Section 4 - The Parasitic Case

What happens when either b or d is smaller than 0?

This isn't exactly mutualism; in fact, it's the ideological opposite, parasitism. But it can be modeled by the same equations as long as either b or d is less than zero.

In a parasitic situation, the quantity bd - ef (the denominator in our equilibrium formula) is always less than zero, since both bd and ef are less than zero. Therefore, a stable equilibrium will usually exist in the first quadrant, where both x and y are positive. Populations in any other quadrant are basically irrelevant because we can't have negative numbers of foxes and rabbits, birds and worms, or fish and anglers.

This equilibrium is similar to the one we found in the mutualistic case, but it's a bit skewed: while the parasite grows beyond its natural bound, the host will approach a population limit that is well below what it could have attained otherwise.

[Phase portrait:
(x is the parasite) {1,2,1,-1,2,1}
(y is the parasite) {1,-1,1,2,1,2}]

This seems like a very good thing for the parasite, but, as with most things, moderation is the key. If the parasite sucks too much life out of the host, the host will become extinct and the parasite will have to look elsewhere for food. In particular, this happens when |ec| is less than or equal to |ad| if y is the host or when |fa| is less than or equal to |bc| if x is the host.

This makes sense because, (assuming y is the host) |ec| represents factors that will help keep the host alive. The parasite's natural death rate, (e), keeps the parasite population under control, while the host's birth rate, (c), replentishs the hosts lost to the parasites. Thus a large |ec| is favorable for keeping the hosts alive.

Likewise |ad| represents factors that could contribute to the host's extinction if allowed to be too large. The parasite's birth rate, (a), allows the parasite population to grow too large, while the factor (d), represents how much the host population is damaged by the existance of the parasites. Thus a large |ad| is unfavorable for keeping the hosts alive.

If the hosts become extinct, the parasites themselves may still survive, depending on their level of dependence on the hosts, but it won't be as easy.

[Phase portrait:
(y dies off) {1,2,1,-1,.5,1}
(x dies off) {.5,-1,1,2,1,1}]

About the Authors