Three Species Population Dynamics Models: Lotka-Volterra : Differential Equations

The aim of this book is to model multiple species food chain in three dimensions using system of Lotka-Volterra equations also known as the predator–prey equations. The consequences of varying parameters in Lotka-Volterra equations which cause changes in population dynamics are examined carefully. The effect on population dynamics as a result of these parametric changes in the systems are studied within invariant surfaces and, in terms of stability, in the neighborhood of equilibrium points. Of course the intention is to investigate the periodic behavior and the initial conditions that lead to it. It is known that the two-dimensional system exhibit stable cyclic behavior for all initial conditions where neither population tends to extinct. However, for the three-dimensional system, the behavior of the system is no longer cyclic all the time but varies, depending on the choice of constants used in the system definition. We might expect for model in chapter 3 that two prey and one predator to coexist; this is not always the case. Our investigation shows that all three species can coexist cyclically only if both prey species are essentially the same otherwise the weaker prey specie dies out eventually. Models discussed in chapters 4 and 5 represent the restrictions imposed on a population by environmental factors, species competition and other limited resources. Our study demonstrates the periodic behavior of the system (4) in chapter 6, which is non-asymptotically stable around the equilibrium point for a particular realistic case. In all cases prey species survive, however the existence of top level predator over time depends solely upon the parameter’s value. The merits and flaws of these models are also discussed.