Chapter 5.3, Problem 29E

In Problems 25-30, use a software package or the subroutine in Appendix F.

Competing Species. Let $p_i\left(t\right)$ denote, respectively, the populations of three competing species $S_i$, $i=1$, 2, 3. Suppose these species have the same growth rates, and the maximum population that the habitat can support is the same for each species. (We assume it to be one unit.) Also suppose the competitive advantage that $S_1$ has over $S_2$ is the same as that of $S_2$ over $S_3$ and $S_3$ over $S_1$.

This situation is modeled by the system

${p^{\prime }}_1=p_1\left(1-p_1-a{p}_2-b{p}_3\right)$,

${p^{\prime }}_2=p_2\left(1-b{p}_1-p_2-a{p}_3\right)$,

${p^{\prime }}_3=p_3\left(1-a{p}_1-b{p}_2-p_3\right)$,

where $a$ and $b$ are positive constants. To demonstrate the population dynamics of this system when $a=b=0.5$, use the Runge-Kutta algorithm for systems with $h=0.1$ to approximate the populations $p_i$ over the time interval $\left[0,10\right]$ under each of the following initial conditions:

a. $p_1\left(0\right)=1.0$, $p_2\left(0\right)=0.1$, $p_3\left(0\right)=0.1$.

b. $p_1\left(0\right)=0.1$, $p_2\left(0\right)=1.0$, $p_3\left(0\right)=0.1$.

c. $p_1\left(0\right)=0.1$, $p_2\left(0\right)=0.1$, $p_3\left(0\right)=1.0$.

On the basis of the results of parts (a)-(c), decide what you think will happen to these populations as $t\to +\infty$.