e-Lecture : Evolutionary Games and Ecosystems

Taksu Cheon

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Evolutionary Games and Ecosystems (6b)

Lotka-Volterra Equations: Its Workings - 2

Competitors

Still more equations. Math allegics, please skip monce more! This time, instead of Zebras and Lions, let's consider Lions and Zulus. Well, treating animals and humans in the same footing might anger some. (I am sure Zulus would have taken pride in comparison to lions though.) So let's imagine there are Jaguars with population x and lions y in a sabanna. Assuming the reproduction rate to be B and D respectively, we have

     dx/dt = B x 、 dy/dt = D y

which is the same equation as before. This time, however, we suppose x and y both command sufficient resources (abundant Zebras for example, which is not explicitely represented) and self-reproducing, whose rates we write b and d, respectively. Environemental factors a (for x) and c (for y) are also considered. Both Lions and Juaguars are notorious aggressors who hinder opponents reproduction. Thus we have

     B = b - a x - R y 、 D = d - c y - S x

where R and S are respective aggression intensities of Lions and Jaguars. As in previous chapter, we obtain

     dx/dt = b x - a x2 - R x y
     dy/dt = d y - c y2 - S x y

which is the Lotka-Volterra equation for two competitors.

Numerical Solutions

The temporal evolution of x(t) and y(t) that satisfy the Lotka-Volterra equation is best solved by computers. The results are represented in the following 'phase diagrams" where no explicit time evolution is shown but the trajectory (x(t), y(t)) are plotted. Horizontal line is x and perpendicular, y.

The graph in the left represents the solution of prey-predator case, and the right, two competitors. In the left, all orbits are absorbed to the centor of vortex, which represents the stable values for populations x and y. In the right, there is no stable location in the orbits other than either x=0 or y=0 (or both). Namely, prey and predator can stably coexist, while one of two competitors is necessarily driven to extinction.
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