Extensions of Verhults Model in Population Dynamics and Extremes

Transcription

1 Extensions of Verhults Moel in Population Dynamics an Extremes Maria e Fátima Brilhante 1, Maria Ivette Gomes 2, an Dinis Pestana 3 1 Universiae os Açores an CEAUL, Ponta Delgaa, Açores, Portugal ( fbrilhante@uac.pt ) 2 CEAUL Centro e Estatística e Aplicações a 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 GEV (General Extreme Value Moel) from the ii 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. 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(1 N(t))] P 1 an the linearization [N(t)] p 1 moeling two very ifferent growth patterns, an factors of the form [ ln N(t)] Q 1 an the linearization [1 N(t)] q 1 leaing to very ifferent ambiental resources control of the growth behavior. Keywors: Verhults logistic moel, Beta an BeTaBoOp moels, population ynamics, extreme value moels, geometric thinning, ranomly stoppe maxima with geometric suborinator. 1 Introuction Let N(t) enote the size of some population at time t. Verhults [16], [17], [18] impose some natural regularity conitions on N(t), namely that t N(t) = A k [N(t)] k, with A 0 = 0 since nothing can stem out from an extinct k=0 population, A 1 > 0 a growing parameter, A 2 < 0 a retroaction parameter controlling sustainable growth tie to available resources, see also Lotka [9]. The secon orer approximation t N(t) = A 1N(t) + A 2 [N(t)] 2 can be rewritten [ N(t) = r N(t) 1 N(t) ] (1) t K

2 where r > 0 is frequently interprete as a Malthusian instantaneous growth rate parameter when moeling natural breeing populations, an K > 0 as the equilibrium limit size of the population. The general form of the solution of the (1) ifferential equation approximation 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), an this is the reason why in the context of population ynamics r x (1 x) is frequently referre to as the logistic parabola. Due to the seasonal reprouction an time life of many natural populations, the ifferential equation (1) is often iscretize, first taking r such that [ N(t + 1) N(t) = r N(t) ] an then α = r + 1, x(t) = r N(t) r +1, to 1 N(t) K obtain x(t + 1) = α x(t)[1 x(t)], an then the asociate ifference equation x n+1 = αx n [1 x n ], (2) where it is convenient to eal with the assumption x k [0, 1], k = 1, 2,... The equilibrium x n+1 = x n leas to a simple secon orer algebraic equation with positive root 1 1 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 (1 2 x k ) > 1 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) may be rewritten x n+1 = α 6 6 x n [1 x n ], an that f(x) = 6 x (1 x) I (0,1) (x) is the Beta(2, 2) probability ensity function. Extensions of the Verhults moel using ifference equations similar to (2), but where the right han sie is tie to a more general Beta(p, q) probability ensity function have been investigate in Aleixo et al. [1] an in Rocha et al. [13]. Herein we consier further extensions of population ynamics first iscusse in Pestana et al. [10], Brilhante et al. [5] an in Brilhante et al. [3], whose inspiration has been to remark that 1 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(1 x). In Section 2, we escribe the BeT aboop(p, q, P, Q), p, q, P, Q > 0 family of probability ensity functions, with special focus on subfamilies for which one at least of those shape parameters is 1. In section 3, some points tying population ynamics an statistical extreme value moels are iscusse, namely iscussing the connection of the instantaneous growing factors x p 1 an [ ln(1 x)] P 1 to moels for minima, an the retroaction control factors (1 x) q 1 an [ ln x] Q 1 to moeling population growth using maxima extreme value moels either in the

3 classical extreme value setting, either in the geo-stable setting, where the geometric thinning curbs own growth to sustainable patterns. 2 The X p,q,p,q BeT aboop(p, q, P, Q) moels, p, q, P, Q > 0 Let {U 1, U 2,..., U Q } be inepenent an ientically istribute (ii) stanar uniform ranom variables, V = U 1 p Q k, p > 0 the prouct of ii Beta(p, 1) ranom variables. As ln V Gamma(Q, 1 p ), the probability ensity function (pf) of V is f V (x) = pq Γ (Q) xp 1 ( ln x) Q 1 I (0,1) (x). Brilhante et al. [5] iscusse the more general Betinha(p, Q) family of ranom variables {X p,q }, p, Q > 0, with pf f Xp,Q (x) = pq Γ (Q) xp 1 ( ln x) Q 1 I (0,1) (x), p, Q > 0 that can be consiere an extension of the Beta(p, q), p, q > 0 family, since (1 x) k 1 x is the linearization of the MacLaurin expansion ln x = k to erive population growth moels that o not comply with the sustainable equilibrium exhibite by the Verhulst logistic growth moel. On the other han, if X q,p Betinha(q, P ), the pf of 1 X q,p is f 1 Xq,P (x) = qp Γ (P ) (1 x)q 1 ( ln(1 x)) P 1 I (0,1) (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(1 x). Having in min Höler s inequality, it follows that x p 1 (1 x) q 1 [ ln(1 x)] P 1 ( ln x) Q 1 L 1 (0,1), p, q, P, Q > 0, an hence f Xp,q,P,Q (x) = xp 1 (1 x) q 1 [ ln(1 x)] P 1 ( ln x) Q 1 I (0,1) (x) 1 0 x p 1 (1 x) q 1 [ ln(1 x)] P 1 ( ln x) Q 1 x is a pf for all p, q, P, Q > 0. Obviously, 1 X p,q,p,q = X q,p,q,p. For simplicity, in what follows we shall use the lighter notation f p,q,p,q instea of f Xp,q,P,Q for the ensity of X p,q,p,q. Brilhante et al. [3] use the notation X p,q,p,q BeT aboop(p, q, P, Q) for the ranom variable with pf (3) obviously the Beta(p, q), p, q > 0 (3)

4 family of ranom variables is the subfamily BeT aboop(p, q, 1, 1), an the formerly introuce Betinha(p, Q), p, Q > 0 is in this more general setting the BeT aboop(p, 1, 1, Q) family. The cases for which some of the shape parameters are 1 an the other parameters are 2 are particularly relevant in population ynamics. In the present paper, we shall iscuss in more epth X p,1,1,q an X 1,q,P,1, an in particular X 2,1,1,2 an X 1,2,2,1. Some of the 15 subfamilies when one or more of the 4 shape parameters p, q, P, Q are 1 have important applications in moeling; below we enumerate the most relevant cases, giving interpretations, for integer parameters, in terms of proucts of powers of inepenent U k Uniform(0, 1) ranom variables. 1. X 1,1,1,1 = U Uniform(0, 1); f 1,1,1,1 (x) = I (0,1) (x). 2. X p,1,1,1 = U 1 p Beta(p, 1); fp,1,1,1 (x) = p x p 1 I (0,1) (x). 3. X 1,q,1,1 = 1 U 1 q Beta(1, q); f1,q,1,1 (x) = q (1 x) q 1 I (0,1) (x). 4. X 1,1,P,1, that for P N is 1 minus the prouct of P ii stanar uniform ranom variables, P X 1,1,P,1 = 1 U k, U k Uniform(0, 1), inepenent. More generally, for all P > 0, f 1,1,P,1 (x) = where Γ (P ) = 0 1 ( ln(1 x))p Γ (P ) x P 1 e x x is Euler s gamma function. I (0,1) (x), 5. X 1,1,1,Q, that for Q N is the prouct of P ii stanar uniform ranom variables, Q X 1,1,1,Q = U k, U k Uniform(0, 1), inepenent; alternatively, X 1,1,1,Q may be escribe in the following hierarchical construction: enote Y 1 = X1,1,1,1 Uniform(0, 1), Y 2 Uniform(0, Y 1 ), Y 3 Uniform(0, Y 2 ),..., Y Q Uniform(0, Y Q 1 ). Then Y Q = X1,1,1,Q BeT aboop(1, 1, 1, Q). ( ln(x))q 1 More generally, for all Q > 0, f 1,1,1,Q (x) = I (0,1) (x). Γ (Q) 6. X p,q,1,1 Beta(p, q), with f p,q,1,1 (x) = xp 1 (1 x) q 1 I (0,1) (x), where B(p, q) 1 as usual B(p, q) = x p 1 (1 x) q 1 Γ (p) Γ (q) x = is Euler s beta 0 Γ (p + q) function.

5 7. X p,1,p,1, with pf f p,1,p,1 (x) = C p,1,p,1 x p 1 [ ln(1 x)] P 1 I (0,1) (x), 1 where C p,1,p,1 = 1 0 xp 1 [ ln(1 x)] P 1 x. For p N, C = p,1,p,1 1 p ( ). p 1 Γ (P ) ( 1) k+1 k 1 k P 8. X 1,q,P,1, with pf f 1,q,P,1 (x) = qp Γ (P ) (1 x)q 1 [ ln(1 x)] P 1 I (0,1) (x). 9. X p,1,1,q, with pf f p,1,1,q (x) = pq Γ (Q) xp 1 [ ln x] Q 1 I (0,1) (x), that for Q N is the prouct of Q ii Beta(p, 1), i.e. stanar uniform ranom variables raise to the power 1 p, cf. also Arnol et al. [2] (we postpone the iscussion of the more complicate moels to the full paper, since they are not iscusse in this shorter version; observe also that the only moels for which an explicit evaluation of raw an of central moments is straightforwar are those with q = P = 1 or with P = Q = 1, an so they are the natural caniates to moel population ynamics). 3 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)]]1+γ (4) obtaining as solutions the three extreme value moels for maxima, Weibull when γ < 0, Gumbel when γ = 0 an Fréchet when γ > 0. The result for γ = 0 has also been presente in Tsoularis [14] an in Waliszewski an Konarski [19], 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+1 = α x n [ ln x n ] 1+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 1. On the other han, if instea of the right han sie N(t) [ ln[n(t)]] 1+γ associate to the BeT aboop(2, 1, 1, 2 + γ) we use as right han sie [ ln[1 N(t)]] 1+γ [1 N(t)], associate to the BeT aboop(1, 2 + γ, 2, 1), t N(t) = r [ ln[1 N(t)]]1+γ [1 N(t)]

6 the solutions obtaine are the corresponing extreme value moels for minima (an bifurcation an chaos 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 (4) an the associate BeT aboop(2, 1, 1, 2 + γ) moel. [1 N(t)] k As ln N(t) = > 1 N(t), for the same value of the k malthusian instantaneous growth parameter r we have r N(t) [1 N(t)] < r N(t) ( ln[n(t))], an hence while (1) moels sustainable growth in view of the available resources, (4) moels extreme value, arguably estructive unsustaine growth for instance cell growth in tumours. The connection to extreme value theory suggests further observations: Assume that U 1, U 2, U 3, U 4 are inepenent ientically istribute stanar uniform ranom variables. 1. The pf of min(u 1, U 2 ) is f min(u1,u 2)(x) = 2 (1 x) I (0,1) (x) an the pf of max(u 1, U 2 ) is f max(u1,u 2)(x) = 2 x I (0,1) (x). Hence the Beta(2, 2) BeT aboop(2, 2, 1, 1) tie to the Verhults moel (1) 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 (0,1) (x) an more generally, the pf of n inepenent stanar uniform ranom variables is a BeT aboop(1, 1, 1, n) an hence the pf of the BeT aboop(2, 1, 1, n) tie to (4) is proportional to the prouct of f max(u1,u 2) by f (U3U 4). Interpreting f max(u1,u 2) f (U3U 4) an f max(u1,u 2) f min(u1,u 2) as likelihoos, this shows that the moel (4) favors more extreme population growth than the moel (1). More explicitly, the probability ensity functions f 1,1,1,2 f (U3U 4)(x) = ln x I (0,1) (x) an f 1,2,1,1 f min(u1,u 2)(x) = 2 (1 x) I (0,1) (x) intersect each other at x , 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 1, U 2 ) < , 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. Rachev an Resnick, [11] evelope a theory of stable limits of ranomly stoppe maxima with geometric suborinator (also calle geo-max stability) similar to what ha been inepenently achieve by Rényi [12], Kovalenko [7] an in all generality by Kozubowski [8], for a panorama cf. also Gneenko an Korolev, [6]. The geo-stable maxima laws are the logistic, the log-logistic an the simetrize log-logistic (corresponing to the Gumbel, Fréchet an Weibull when there is no geometric thinning, an with similr characterization

7 of omains of attraction). Hence, the classical Verhulst (1) population growth moel can also be looke at as an extreme value moel, but in a context where there exists a natural thinning that maintains sustainable growth. 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 t [ 1 N(t) ] γ, γ < 2 (5) K is K N(t) = { ( ) } 1 1 γ 1 + (γ 1) r K 1 γ 1+γ K t + N 0 1 as shown by Turner et al. [15], cf. also Tsoularis [14]. References 1. 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 Maurício Peixoto an Davi Ran, vol II, Ch. 5, pages 79 95, New York, Springer Verlag. 2. Arnol, B. C., Carlos A. 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, 2012, oi: /j.jmva Brilhante, M. F., Gomes, M. I., an Pestana, D., BetaBoop Brings in Chaos. Chaotic Moeling an Simulation, 1: 39 50, Brilhante, M. F., Menonça, 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 2010, 32n International Conference on Information Technology Interfaces, IEEE CFP10498-PRT, pages , Zagreb, Brilhante, M. F., Pestana, D. an Rocha, M. L., Betices, Bol. Soc. Port. Matemática, , Gneenko, B. V., an Korolev, V. Yu., Ranom Summation: Limit Theorems an Applications, Bocca Raton, CRC-Press. 7. Kovalenko, I. N., On a class of limit istributions for rarefie flows of homogeneous events, Lit. Mat. Sbornik 5, , (Transation: On the class of limit istributions for thinning streams of homogeneous events, Selecte Transl. Math. Statist. an Prob. 9, 75 81, 1971, Provience, Rhoe Islan.) 8. Kozubowski, T. J., Representation an properties of geometric stable laws, Approximation, Probability, an Relate Fiels, New York, , 1994, Plenum.

8 9. Lotka, A. J., Elements of Physical Biology, Baltimore, 1925, Williams an Wilkins Co (reprinte uner the title Elements of Mathematical Biology, New York, 1956, Dover; original eition ownloaable at elementsofphysic017171mbp.pf). 10. 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 Worl Scientific. 11. Rachev, S. T., an Resnick, S., Max-geometric infinite ivisibility an stability, Communications in Statistics Stochastic Moels, 7: , Rényi, A., A characterization of the Poisson process, MTA Mat. Kut. Int. Kzl. 1, , (English translation in Selecte Papers of Alfre Rényi, 1, , P. Turán, eitors, Akaémiai Kiaó, Buapest, with a note by D. Szász on posterior evelopments untill 1976). 13. 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, 2011, Worl Scientific. 14. A. Tsoularis. Analysis of logistic growth moels. Res. Lett. Inf. Math. Sci., vol. 2:23 46, M. E. Turner, E. Braley, K. Kirk, K. Pruitt. A theory of growth. Mathematical Biosciences, vol. 29: , Verhults, P.-F. Notice sur la loi que la population poursuit ans son accroissement. Corresp. Math. Physics 10: , Verhults, P.-F. La loi e l accroissement e la population, Nouveaux Mémoires e l Acaémie Royale es Sciences et Belles-Lettres e Bruxelles, 18:1 42, (Partially reprouce in Davi, H. A., an Ewars, A. W. F. Annotate Reaings in the History of Statistics, New York, 2001, Springer Verlag. 18. Verhults, 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 20:1 32, 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/UI0006/2011, an PTDC/FEDER.