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B1605
Title: Random growth rate and carrying capacity in Verhulst population dynamics Authors:  Maria Brilhante - FCiencias.ID (Portugal) [presenting]
Dinis Pestana - FCiencias.ID, Universidade de Lisboa and CEAUL (Portugal)
Maria Rocha - University of the Azores (Portugal)
Abstract: Either in biological or in economic settings it makes sense to consider that the carrying capacity $M$ and the growth rate $\rho$ in Verhulst population model ${\rm d}\,N(t)/{{\rm d} t} = \rho\, N(t) [M-N(t)]$ are random. The difference equation $x_{n+1}= r\, x_n [1-x_n], ~x_n \in[0,1],~n=1,2,\dots$ obtained from the discretization of ${\rm d}\, \alpha(t)/{{\rm d} t}= \rho M\, \alpha(t) [1-\alpha(t)], ~\alpha(t)=N(t)/M \in[0,1]$, with $\alpha(n)=(1+\rho M)\,x_n/(\rho M)\, ,~r=1+\rho M$ has been thoroughly investigated, namely in what concerns the fixed point search of the stability rate ($x_{n+1}\approx x_n$), with overwhelming success in explaining gross fluctuations of generations size. We investigate the dynamics of the ``randomized'' logistic parabola $R \,x\,(1-x), ~x\in[0,1], ~R=1+\rho M$, namely the possibility instabilities or of extinction when the growth rate is high. Interesting models, such as generalized Pareto, Morris natural exponential family, extended Panjer basic count models are used to fit $R$ from available time series, and this is the basis for a simulation study on the evolution dynamics and namely on the settings bringing in chaos.