Generalised Lotka-Volterra models

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 

Here x,(0 i(0 are the population sizes for prey and predators, respectively, a 0 and 6 0 represent competitive effects between two prey, e, 0 and Uj 0 are coefficients of decrease of prey due to predation, and the 4 are the transformation rates of predator. 

The two-species-competing system (7.25) cannot show oscillatory behaviour (in consequence of the Hanusse-Tyson-Light-Pota theorem it is different from the Lotka-Volterra model). The conditions of stability of equilibria were studied by Takeuchi Adachi (1983b). Accordingly, the ( ++) is globally stable if and only if a I, 6 1. ( +0) and ( 0+) are globally stable if and only if a , I and, b respectively. For the case of a , b ( +0) and ( 00) are locally stable. 

The combined analytical and numerical analysis of the two-prey, two-predators, system (Takeuchi Adachi, 1983a, b) showed similar the following qualitative results a stable equilibrium also can bifurcate to a stable limit cycle and the limit cycle turns into a nonperiodic oscillation of bounded amplitude with increasing cycle time. 

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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. 

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