Final answer:
In the Lotka-Volterra model, no equilibrium can exist with one species' nonzero population and the other extinct because the populations are interdependent. Equilibria are found by setting the population growth rates to zero and solving for prey (N) and predator (P) numbers.
Step-by-step explanation:
The Lotka-Volterra model is a mathematical model for the dynamics of predator-prey interactions. Part a of the question relates to why there can be no equilibrium with one species having a nonzero population while the other is extinct. Without any algebra, we can argue that as per the Lotka-Volterra model, both predator and prey species are interdependent for their population dynamics. Prey are necessary for predators to survive (as a food source), and predator presence influences prey population through predation. Thus, there cannot be an equilibrium where only one species persists while the other is nonexistent since their population rates are directly linked by the coefficients in the model equations.
To find the equilibria in part b, we set the derivatives of both populations to zero, resulting in the following two equations: N' = rN - aNP = 0 and P' = caNP - δP = 0. Solving these equations for N and P will give us the equilibrium points, where the population sizes of both species do not change over time. An equilibrium occurs when N = δ/(ca) and P = r/a, given that neither N nor P is zero, indicating a coexistence equilibrium where both species have nonzero populations.