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Title: Population dynamics: Risk modeling and extremes in the generalized Verhulst model Authors:  Maria Brilhante - FCiencias.ID (Portugal) [presenting]
Ivette Gomes - FCiencias.ID, Universidade de Lisboa and CEAUL (Portugal)
Dinis Pestana - FCiencias.ID, Universidade de Lisboa and CEAUL (Portugal)
Sandra Mendonca - Universidade da Madeira and CEAUL (Portugal)
Abstract: The Verhulst Model $dN(t)/dt=rN(t)[1-N(t)/K]$, for a carrying capacity $K$, whose logistic solution describes sustainable population growth when the Malthusian reproduction rate $r$ belongs to [1,3], and is proportional to the Rachev and Resnick geo-max-stable logistic law. Observe that this sustainable growth results form the equilibrium between the increase factor $N(t)$ and the retroaction control $1-N(t)/K$. Unstabilities that arise when the Malthusian parameter $r$ falls out of the range [1,3] have been thoroughly described in the chaos literature. The Verhulst model can be looked as an approximation of $dN(t)/dt=r [-ln[1-N(t)/K] [ln N(t)]$; other approximations of the form $dN(t)/dt=r N^{p-1}(t)[1-N(t)/K]^{q-1}$ - and more generally of the form $dN(t)/dt=r N^{p-1}(t)[ln[1-N(t)/K]^{P-1}[1-N(t)/K]^{q-1}[-ln N(t)]^{Q-1}$ - for appropriate choices of equilibria of the growth factor and of the retroaction control, have solutions that are proportional to a Rachev and Resnick geo-max-stable law (respectively to a Fisher and Tippett extreme value stable law), either of maxima or of minima. In particular, the solution of $dN(t)/dt=r N(t)[-ln N(t)]$ is the Gompertz function, proportional to the Gumbel extreme value law, and this has been studied as a population growth model of cells in cancer tumors. Our aim is to investigate how far we may use evaluations of growth and of control factors to model risks of extreme events in population dynamics.