Conclusion

The equation to simulate the population in the case we have considered is quite simple because it is a quadratic function: $x_{i+1}=\lambda_0 x_i(1-x_i)$. But it shows a very rich variety of different behaviours if we change the parameter (maximum growth rate) $\lambda _0$. It can be shown that for a certain value for $\lambda _0$ (which is approximately $3.570$) the attractor is a Cantor set. That is why we obtain a fractal for some values of $\lambda _0$.