Non-linear Biological Systems

In my last post I introduced Mathematical biology by explaining simple, 1st order, linear, differential systems.

Now, I’d like to get into non-linear systems, which better represent biological phenomenon, because the solutions output is rarely in direct proportion to the input.

Consider, for example, an enzymatic reaction. At low levels of substrate concentration, adding more substrate has a direct correlation with the amount of product form. However, the concentration of enzyme in this case limits the maximum velocity of the reaction, because the enzyme gets saturated with substrate. Therefore consider the changing population of Substrate and product like this:

-dS/dt = dP/dt = Vmax x [S] / (Km + S)

This is the Michaelis-Menten equation. Plotted it looks like this:

On the y axis is reaction velocity (dS/dt) and the y axis is substrate concentration.  While this equation is specific for enzymatic reactions, a similar model for logistic growth can be applied to natural populations, growing in a limiting environment.

Consider first a continously growing population of a single species.  As we went over last time, the size of population N at time n determines the population at Nn+1.  Although the population can only be assesed at censuses, let us consider a continuosly reproducing species.

If a population doubles during every time period of τ  then we get population growth that looks like this:

Time Number

0                    N0

τ                    2N0

2τ                  4N0

3τ                  8N0

This is a clear example of exponential growth – the coefficient of xN0 doubles for every linear increase of τ.   The general equation which fits this exponential growth is N(t) = N0 2^(t/τ) -  In the above chart, t is always given as a function of the doubling time.

Let us now consider natural death in a population (assume no competition).  In this case, contrary to the growth rate, the death rate is directly proportional to the size of the population.

dN/dt  = -γN where γ is the per capita death rate which, when solved, gives   N(t) = N0 e^(-γt) an exponentially decaying function.

To combine terms and get a useful model for a growing species, lets first put the birth equation into a more usable form:

N(t) = N0 2^(t/τ) = N0 e^ln 2 * (t/τ)  and let β = (ln2)/τ   so that N(t) = N0 e^(βt) and dN/dt = βN(t)

Combining the growth and death terms, we get:

dN/dt = βN(t) – γN(t) = (β – γ) N(t)  => N(t) = N0 e^(β – γ)t

for all β > γ you get an exponentially growing population.

But of course, exponential growth is never seen in nature (with people being the exception).  This is because finite resources limit growth.  Therefore competition ensures that the death rate will exceed the birth rate under certain conditions.  Rather than defining these rates as complex functions, we can substitute the logistics equation of growth (which pretty accurately describes population growth).

dN/dt = (β – γ) N(t) [1-(N/k)]; where the new term (k) is the carrying capacity of the population (which can usually be determined experimentally and depends on the most limiting resource).

solving the logistics equation gives:

N(t) = kNoe^(Rot) /  [K - No+ No^(Rot)]  where Ro = (β – γ).

The logistics curve, N(t),  looks something like the above (ignore the negative portion).  Here growth is initially exponential but flattens out to K.  (K equals Nmax).

In our next episode, we will consider growth on non-continuously reproducing species and intraspecific competition.

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Introduction to Mathematical Biology

I’m taking my first mathbio class this semester and I would like to share with you all the insights I’ve learned so far.

Most biological processes are 1st order dynamic systems.  Meaning that, unlike many physical phenomena, the ‘momentum’ of a system has no affect on the trajectory.  The only state that plays a role on the current state is the one directly preceding it.

Consider 2 changing populations with the same birth rate and no natural deaths.  Lets say one starts at a 10 individuals and the other starts at 15.  In the next generation, both populations double; 20 and 30.  In population two, I now remove 10 individuals and allow the populations to double again.  Both populations will end up with 40 individuals.  Even though the second population started out with more, the only state that matters is the one directly preceding it.

Lets consider a linear, 1st order, system with two variables;

dx/dt = ax + by  and dy/dt = cx/dy

What are some properties of this system (that we can extend to linear dynamic systems in general)?

1) at x = y = 0, the velocities are zero

2) If velocity at (x,y) is (Vx,Vy)

AND velocity at (x2,y2) is (Vx2,Vy2)

Then V(x1+x2,y1+y2) = (Vx1+Vx2,Vy1+Vy2)

3) If V @ (x,y) is (Vx,Vy)

then V @ (cx,cy) is (cVx,cVy)

Here is an example of the above.  Consider:

dx/dt = -x   &   dy/dt = -y

In the graph we see a description of the differential equation system.   Every arrow represents the velocity; direction and magnitude.  All paths point towards the origin, and spot at the stationary point at (0,0).

Lets know consider a more complicated system with two variables.

dx/dt = x + y    &    dy/dt = -x + y

Again, at (x,y), V = 0.  When y = 0, dx/dt = x and dy/dt = -x

In this scenario, we have a unique case at x = y; dx/dt = 2x and dy/dt = 0

Graphing this system of equations, we get something like:

In this system, we get what looks like an destabilizing stationary point at the origin, with a spiral shape pushing outwards.

If you were to imagine x and y as 2 populations over time, it would look as if x and y alternating maximums, but both maximums getting higher as time progresses:

While its doubtful that a population of two species look like this in nature, it helps demonstrate how changing populations over time can be modeled with differential equations.  For example, if species y is a predator of species x and species x has ample resources to grow, a growth pattern like this could occur, as blooms in one species causes a bloom in the other, and species x is able to take advantage of crashing y populations, and ample other resources to rebuild its own numbers even greater than before.

I continue this series next time with more modeling of populations.

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