I tried to create very simple population model with only two variables; population and food.
I want food to be a function of population, every person is working on making food and creating some small surplus. However, food production is limited by area, or carrying capacity.
On the other hand, population should be function of population(reproduction) and food surplus, and probably also limited by carrying capacity.
I tried to create some system of differential equation for this problem, but i am struggling with it. Lotka–Volterra model cant be used here, since food is not reproducing on its own, but is function of population.
The equations i came up with are as follows:
$$\frac{dF}{dt} = sP(1 - \frac{P}{K_F}),$$ where $F$ is food, $s$ is surplus food constant(food production - food consumption), and $K_F$ is carrying capacity of food production.
The population equation is came up with is just logistic equation multiplied by food: $$\frac{dP}{dt} = rPF(1-\frac{P}{K_P}),$$ where $P$ is population, $r$ is population reproduction rate and $K_P$ is population carrying capacity.
Two equations can be divided to yield:
$$\frac{dP}{dF} = \frac{r}{s} F \frac{(1 - \frac{P}{K_F})}{(1-\frac{P}{K_P})}.$$
I was only able to get exact solution if I assumed $K_F = K_P$
and the solution I got is
$$\sqrt{2 r s} t = 2 \tanh^{-1}(\sqrt{\frac{P}{K_P}}) - \frac{2}{\sqrt{P}}.$$ The result seems reasonable, however it seems that what i got is similar to of logistical equation, which results in sigmoid growth.
The question is, what do you think of this system? How would you do it. Do you know of some established models that are applicable to this problem, that is population food model?
edits: as was pointed out, originally I made mistake when I integrated food equation, not realising P depends on time. I removed the associated context from the question.
