A Simple Look at Predator-Prey Models

• Prey grow in an unlimited way in absence of predators.
• Predators depend on prey to survive.
• The rate of predation depends on the likelihood of encounter.
• The predator growth rate is proportional to rate of predation.

              dx/dt = ax – bxy

              dy/dt = – cy + dxy

     The xy term is an approximation of the likelihood that an 

     encounter will take place.  This comes from a rate of molecular

     collisions of two types of molecules which move about randomly

     and are uniformly distributed.

      It turns out that the two possible steady states are at

          ( x ,  y ) = ( 0 , 0 )      and    ( x ,  y ) = ( c/d , a/b )  

      Let a = .25,  b = .01,   c = 1, and d = .01

     Lets try a few using this Java applet.

     Now change assumption 1. to a slightly more realistic assumption of logistic growth to arrive at the following.

              dx/dt = ax (K–x)/K – bxy

              dy/dt = – cy + dxy

Again use the applet.

The interesting steady state point becomes: 

                       ( x ,  y ) = ( c/d , a/b – ca/dbK) 

Now we look at x and y versus t for

              dx/dt = ax – bxy

              dy/dt = – cy + dxy

without predators we get

Now x and y versus t for

               dx/dt = ax (K–x)/K – bxy

              dy/dt = – cy + dxy

without predators we get

We can see how this looks in another simulation in this applet