0.6 Second order model  (Page 4/9)

If a different situation is considered where one population has a growth rate of 20% and the other 5% $$ , but the interactions are still at a ratio of 100 $$ , the equations become

The solutions for this case are shown in Figure H. Here the results of the different rates are rather startling.The trajectory number 1 starts at $p_1=10$ and $p_2=1$ , yet $p_2$ overcomes $p_1$ . Trajectory number 2 startsat $p_1=16$ and $p_2=1$ , and $p_2$ still wins; but when the initial valuesare $p_1=17$ and $p_2=1$ , trajectory number 3 shows $p_1$ wins For $p_2=500$ and $p_1=200$ or 240, trajectories numbers 4 and 5 show $p_2$ wins; but with $p_2=500$ and $p_1=250$ or 300, trajectories 6 and 7 show $p_1$ wins. This exemplifies the very large difference a four-to-one growth rateratio can make, and how critical the outcome depends on the initial values.It also illustrate the power of the phase plane in describing the model.

In the basic competition model described by [link] , and when normalized, described by [link] , we see that even if the interactive terms are very small, one population always grows without limit and the other becomesextinct. This describes a "survival of the fittest" model, but the unlimitedgrowth and no possibility of coexistence seems unreasonable.

The next level of complication is the addition of a limit to growth in the same manner that the exponential was changed to a logistic.A crowding or self-competition term is added to the simple competition model.Consider now

Using the normalizing procedure that was used before on [link] reduces the number of parameters from six to four:

where

Consider the partial equilibrium curves for this model.

On the phase plane, this becomes

It is obvious that the character of this system depends on the relative values of $a,$ $b,$ $K$ and $L$ , and indeed these are from rather different possible systems.

We will first consider the case illustrated above where both limiting factors are relatively small.

Note that as $K$ and $L$ approach zero, the system approaches the previously studied system.For this case, there are three possible equilibrium or singular points. There is an unstable point at the intersection of the two partialequilibrium curves, and a stable point at $y=0$ , $x=\frac{a}{K}$ , and another stable point at $y=\frac{c}{L}$ , $x=0$ . In this case, as before, one or the other population always wins,depending on initial conditions, and the remaining population dies to zero.There is now a limit reached by the winner and indeed, the time plot of the winning population looks very similar to a logistic.For example, for particular initial $x$ and $y$ , we have

The phase plane trajectories are illustrated for the normalized variables $x$ and $y$ in Figure J. The terms $a$ and $c$ are set equal to one, with $K$ and $L$ set equal to one-half.The winning population approaches 2 as its equilibrium value, and the loser becomes extinct.

The second case to consider has strong self-limiting factors relative to the interactive terms