e-Lecture : Evolutionary Games and Ecosystems

Taksu Cheon

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Evolutionary Games and Ecosystems (6a)

Lotka-Volterra Equations: Its Workings - 1

Prey and Predator

You must have realized that full understanding of the game theory is not possible without dealing equations. So we shall do some heavy lifting, and advise those who are allergic to differetiations and integration to skip.
Now, we consider the temporal evolution of the population of Zebras (prey) x, and that of Lions (predator) y. The simplest case of constantreproduction rates B fro Zebra and D for Lions is described by

     dx/dt = B x 、 dy/dt = D y

As a reminder, we note that the differential dx/dt represents the rate at which x increase per unit time (1 year, for example). Assumption of B and D being constant is unrealistic, however. First of all, as for B, its number are expected to fall when x is large, because of the environmental degredation. That should be represented by part of B negatively proportional to x. More importantly the presence of the Lions preying the Zebra would cause decrease of B which is represented as a term negatively propotional to y. Thus we have

     B = b - a x - R y

where b is a natural reproduction rate of Zebras at small x, a, the limiting factor of environment, and R, the agression intensity of Lions against Zebras. Lions, left by themself would die out revealing the negative natural reproduction rate, which we write -d. Their survival depends on a term proportional to x that represents the preying activiy on Zebras. We neglect environmental degredation of D because their number is small compared to Zebras. We have

     D = - d + S x

Inserting last two equations to the two equations at the top, we obtain

     dx/dt = b x - a x2 - R x y
     dy/dt = -d y + f R x y

Here, we have made replacement S = f R whose meaning is easy to comprehend: Hunted-down prey resouce R x is utilized for predator with efficiency f. This set is called Lotka-Volterra equations for prey and predator.

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