Now let's try the predator isocline to the left of the hump.įig.
What similar sort of assumption-for-the-purposeof stability analysis did we make when considering matrix models? Question to ponder: In this stability analysis, we will restrict our attention toperturbations near the equilibrium. Now, instead of sensitivedependence on initial conditions, many different starting points will all tend to spiral inward to the same destination - a stable equilibrium at the intersection of the isoclines.ĭensity-dependence for the prey and inefficiency of the predator combine to stabilize the system. That will reduce(actually eliminate) the severity of the "crash" that we see on the left-hand side of the trajectories in Fig. 23.1 we are close to the prey isocline - they begin to increase even before the predator has declined to its equilibrium density. Now continue on to the arrow pointing SW. 23.1we are quite close to the prey isocline - that is, the prey will start declining again, even before we have reached the equilibrium predatordensity. 24.1 are the one pointing NE and the one pointing SW.Imagine that we start (initial condition) at the bottom of the arrow pointing NE. 23.1 Note that the blue arrows all point at one or more isoclines, leading to stability.What has this done for us? The two most important blue arrows in Fig. Blue arrows are the summed predator and prey vectors, as in Fig. Red solid curve: prey isocline green dashed line: predator isocline.
An inefficient predator - prey occur at high densities at equilibrium. Predator-prey phase plot incorporating density-dependence for the prey - with predator isocline to the right of the hump. Let's start to the right of the hump, as shown in Fig. With a "humped" prey isocline we have two qualitatively different places we could place the vertical predator isocline (wewon't change the predator isocline for the moment). The result of the combination of intraspecific competition (pulling down RHS) and excess predation (pulling down LHS) will be a "humped" prey isocline. The result will be to pull the left-hand side (LHS) of the prey isocline downward, also. ratios like 2:10 or 3.1:10)? Excess predators will eat more prey, and reduce prey numbers. What if we have an excess of the predators relative to prey (e.g. The "equilibrium" predator-prey ratio is 1:10 using the parameters in Fig. Interspecific predator/prey ratio effect. That "crowding" effect would pull the right-hand side of the red isocline in Fig. What would a density-dependent (logistic-based) prey isocline look like? Most obviously perhaps, we'd expect the prey growth rate to decline at high levels of its own population, V. Again, the vector-sum method will be a very useful tool. Our approach will be graphical - that is, rather than try to solve increasingly complex equations, we will make inferences aboutthe system's behavior based on examination of graphs for a variety of shapes and placements of the isoclines. We will include density -dependence for the prey. Let's see what would happen to the system if we made one reasonableimprovement in the predator-prey model. We would like our model to be a little more robust (less sensitive to initial conditions and less sensitive to reasonable changes in the assumptions).įirst "improvement" - logistic prey growth. That means that if we tweak parameters a little (e.g., by changing the starting population sizes, which we call the "initial conditions") then we get very different outcomes. The basic model makes some rather unrealistic simplifying assumptions.Ĭan we do better? The Lotka-Volterra predator-prey model is what is called "structurally unstable". Last time I introduced the classic Lotka-Volterra equations for jointly analyzing predator and prey dynamics. Return to Main Index page Go back to notes for Lecture 23, 13-Mar Go forward to lecture 25, 27-Mar-13 PopEcol Lect 24 Lecture notes for ZOO 4400/5400 Population Ecology