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.

This entry was posted on Thursday, March 12th, 2009 at 5:25 pm and is filed under ecology, mathematics, medicine, musings. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.