Big bang bifurcations and Allee effect in Blumberg’s dynamics

Big Bang Bifurcation Analysis and Allee Effect in Generic Growth Functions

“…generic growth functions with Allee effect. We remark that this definition of Allee's functions is similar to the class of weak Allee's functions analyzed in [Rocha et al, 2013[Rocha et al, , 2015a[Rocha et al, , 2015d[Rocha et al, , 2016, see also [Elaydi & Sacker, 2010;Gyllenberg et al, 1996;Rocha et al, 2014aRocha et al, , 2015bSchreiber, 2001]. The next results describe the possible initial population densities for which the extinction occurs and also where the dynamics remains inside, i.e.…”

“…(i) for β = γ = 1, f (x; r, 1, 1) = rx(1 − x) is the logistic function; (ii) for β = 1 and 0 < γ < 2, f (x; r, 1, γ) = rx 2−γ (1 − x) γ is a particular case of Blumberg's functions, see [Rocha & Aleixo, 2013a] and [Rocha et al, 2014a]; (iii) for β > 0 and γ = 1, f (x; r, β, 1) = rx(1 − x β ) are Richards' functions, see [Rocha et al, 2013b] and [Rocha et al, 2014b], with two fixed points x = 0 and x = r−1 r 1/β . The degenerate limit case is verified,…”

“…4, [Mira, 1987;Mira et al, 1996] for more details. See also [Rocha et al, 2014a[Rocha et al, , 2015b[Rocha et al, , 2015c[Rocha et al, , 2015d[Rocha et al, , 2016 for analogous studies. In Figs.…”

“…Surprisingly, the models studied in this work, given by Eq. (2), naturally incorporate the Allee effect, see similar studies in [Rocha et al, 2014a] and [Rocha et al, 2015b]. In the works cited previously, in [Elaydi & Sacker, 2010] and the references therein can be consulted for more information on Allee effects for several growth models.…”

“…The previous works are examples of different approaches in the use of the population size function f (N ) associated to the differential equation that defines the growth model under study. Similarly, considering population size functions, Rocha et al presents probabilistic and dynamical studies in different logistic growth models, which can be seen in [Rocha & Aleixo, 2013a;Rocha et al, 2013;Rocha & Aleixo, 2013b;Rocha et al, 2014a;Rocha et al, 2014b;Rocha et al, 2015aRocha et al, , 2015bRocha et al, , 2015cRocha et al, , 2015dRocha et al, , 2016.…”