Competition Models

Question

Need help getting started on this homework problem and I am really lost. The notes given on this subject are really sparse and I haven't found anything online that was useful. Sorry about the lack of LaTeX

Let $N(t)$ be the total population of hominids, which consists of a population of Neanderthals, $x(t)$ and humans $y(t)$: $N(t) = x(t) + y(t)$.

Suppose the two speciies lived in the same resource-limited environment and therefore the total population satisfies the logistic equation: $dN/dt = rN(1-(N/K)) - \beta N$ where K is the total carrying capacity for all hominids combined and beta is their mortality rate. We assume $r > \beta > 0$ becuase the net growth rate should be positive for small populations.

a) suppose there is no difference in the two species' survival skills. Write down two coupled equations for $x(t)$ and $y(t)$ in the form $$ \frac{dx}{dt} = x(F(x,y) - \beta)$$ $$\frac{dy}{dt} = y(F(x,y) - \beta)$$ where $F(x,y)$ is the same in both.

I'm really not sure how to find these initial equations. I don't feel like I'm understanding the problem outside of the initial logistic equation.

Answer 1

Write $N=x+y$ and note that $\dot{N} = \dot{x}+\dot{y} = x(F-\beta)+y(F-\beta)=N(F-\beta)$. Then you can put this into the original logistic equation and solve for $F$:

$$ N(F-\beta) = rN(1-N/K)-\beta N,$$ $$ F = r(1-N/K).$$

Then replace $N$ with $x+y$ and you get $F(x,y) = r(1-(x+y)/K)$.