Differential equation for population dynamic without and with carrying capacity are discussed here in brief.

b-d rate flows

Suppose population size at time $t$ is define as $N$. Rate flows of birth $B$ and death $D$ can be defined as ¹

$$\tag{1} B = b N $$

and

$$\tag{2} D = d N $$

with $b$ adan $d$ are per-capita birth and date rates, respectively.

n change

Without considering carrying capacity, population changes only due to two basic flows births $B$ and deaths $D$, where the former add individuals to the population and the later remove individuals from the population, which gives ²

$$\tag{3} \frac{dN}{dt} = B - D, $$

which can be simplified further into

$$\tag{4} \frac{dN}{dt} = (b-d) N. $$

It will lead to exponential population growth when $b > d$. Usually, net growth rate $r = b - d$ is defined to simplify previous equation.

carrying capacity

In order to limit the growth of population $N$ to a saturation value, e.g. $N_\infty$, carrying capacity $K$ is introduced and it modifies previous equation into ³

$$\tag{5} \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right), $$

which makes growth exponential when $N « K$, net growth tends to zero when $N \rightarrow K$, and growth becomes negative when $N > K$.

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