Chapter 28, Problem 32P

The Lotka-Volterra equations described in Sec. 28.2 have been refined to include additional factors that impact predator-prey dynamics. For example, over and above predation, prey population can be limited by other factors such as space. Space limitation can be incorporated into the model as a carrying capacity (recall the logistic model described in Prob. 28.16) as in

$\frac{dx}{dt}=a\left(1-\frac{x}{K}\right)x-bxy$

$\frac{dy}{dt}=cy+dxy$

where $k=\text{the}$ the carrying capacity. Use the same parameter values and initial conditions as in Sec. 28.2 to integrate these equations from $t=0\text{to 100}$ using ode45.

(a) Employ a very large value of $k={10}^8$ to validate that you obtain the same results as in Sec. 28.2.

(b) Compare (a) with the more realistic carrying capacity of $k=200$. Discuss your results.