what is the meaning of Ricker model?

Question

How can I interpret Ricker model? It is about population model but it has the term $e^{1-x}$ see:

$$a_{t+1} = a_t e^{r\left(1-\frac{a_t}{k}\right)}.$$

http://www.dma.uvigo.es/~eliz/pdf/oman.pdf http://en.wikipedia.org/wiki/Ricker_model

Answer 1

When the population is small, it will grow with a factor $e^r$ from one generation to the next.

However, the environment (or other conditions) limits the population size. The population capacity of the system is $k$, so if the population grows beyond $k$ this cannot be maintained and the population will start decreasing. Also, as the population increases towards $k$, the growth from one generation to the next will be reduced.

Depending on the value of $r$, there seems to be three different scenarios of population growths when starting with a population $a$ less than the capacity $k$. In the first scenario ($r\le1$), the population will grow monotonously towards $k$. The second alternative ($r\in(1,2]$) is that the population will grow beyond $k$ in some generation, but then drop below $k$ in the next, and subsequently fluctuate around $k$ while converging towards $k$. The third alternative ($r>2$) is that the population will grow beyond $k$, and then as in alternative two fluctuate around $k$, but without ever converging to $k$.