A "movie" of the trajectories would show dots moving around each of the loops above in endlesscycles. Here, in contrast, every initial condition will have itsown closed loop around which it cycles endlessly. Previously, with our competition equationanalyses, many different initial conditions would converge on a single equilibriumpoint (either stable coexistence or extinction of one or the other of the two species). 23.2 is sensitive to/dependent upon initial conditions. Springer-Verlag, NY and its ODE.m package.The behavior shown by Fig. Introduction to Ordinary Differential Equations with Mathematica. I calculated the trajectories with Mathematica routines, aided by the Mathematica-based book: Gray, A., M. Note that for these parameter values, and starting with just 20 prey as the initial condition, the predator crash occurs along a line that is very close to prey extinction. The cycle is: prey explosion, prey crash, predator crash (with a slower rise in the predators happening along the "width" of the egg-shaped orbit). Note that the trajectories are not circular because the prey grow more quickly than do the predators (maximum prey abundance is 5 times equilibrium, whereas maximum predator abundance is three times equilibrium). would stay at the point of intersection of the two isoclines (because that is the equilibrium point that is the focus for all the orbits). Again, our previous numerical response example (Fig. Numerical response by the predator that is a constant times the functional response (tells us how prey consumption is converted into baby predators). Linear functional response by the predator (more prey mean more eaten) - remember we had a curved functional response in the Bay-breasted Warbler example (Fig. The Lotka-Volterra predator-prey equations/graphs (continuous, differential equations)Įxponential growth of the prey in the absence of predation (like Eqn 4.5 and Fig. The first will be unrealistically simple. We will concentrate on just a couple of models. Why analyze predators and prey jointly?It seems reasonable to expect that abundant prey lead to increased predator numbers and that the converse is true for prey. Now we will turn to the classic Lotka-Volterra equations for jointly analyzing predator and prey dynamics. Last time I introduced the topic of predation (types, importance and indirect effects) and explored:ġ) Mesopredator effects on birds in California scrublandsĢ) Functional and numerical responses and the case history of Bay-breasted Warblers as predators of spruce budworm Return to Main Index page Go back to notes for Lecture 22, 13-Mar Go forward to lecture 24, 25-Mar-13 PopEcol Lect 23 Lecture notes for ZOO 4400/5400 Population Ecology
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