# How fast will a shape grow if it can grow exponentially only at the border, and growth is limited by crowding?

Take a hypothetical bacterium which divide once per minute. After $n$ minutes there will be $2^n$ bacteria, assuming no constraints.

But what if its growth is constrained by resources and space? I am imagining a simple model in which "food" is distributed uniformly across the plane/volume.

• There is enough food in the vicinity of any bacterium to divide provided that it is not surrounded by other bacteria. In which case it will divide and the space and food around it will be consumed.

• Otherwise there is enough food in the vicinity of any enclosed bacterium to keep it alive for as long as the experiment is running.

• Neither food nor bacteria can move.

• Assuming circular bacteria forming a honeycomb, imagine a "dent" in the border -- there will be four neighbours to the space and two gaps, therefore only two of the neighbours will be able to divide.

It's tempting to say that, although the central part is static, the perimeter will still grow exponentially and therefore the area/number will still be exponential, but I'd like an analysis which takes into account the crowding at the border. ideally for non-infinitesimal bacteria.

(I don't have a use for this, it just seems like an interesting exercise.)

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Your discussion could take place in an alternative geometry where no crowding ever takes place: the geometry of the hyperbolic plane $\mathbb{H}$, where the perimeter of a circle of radius $r$ indeed grows exponentially as a function of $r$. But your assumption that the "isoperimetric ratio" area/number is exponential is incorrect for this example, indeed the isoperimetric ratio is actually bounded for large circles the hyperbolic plane. –  Lee Mosher May 31 at 14:05
This sounds like "a new border once per division cycle" which is one bacterium in size. In other words, it sounds like growth matching the parameters of the shape (which I think eventually approximates a circle) and is thus quadratic for planar shapes... –  abiessu May 31 at 14:06

If the bacteria all multiply at the same rate then you are in effect adding "area" to the bacterial colony at a rate which is proportional to the perimeter. Say the colony has a radius $R$ then the perimeter is $2\pi R$ so the rate of bacteria growth is $\lambda 2\pi R$ for some $\lambda >0$. Thus we have a differential equation
$$\frac{dA}{dt} = 2\pi \lambda R = 2\lambda \sqrt{\pi\,A}$$
$$A = \frac{1}{4}(4 B\lambda\sqrt{\pi}\, t +C^2+4\lambda^2\pi\,t^2)$$
where $B$ and $C$ are constants. So you can see that the growth rate of the number of bacteria (which is $\propto A$) is quadratic in time not exponential.