A Lotka-Volterra Three-species Food Chain

this paper, we completely characterize the qualitative behavior of a linear threespecies food chain where the dynamics are given by classic (nonlogistic) LotkaVolterra type equations. The Lotka-Volterra equations are typically modified by making the prey equation a logistic (Holling-type [5]) equation to eliminate the possibility of unbounded growth of the prey in the absence of the predator. We study a more basic nonlogistic system that is the direct generalization of the classic Lotka-Volterra equations. Although the model is more simplified, the dynamics of the associated system are quite complicated, as the model exhibits degeneracies that make it an excellent instructional tool whose analysis involves advanced topics such as: trapping regions, nonlinear analysis, invariant sets, Lyapunov-type functions (F and G in what follows), the stable/center manifold theorem, and the Poincar e-Bendixson theorem. Figure 3 Historical plots of hare and lynx pelts collected by the Hudson's Bay Company The model The ecosystem that we wish to model is a linear three-species food chain where the lowest-level prey x is preyed upon by a mid-level species y, which, in turn, is preyed upon by a top level predator z. Examples of such three-species ecosystems include: mouse-snake-owl, vegetation-hare-lynx, and worm-robin-falcon. The model we propose to study is # # # # # # # # # # # # # bxy dy dz gyz, (2) for a, b, c, d, e, f, g > 0, where a, b, c and d are as in the Lotka-Volterra equations and: . e represents the effect of predation on species y by species z, . f represents the natural death rate of the predator z in the absence of prey, . g represents the efficiency and propagation rate of the predator z in the presence of prey. Since populations are nonnegative, we will r..