Lotka-Volterra “prey-predator” interaction

The system rotates irreversibly in a direction determined by the sign of /. An example of such a system is the well-known Lotka-Volterra prey-predator interaction given as an exercise (exc. 18.9). We can also apply this inequality to derive a sufficient condition for the stability of a steady state. If all fluctuations fipP > 0 then the steady state is stable. But here it is more expedient to use the Lyapunov theory of stability to which we turn now. 

First model for oscillating system was proposed by Volterra for prey-predator interactions in biological systems and by Lotka for autocatalytic chemical reactions. Lotka s model can be represented as... 

The model equations for simple prey-predator interactions, initially according to Lotka and Volterra and later refined by Bungay and Bungay (1968), are as follows ... 

Far from equilibrium, we only keep the forward reactions. Using the linear stability theory, show that the perturbations around the nonequilibrium steady state lead to oscillations in [X] and [Y], as was discussed in section 18.2. This model was used by Lotka and Volterra to describe the struggle of life (see V. Volterra, Theorie mathematique de la Lutte pour la Vie, Paris Gauthier Villars, 1931). Here X is the prey (lamb) and Y is the predator (wolf). This model of the prey-predator interaction shows that the populations X and Y will exhibit oscillations. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

In biological dissipative structures, self-oiganization may be related to the attractors in the phase space, which correspond to ordered motions of the involved biological elements (De la Fuenta, 1999). When the system is far from equilibrium, ordering in time or spontaneous rhythmic behavior may occur. The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatalytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes ... 

The classical theory of predator-prey interaction as formulated by Volterra involves two equations which express the growth rate of the prey and the predator (57). Within the context of phytoplankton and zooplankton population, the prey is the phytoplankton and the predator the zooplankton. In the notation of the previous sections, for a one-volume system, the Lotka-Volterra equations are ... 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

Let us now consider in detail the classical predator-prey problem, that is, the interaction between two wild-life species, the prey, which is a herbivore, and the predator, a carnivore. These two animals coinhabit a region where the prey have an abundant supply of natural vegetation for food, and the predators depend on the prey for their entire supply of food. This is a simplification of the real ecological system where more than two species coexist, and where predators usually feed on a variety of prey. The Lotka-Volterra equations have also been formulated for such... 

This section is meant to contribute to the old problem of the interaction between two biological species. To be or not to be is the essential question decided by the predator-prey interaction for the members of certain species. The famous Volterra-Lotka model for this problem [4.21,22] has attracted many researchers who have tried to generalize it in many respects [4.1, 6, 8, 9, 23-26]. 

The extension of the pure Volterra-Lotka model by adding non-linear migration leads to a complex picture with a relatively large variety of different solution cases. In particular, the result has been obtained that with an appropriate mixture of migration and predator-prey interaction a limit cycle appears which is similar to the original Volterra-Lotka cycles but which is stable and independent of the initial conditions. 

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