We assume that in a battle army #1 has resources A at a given time and army #2 has resources B at a given time.

• We assume that the resources of army #1 are destroyed at a rate proportional to the resources of army B, and that the resources of army #2 are destroyed at a rate proportional to the resources of army A.
• Thus we have dA / dt = - k1 B and dB / dt = -k2 B.
• The constants k1 and k2 of proportionality can be thought of, respectively, as the 'kill efficiencies' of armies #2 and #1.

Suppose that army B has 'kill efficiency' .03 while army A has 'kill efficiency' .006.

• In this case we obtain the differential equations shown that the bottom of the figure below.
• We note that the efficiency of army B is 5 times that of army A.

We can plot a 'direction field' for this situation on a graph representing B vs. A.

• At each point on this graph we will move a distance proportional to dA / dt in the A direction and a distance proportional to dB / dt in the B direction.
• The slope of our segment will therefore be [ dB / dt ] / [ dA / dt ].
• Some representatives slopes are shown the figure below.

Since the magnitude of dB / dt increases with increasing A, for given B, the slopes will tend to increase as we move to the right.

Since the magnitude of dA / dt increases with increasing B, the slopes will tend to decrease as we move upward.

The figure below shows these tendencies.

For example at the point (100, 100), we see that dB / dA = .2, so the line segment would have slope .2.

More generally we see, as in the second figure below, that dB / dA = A / B.

Video Clip #01

Since the slope dB / dA is equal to .2 A / B, we see that constant slope will occur when slope = .2 A / B = c = constant.

• This occurs when B = .2 / c * A.
• For various values of c we sketch the lines B = .2 / c * A, and on each line we sketch a segment is slope is c.
• This slope of a line on the B vs. A graph is .2 / c; the slopes of the segments on this line are c.
• For example, if c = .3, this slope of the B vs. A line (the 'blue' line) will be .2 / c = .2 / .3 = .7 (approx), while the slopes of the 'red' segments will be c = .3.

Below the line with a slope of the line and that of the segments on the line are equal, a graph of B vs. A will curve more and more sharply downward and will intercept the A the axis, corresponding to a victory for army A (which will have resources left when resources of army B are completely depleted).

Above this line the graph will curve in such a way as to intercept the B axis, corresponding to victory by army B.

Thus army B wins for situations in which the initial resources of the two armies correspond to points above the designated line, while army A wins for situations corresponding to points below the designated line.

• We see that to win army A must have much greater initial resources that army B.
• This is due to the fact that army B is much more efficient at destroying resources of army A.

Video Clip #02

The figure below depicts the system of equations for the populations of two competing species.

The behavior of each species is assumed to be a logistic model, minus a competition term.

• In the absence of competition, species 1 would follow a logistic model with limiting population L = 10 and unrestricted growth rate .07, while species t would have a higher limiting population of L =15 and higher unrestricted growth rate .12.
• The second species, with the higher unrestricted growth rate and higher limiting population, is however more susceptible to competition due to the competition term -.018 P1 P2 (compare with the competition term -.003 P1 P2 for the first species).

We perform what is called a phase plane analysis on the system.

• In a phase plane analysis we find nullclines, where at least one of the rates of change dP2 / dt and dP1 / dt are 0, and stable points, where both rates of change dP2 / dt and dP1 / dt are 0.
• In the figure below we note that when dP2 / dt = 0, the curve representing the orbit of the system must be horizontal, while when dP1 / dt = 0 the curve is vertical.
• A curve can 'fall in' to a stable point but cannot escape.

A nullcline for P1 is the locus of points found by solving dP1 / dt = 0.

• In the figure below we solve this equation to obtain a the equation of a straight line P2 = -7/3 P1 + 29.33..., and the vertical line P1 = 0.

We find a nullcline for P2 in a similar manner, obtaining the equation of another straight line and the horizontal line P2 = 0.

Video Clip #03

We plot the nullcines as shown in the figure below.

• On the sloping nullcline for P1 we sketch vertical segments indicating the vertical direction of the orbit as it crosses that line.
• On the sloping nullcline for P2 we sketch horizontal segments indicating the horizontal direction of the orbit as crosses that line.
• We also note the horizontal motion along the P1 axis and the vertical motion along the P2 axis.

We see that there are two stable points, one at (10, 0) and the other at (0, 15), corresponding to one population or the other being at its limiting population with the other population nonexistent.

We see also that in the three regions defined by the two sloping nullclines, the direction of the curve will be as indicated by the arrows in those regions.