Let’s go back to the world’s simplest model $ X' = bX $, where $ b$ is the (positive) per capita growth rate of the population. The population grows without bound, to infinity.

If we just look at the vector field, we see why. The change vectors get bigger and bigger in direct proportion to $ X $.

but that’s absurd. So what would a more reasonable model look like? The concept of “carrying capacity” leads us to a nonlinear model $$ X' = bX\times \left(1-\dfrac{X}{k} \right) $$

here the change arrows point to the right (increasing) for all $ X < k $, and point to the left (decreasing) for $X > k$.

This results in the population approaching the limiting value of $ X = k $ for all non-zero initial conditions.