4. [Predator-prey model]

Volterra (1926) proposed a simple model for the predation of one species (N(t), the prey) by another (P(t), the predator) to explain the oscillatory levels of fish catches in the Adriatic. The assumptions in the model are: (i) the growth rate, a (i.e. the difference between the natural birth and death rates), is a positive constant for the prey population in the absence of the predator; (ii) the growth rate of the predator population, -d (d>0) is negative in the absence of the prey; (iii) in the presence of the predators, the chance for each prey to encounter a predator and become a victim of it is bP (where b>0 is typically small); (iv) in the presence of the prey population, the growth rate of the predator is increased by, cN, where c>0 is a constant.

(a)Write down equations for the dynamics of both the predator and the prey populations when the two exist in distinct regions (i.e. when they are separated from each other).

(b)Write down equations for the dynamics of this predator-prey system when both populations co-exist in the same region.

(c)Describe the qualitative behavior of the dynamical system in (a).

(d)Describe the quantitative behavior of the system in (b). Show that the linearized system fails to predict the behavior of the nonlinear system.