In my ODE course, we have a final project, and I'm looking to do something involving bio math, I know, a very broad topic, but I'm looking at population modeling, though I'm decent enough with with the normal ODE stuff, the analytics is beyond me, as is using matlab to help graph things. I was looking at Hassel's model, where $x_n ≥ 0$, and $r > 0$ is a positive parameter:
$x_{n+1}=\frac{rx_n}{(1+x_n)^3}$
When trying to determine equilibria here, is it simply -1,0? Because its a population, I think I should only consider nonnegative equilibriums. For the r parameter, I believe that it's range should be $r \neq 0$, as that would copy the equilibrium.
I was looking for help using matlab to plot the time series of solutions with $r = 0.9, r = 2$ and $r = 30$ with $x_1 = 1$. Though I know how to use matlab to plot, say the next 100 solutions, but is that the same as plotting a time series? Further, how do you determine stability of the equilibria for each r? Can you tell simply from the plot?
Also, how can you tell the bifurcation points of r where the stability of the equilibria change? How can you classify the bifurcations? What are the bifurcation points of r where the stability of either equilibrium changes? Any help sketching a bifurcation diagram with r as the bifurcation parameter would be helpful. Lastly any help ploting a bifurcation diagram over a range of r, say $x \leq r \leq y$
Sorry for huge post. I'm just looking to get a jump start on thinking about problems like this but I feel that it's all going over my head.
Equilibria for the discrete dynamics $x_{n+1}=g(x_n)$ are fixed-points $x=g(x)$. In your recursion that leads to $x=0$ or $$ (1+x)^3=r\implies x=\sqrt[3]{r}-1. $$
An equilibrium is stable if $|g'(x)|<1$.
Bifurcations are situations where equilibria as functions of the parameter $r$ cross or the number of equilibria changes.
Now $g'(x)=\frac{r(1-2x)}{(1+x)^4)}$ so that $g'(0)=r$ and $g'(\sqrt[3]{r}-1)=\frac{3}{\sqrt[3]{r}}-2$. This makes $x=0$ attracting or stable for $r<1$ and $x=\sqrt[3]{r}-1$ stable for $r>1$. At $r=1$ you get a crossing of the fixed points, thus a bifurcation.