Model
There are three types of animals:
- $Y$, young (0-5 years old)
- $A$, adult (5-10 years old)
- $O$, old (10 years old or more)
The initial conditions of the system are $Y_0=2500$, $A_0=1200$, $O_0=500$.
The probability that:
- an young animal turns into an adult one is $0.68$
- an adult animal turns into an old one is $0.30$
- an old animal dies is $0.8$
The adult and old animals produce together $90$ offspring per $100$ animals.
Problems
- Using the information above describe the system with difference equations.
$Y(t)=$
$A(t)=$
$O(t)=$
where $t$ is a time constant. $t=0$ is 2006. Each $t$ increment by $1$ corresponds to $5$ years in real world.
- Find an $\frac{offsprings}{animals}$ ratio for which the becomes stable
That's what I've tried, however it doesn't seem to describe this system well:
1. $$Y(t)=0.9(O(t-1)+A(t-1))$$ $$A(t)=0.68Y(t-1)$$ $$O(t)=0.3A(t-1)+0.2O(t-1)$$
Any ideas and hints are appreciated, feel free to ask questions in the comment section.
Question 1: \begin{align*} Y_{n+1}&=0.90O_n+0.90A_n\\ A_{n+1}&=0.68Y_n\\ O_{n+1}&=0.30A_n-0.80O_n \end{align*}