Section 2 - Without Predators

What happens to x if y = 0?

The first thing to do is to examine the dynamics of the prey (x) population with no predation involved (i.e. y = 0).

c' = ba - gc - dc
a' = gc - da
[ x' = ba - d(a + c) ]

Take the Jacobian:

        [ - g - d      b ]
        [   g        - d ]

trace = - g - 2d < 0
det = d (g + d) - bg

Throughout these documents, it is very helpful to use DSTool to examine the phase portraits of systems with particular parameters. If you are familiar with DSTool, you can simply launch it now (if it isn't already running) and, for each of the sets of sample parameters given in this system and the next, examine the respective phase portrait. If you are not entirely comfortable with DSTool, jump to DSTool Help for assistance.

when Det = bg - d(g + d) = 0 [multiplied through by -1]

a bifurcation occurs -- this is the transition between saddle and node
the population stabilizes at a certain point, which depends on the initial concentration of a and c
the set of possible convergence points forms a line through the point (0,0)
that line is similar to the line-like node which we will see with other parameters

[phase portrait(s): {b,g,d} = {1.5,2,1} and/or {6,1,2}]

when bg - d(g + d) > 0

the population x increases without bound
ratio of a/c is constant
the phase portrait is a saddle

[phase portrait(s): possible {b,g,d} = {2,2,1} and {6,3,2}]

when bg - d(g + d) < 0

the population x becomes extinct
ratio of a/c is constant
the phase portrait is a node

[phase portrait(s): {.5,1,1} and {1,1,2}]

If a population becomes extinct without predation, it will become extinct with predation

[note that there is no "crowding factor" in these differential equations; the number of deaths is directly proportional to the number of prey].

Therefore, for the rest of this problem, it is assumed that bg - d(g + d) > 0.

We now have a viable system to work with.

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