Section 3 - The Mutualistic Case

What happens when b and d are both larger than zero?

Here, the system is actually mutualistic. The larger population y gets, the faster population x grows, and vice versa. Of course, they're still inhibited by death. This system is represented by the equations we had in the first place:

x' = ax + bxy - ex^2
y' = cy + dxy - fy^2

Does this system have a "resting point" as the last one did? In order to find the equilibria, we will again set the growth rates equal to zero and solve for x and y:

x') 0 = ax + bxy - ex^2
y') 0 = cy + dxy - fy^2

There is clearly an equilibrium at (x,y)=(0,0). To find the other equilibrium, we assume that x and y are not equal to zero.

After a bit of algebra , we get:

x = (-bc - fa)/(bd - ef)
y = (-ec - ad)/(bd - ef)

So, in order to have an equilibrium where where both populations are alive, the only requirement is that the quantity bd - ef be less than zero. Why is this?

Well, since all of the parameters are greater than zero in this case, the numerators of both x and y are negative: -(bc+fa) and -(ec+ad). Thus, if the denominator is also negative, x and y will be positive and we will have a result that makes sense.

Take the Jacobian of the system:
[ a + by - 2ex	                  bx	]
[ dx			c + dx - 2fy	]

The determinant of the matrix:
x (ad - 2ec) + 4xyef + y (bc - 2af) - 2bfy^2 - 2dex^2 + ac

When we substitute the x and y equilibria into this equation, we get:
det = - (bc^2e + abcd + acef + a^2df) / (bd-ef)

Meanwhile, trace = a + by - 2ex + c + dx - 2fy

When we substitute the x and y equilibria into this equation, we get:
trace = (aef + bce + cef + adf) / (bd - ef)

Assuming bd - ef < 0 (so that we have a viable system), the determinant is positive and the trace is negative.

In these cases, the equilibrium is a node. This (stable) equilibrium represents the maximum attainable level for the population. It will be farther from the origin than {a/e, c/f}, since both species' growth rates are augmented by mutualism and they are growing more than the environment would otherwise allow.

[Phase portrait: {a,b,c,d,e,f} = {1,1,1,1,2,2}]

When bd - ef = 0, a birfurcation occurs. Here, the equilibrium in the first quadrant disappears and both species can grow wihout bound. In this circumstance, the mutualism rates effectually cancel out the death rates of the two species. In reality, this would probably not happen, so this case is more of a reflection of the limiting condition as bd - ef approaches zero.

[Phase portrait: {1,1,1,1,1,1}]

If bd - ef goes above zero, a saddle equilibrium appears in the third quadrant, but the populations in the first quadrant grow without bound. (The mutualism factor has overcome the death factor; they help each other so much that they never become extinct.)

[Phase portrait: {1,2,1,2,1,1} -- reset range]


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