Practice question

A prey zero-growth isocline contains combinations of prey and predator densities for which the prey population’s instantaneous net growth is zero. In the basic Lotka–Volterra equation dN/dt = rN − aNP, setting the derivative to zero for positive N gives P = r/a. Below that predator density, prey births exceed losses to predation and prey increase; above it, prey decline. The isocline therefore summarizes how prey growth changes with predator density, which is the intended meaning of “prey population growth versus predator density.” It is not itself a predator-mortality relation or merely a predator–prey ratio. Nor is it automatically an extinction threshold, because a zero derivative at a particular state can be crossed in either direction and trajectories depend on both equations. If logistic prey growth is added, the isocline slopes downward with prey density because crowding also limits growth. Isoclines are phase-plane tools: their intersection locates an equilibrium, and the direction of change around them reveals the system’s dynamics.