In a closed ecosystem, we have that Frogs,represented by $F(t)$, eat fleas, represented by $f(t)$, and the fleas eat fungus, represented by $g(t)$.
Assuming that fungus grows at rate A, fleas eat the fungus at rate B, frogs eat the fleas at rate C, and frogs die at rate D, write the system of equations describing the populations in this system:
My attempt:
Fungus grows at rate A, which means $dg/dt = A$
Frog = $dF/dt = C-D$ as frogs thrive on eating fleas and dying at rate D
Fleas = $df/dt = -C $ as frogs are eating fleas at rate C
Why is this wrong?
The rates aren't fixed; they are proportional to the size of the population. For the fungus, for example, you might have $\frac{dg}{dt}=Ag(t)-Bf(t)$, rather than $\frac{dg}{dt}=A-B$. This should make sense if you think about it for a second. If the flea population was zero, would they be able to eat the fungus? If the fungus population was zero, how would they reproduce?
It is also generally assumed that a population grows in a way proportional to both the current population, and the available food. So the flea equation would be $\frac{df}{dt}=Bf(t)-CF(t)$, rather than $\frac{df}{dt}=-CF(t)$.