2, 2) BeT aboop(2, 2,, ) tie to the Verhulst moel () is proportional to the prouct of the pf of the maximum an the pf of the minimum of inepenent stanar uniforms. 2. The pf of the prouct U 3 U 4 is f U3U 4 (x) = ln x I (0,) (x) an more generally, the pf of n inepenent stanar uniform ranom variables is a BeT aboop(,,, n) an hence the pf of the BeT aboop(2,,, n) tie to (2) is proportional to the prouct of f max(u,u 2) by f U3U 4. Interpreting f max(u,u 2) f U3U 4 an f max(u,u 2) f min(u,u 2) as likelihoos, this shows that the moel (2) favors more extreme population growth than the moel (). More explicitly, the pfs f,,,2 f U3U 4 (x) = ln x I (0,) (x) an f,2,, f min(u,u 2)(x) = 2 ( x) I (0,) (x) intersect each other at x 0388, an scrutiny of the graph shows that the probability that U 3 U 4 takes on very small values below that value is much higher than the probability of min(u, U 2 ) < 0388, an therefore the controlling retroaction tens to be smaller, allowing for unsustainable growth. For more on prouct of functions of powers of proucts of inepenent stanar uniform ranom variables, cf. Brilhante et al., [4], an Arnol et al., [2]. 3. The max-geo-stable laws are the logistic, the log-logistic an the symmetrise log-logistic (corresponing to the Gumbel, Fréchet an max- Weibull when there is no geometric thinning, an with a similar characterisation of omains of attraction). Hence, the classical Verhulst population growth moel, in (), can also be looke at as an extreme value moel, but in a context where there exists a natural thinning that maintains sustainable growth. As shown in [3], non-stable extreme values (arising when the hypothesis of ientically istribute ranom variables is roppe out) may arise when the retroaction factor is elaye. More involve population ynamics growth ifferential equation moels o have explicit solution for special combinations of the shape parameters. For instance, the solution of [ N(t) = r [N(t)]2 γ N(t) ] γ, γ < 2, (3) t K is K N(t) = { ( ) } γ + (γ ) r K γ +γ K t + N 0 as shown by Turner et al., [2], cf. also Tsoularis, [20].

15 As in the case of the Verhulst parabola, the ifference equations corresponing to the ifferential equations with BeTaBoOp kernel escribing other population growth equilibria o exhibit bifurcation an ultimate chaos when the corresponing unimoal curve slope brings in instability to the fixe point algorithm, inicating that the reprouction rate an ensuing growth rate is too high (an therefore resources an sustainability are enangere, an growth rate of competing species rises concomitantly). In the figure below we show the bifurcation graphs for some combinations of the parameters: Betaboop(,, 2, 2).5 Betaboop(,, 2, 3) BetaBoop(, 2, 3, 2) Betaboop(2,, 2, 2) BetaBoop(, 3, 2, 2) Betaboop(2,, 2, 3)

16 Betaboop(2, 2,, 2) Betaboop(2, 3, 2, ) Betaboop(2, 2,, 3) Betaboop(3, 2, 2, ) References. Aleixo, S., Rocha, J.L., an Pestana, D., Probabilistic Methos in Dynamical Analysis: Population Growths Associate to Moels Beta (p,q) with Allee Effect, in Peixoto, M. M; Pinto, A.A.; Ran, D.A.J., eitors, Dynamics, Games an Science, in Honour of Maur ıcio Peixoto an Davi Ran, vol II, Ch. 5, pages 79 95, New York, 20, Springer Verlag. 2. Arnol, B.C., Coelho, C.A., an Marques, F.J., The istribution of the prouct of powers of inepenent uniform ranom variables A simple but useful tool to aress an better unerstan the structure of some istributions, J. Multivariate Analysis, 202, oi:0.06/j.jmva Brilhante, M.F., Gomes, M.I., an Pestana, D., BetaBoop Brings in Chaos. Chaotic Moeling an Simulation, : 39 50, Brilhante, M.F., Menon ca, S., Pestana, D., an Sequeira, F., Using Proucts an Powers of Proucts to Test Uniformity, In Luzar-Stiffler, V., Jarec, I. an Bekic, Z. (es.), Proceeings of the ITI 200, 32n International Conference on Information Technology Interfaces, IEEE CFP0498-PRT, pages , Zagreb, Brilhante, M.F., Pestana, D., an Rocha, M.L., Betices, Bol. Soc. Port. Matem atica, 77 82, Cline, D., Convolution tails, prouct tails an omains of attraction, Probab. Theor. 72, , 986.

17 7. Fisher, R.A. an Tippett, L.H.C., Limiting form of the frequency istribution of the largest or smallest member of a sample, Proc. Cambrige Philos. Soc. 24, 80 90, Fréchet, M., Sur la loi e probabilité e l écart maximum, Ann. Soc. Polonaise Math., 6, 93 6, Gneenko, B.V., Sur la istribution limite u terme maximum une série aléatoire, Ann. Math. 44, , Gneenko, B.V., an Korolev, V.Yu., Ranom Summation: Limit Theorems an Applications, Bocca Raton, 996. CRC-Press.. Haan, L. e, On Regular Variation an its Application to the Weak Convergence of Sample Extremes, Math. Centre Tracts 32, Amsteram, Kovalenko, I.N., On a class of limit istributions for rarefie flows of homogeneous events, Lit. Mat. Sbornik 5, , 965. (Transation: On the class of limit istributions for thinning streams of homogeneous events, Selecte Transl. Math. Statist. an Prob. 9, 75 8, 97, Provience, Rhoe Islan.) 3. Kozubowski, T.J., Representation an properties of geometric stable laws, Approximation, Probability, an Relate Fiels, New York, , 994, Plenum. 4. Lotka, A.J., Elements of Physical Biology, Baltimore, 925, Williams an Wilkins Co (reprinte uner the title Elements of Mathematical Biology, New York, 956, Dover; original eition ownloaable at elementsofphysic077mbp.pf). 5. Pestana, D., Aleixo, S., an Rocha, J.L., Regular variation, paretian istributions, an the interplay of light an heavy tails in the fractality of asymptotic moels. In C. H. Skiaas, I. Dimotikalis an C. Skiaas, eitors, Chaos Theory: Moeling, Simulation an Applications, pages , Singapore 20. Worl Scientific. 6. Rachev, S.T., an Resnick, S., Max-geometric infinite ivisibility an stability, Communications in Statistics Stochastic Moels, 7:9-28, Rényi, A., A characterization of the Poisson process, MTA Mat. Kut. Int. Kzl., , 956. (English translation in Selecte Papers of Alfre Rényi,, , P. Turán, eitors, Akaémiai Kiaó, Buapest, with a note by D. Szász on posterior evelopments untill 976). 8. Resnick, S., Proucts of istribution functions attracte to extreme value laws, J. Appl. Prob. 8, , Rocha, J.L., Aleixo, S.M. an Pestana, D.D., Beta(p, q)-cantor Sets: Determinism an Ranomness. In Skiaas, C. H., Dimotikalis, Y. an Skiaas, C., eitors, Chaos Theory: Moeling, Simulation an Applications, pages , Singapore, 20, Worl Scientific. 20. Tsoularis, A., Analysis of logistic growth moels. Res. Lett. Inf. Math. Sci., vol. 2:23 46, Turner, M.E., Braley, E., Kirk, K., an Pruitt, K., A theory of growth. Mathematical Biosciences, vol. 29: , Verhulst, P.-F., Notice sur la loi que la population poursuit ans son accroissement. Corresp. Math. Physics 0:3 2, 838. (easily available in http: //books.google.pt/books?i=8gseaaaayaaj&printsec=frontcover&hl= pt-pt&source=gbs_ge_summary_r&ca=0#v=onepage&q&f=false

18 23. Verhulst, P.-F., La loi e l accroissement e la population, Nouveaux Mémoires e l Acaémie Royale es Sciences et Belles-Lettres e Bruxelles, 8: 42, 845. (Partially reprouce in Davi, H. A., an Ewars, A. W. F. Annotate Reaings in the History of Statistics, New York, 200, Springer Verlag; available at Verhulst, P.-F., Deuxième mémoire sur la loi accroissement e la population. Mémoires e l Acaémie Royale es Sciences, es Lettres et es Beaux-Arts e Belgique 20: 32, 847. ( img/) 25. Waliszewski, J., an Konarski, J., A Mystery of the Gompertz Function, in G.A. Losa, D. Merlini, T. F.Nonnenmacher an E.R. Weibel, eitors, Fractals in Biology an Meicine, Basel, , 2005, Birkhuser. This research has been supporte by National Funs through FCT Funação para a Ciência e a Tecnologia, project PEst- OE/MAT/UI0006/20, an PTDC/FEDER.