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Dynamic stability and spatial heterogeneity in the individual-based modelling of a Lotka-Volterra gas

Waniewski J., Jędruch W., Żołek N. S.

International Journal of Applied Mathematics and Computer Science

2004

Vol. 14, no 2

139-147

Computer simulation of a few thousands of particles moving (approximately) according to the energy and momentum conservation laws on a tessellation of 800 x 800 squares in discrete time steps and interacting according to the predator-prey scheme is analyzed. The population dynamics are described by the basic Lotka-Volterra interactions (multiplication of preys, predation and multiplication of predators, death of predators), but the spatial effects result in differences between the system evolution and the mathematical description by the Lotka-Volterra equations. The spatial patterns were evaluated using entropy and a cross correlation coefficient for the spatial distribution of both populations. In some simulations the system oscillated with variable amplitude but rather stable period, but the particle distribution departed from the (quasi) homogeneous state and did not return to it. The distribution entropy oscillated in the same rhythm as the population, but its value was smaller than in the initial homogeneous state. The cross correlation coefficient oscillated between positive and negative values. Its average value depended on the space scale applied for its evaluation with the negative values on the small scale (separation of preys from predators) and the positive values on the large scale (aggregation of both populations). The stability of such oscillation patterns was based on a balance of the population parameters and particle mobility. The increased mobility (particle mixing) resulted in unstable oscillations with high amplitude, sustained homogeneity of the particle distribution, and final extinction of one or both populations.

Extinction, Weak Extinction and Persistence in a Discrete, Competitive Lotka-Volterra Model

Chan D. M., Franke J. E.

International Journal of Applied Mathematics and Computer Science

2000

Vol. 10, no 1

7-36

In a discrete Lotka-Volterra model, the set of points where a population remains unchanged over one generation is a hyperplane. Examining the relative position of these hyperplanes, we give sufficient conditions for a groupof species to drive another species to extinction. Further using these hyperplanes, we find necessary and sufficient conditions where every w-limit point of the model has at least one species missing. Building on the workof Hofbauer et al. (1987) involving permanence, we obtain a sufficient condition for one or more species to persist. Additionally, in the presence of extinction occurring, we take these persistence results and the previously mentioned extinction results and extend them to subsystems of the full model. Finally, we combine the ideas of persistence and weak extinctionto obtain another extinction result.