Evolutionary Games and Ecosystems (9a)

Lotka-Volterra Equation with Evolving Coefficients- 1

Prey & Predator Population as Function of Aggression Intensity

The qualitative understanding in previous chapters (8) and (9) are not really satisfactory, and it is highly desired that we achieve quantitative descriptions. We ask those who so inclined to skip this one again, and look into the detail. We rewrite the prey & predator Lotka-Volterra:

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

The location of the center of attracting vortex (X,Y) is given by the condition for its being fixed point (namely stay there forever after arrival); namely, dX/dt=0、dy/dt = 0. Thus we get

　　　　　0 = b - a X - R Y
　　　　　0 = - d + f R X

(In truth, we have to prove that this fixed point is of "stable' type. For that we follow the standard procedure of "linearizing" the equation around the stable point and obtain the condition for its having two negative eigenvalues. This is far too technical for current format, and we just note that the condition is indeed met with sufficiently large b) Now we regard R, the aggression intensity of the predators to be at the desposal of the predators themselves. Regarding both X 、Y as functions of R

　　　　　X[R] = d/(fR)
　　　　　Y[R] = 1/R - (ad)/(fR²)

we maximize Y[R] to obtain the optimal R* for the predator, which is specified by the condition

　　　　　dY[R]/dR | _R* = 0

Explicit expressions are easily obtained as

　　　　　R* = (2ad)/(fb)
　　　　　X* = X[R*] = b/(2a)
　　　　　Y* = Y[R*] = (fb²)/(4ad)

A notable fact is that the resulting prey population X* is given by half of the "natural population, b/a. This could be regarded as the mathematical expression of the oceanographic concept of "maximal sustainable yield".