Difference between Population Growth Models (Differential Equations)

Question

I'm currently studying population growth models in Math class right now and is presented with different equations for different models.

I think I understand that we use $dP/dt = rP$ (where $r$ is the intrinsic growth rate and $P$ is the population) when we have infinite growth.

However, I'm struggling to find the difference between the models:

$dP/dt = rP(1-P/k)$ --> (where $r$ is the intrinsic growth rate, $P$ is the population and $k$ is the carrying capacity)

and

$dP/dt = r(k-P)$ -->(where $r$ is the intrinsic growth rate, $P$ is the population and $k$ is the carrying capacity)

as they both pertain to rate of growth with carrying capacity...

Any explanation/clarification is greatly appreciated!

Answer 1

In the second, $r$ is the intrinsic death rate, growth happens at $rk$ units per time unit.

Note that the first one of the problematic models is quadratic while the second one is linear. Thus the first has two fixed points at $0$ and $k$, while the second only has the one at $k$.

Answer 2

The logistic model$$dP/dt = rP(1-P/k)$$ has two equilibrium points namely $P=0$ and $P=k$ with $P=k$ being the stable one.

The $$ dP/dt = r(k-P)$$ model has only one equilibrium point which is stable at $P=k$

The behavior at $P<0$ is significantly different for the two models but they behaves similar around $P=k$

Answer 3

Probably the best way to analyse your models is to plot the r.h.s., let denote it $f(P)$, versus $P$.

In the first case, the population grows a little when close to $0$ (population is too small), then the growth rate increases and attains its maximum at $P=k/2$ (optimal conditions). As $P$ continues to grow, $f(P)$ decreases and reaches $0$ at $P=k$ (due to the competition for resources). This picture agrees pretty well with our intuition.

In contrast to that, in the second model $f(P)$ is maximal when $P=0$ and gradually decreases as $P$ grows. That is to say, the smaller the population, the faster it grows. In this sense, this model gives wrong results for small population sizes.