Lotka-Volterra problem

It can be shown that whenever the Lotka-Volterra problem ha.s the form of Eqs. (1) and (2) in Prob. 5.7, the real parts of the eigenvalues of the Jacobian matrix are zero. This implies that the solution always has neutrally stable oscillatory behavior. This is explained by the fact that assumptions (a) to (d) of Prob. 5.7 did not include the crowding effect each population may have on its own fertility or mortality. For example, Eq. (I) can be rewritten with the additional term... 

More interesting aspects of stochastic problems are observed when passing to systems with unstable stationary points. Since we restrict ourselves to mono- and bimolecular reactions with a maximum of two intermediate products (freedom degrees), s — 2, only the Lotka-Volterra model by reasons discussed in Section 2.1.1 can serve as the analog of unstable systems. 

Going back to D Ancona s problem of inhibited fishing in the Adria, we include the effects of fishing into the Lotka-Volterra model (15.1) by introducing a linear loss term with harvesting rate h for both predator and prey species... 

The examination of competitive interactions among different species has been one of the main topics of mathematical biology. The most often used mathematical model is still a generalisation of the Lotka-Volterra model systems of polynomial ordinary differential equations expressible in terms of formal chemical reactions have also been investigated. The main problem is to find criteria for the coexistence of species. All species in the communities... 

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... 

Some workers approached the problem in a different way and suggested the existence of a Lotka-Volterra oscillator acting between some active and passive regions at the steel surface, the nature of which being not clearly identified. Others considered the appearance of the current transients as a result of the potential-assisted formation of a salt film... 

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]. 

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