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, Tome 14 (2004) no. 2, pp. 139-147.

Résumé

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.

Détail

Keywords: predator-prey system, entropy, correlation coefficient
Mots-clés : równanie Lotka-Volterra, entropia, współczynnik korelacji

Bibliographie

[1] Adami C. (1998): Introduction to Artificial Life. — New York: Springer.

[2] Boccara N., Cheong K. and Oram M. (1994): A probabilistic automata network epidemic model with births and deaths exhibiting cyclic behaviour. — J. Phys. A: Math. Gen., Vol. 27, pp. 1585–1597.

[3] Boccara N., Roblin O. and Roger M. (1994): Automata network predator-prey model with pursuit and evasion. — Phys. Rev. E 50, Vol. 50, No. 6, pp. 4531–4541.

[4] McCauley E.,Wilson W.G. and de Roos A.M. (1993): Dynamics of age-structured and spatially structured predator-prey interactions: Individual-based models and population-level formulations. — Am. Naturalist, Vol. 142, No. 3, pp. 412–442.

[5] McCauley E., Wilson W.G. and de Roos A.M. (1996): Dynamics of age-structured predator-prey populations in space: Asymmetrical effects of mobility in juvenile and adult predators.—OIKOS, Vol. 76, pp. 485–497.

[6] Durrett R. and Levin S. (1994): The importance of being discrete (and spatial).—Theor. Popul. Biol., Vol. 46, pp. 363–394.

[7] Lipowski A. (1999): Oscillatory behavior in a lattice prey-predator system. — Phys. Rev. ER 60, No. 5, pp. 5179–5184.

[8] Lipowski A. and Lipowska D. (2000): Nonequilibrium phase transition in a lattice prey-predator system. — Physica A 276, pp. 456–464.

[9] Poland D. (1989): The effect of clustering on the Lotka-Volterra model.—Physica D 35, pp. 148–166.

[10] Rand D.A. (1994): Measuring and characterizing spatial patterns, dynamics and chaos in spatially extended dynamical systems and ecologies. — Phil. Trans. R. Soc. Lond. A 348, pp. 497–514.

[11] Rand D.A., Keeling M. and Wilson H.B. (1995): Invasion, stability and evolution to criticality in spatially extended, artificial host-pathogen ecologies. — Proc. R. Soc. Lond. B 259, pp. 55–63.

[12] Rand D.A. and Wilson H.B. (1995): Using spatio-temporal chaos and intermediate scale determinism to quantify spatially extended ecosystems. — Proc. R. Soc. Lond. B 259, pp. 111–117.

[13] Ranta E. and Kaitala V. (1997): Travelling waves in vole population dynamics.—Nature 390, pp. 456.

[14] Ranta E., Kaitala V. and Lundberg P. (1997): The spatial dimension in population fluctuations.—Science 278, pp. 1621–1623.

[15] Renshaw E. (1991): Modeling Biological Populations in Space and Time. — Cambridge: Cambridge University Press.

[16] De Roos A.M., McCauley E. and Wilson W.G. (1991): Mobility versus density-limited predator-prey dynamics on different spatial scales.—Proc. R. Soc. Lond. B 246, pp. 117–122.

[17] De Roos A.M., McCauley E. and Wilson W.G. (1998): Pattern formation and the spatial scale of interaction between predators and their prey.—Theor. Popul. Biol. 53, No. 2, pp. 108–130.

[18] Satulovsky J.E. (1996): Lattice Lotka-Volterra models and negative cross-diffusion. — J. Theor. Biol. 183, pp. 381–389.

[19] Satulovsky J.E. and Tome T. (1994): Stochastic lattice gas model for a predator-prey system. — Phys. Rev. E 49, No. 6, pp. 5073–5079.

[20] Satulovsky J.E. and Tome T. (1997): Spatial instabilities and local oscillations in a lattice gas Lotka-Volterra model.— J. Math. Biol. 35, pp. 344–358.

[21] Tainaka K. and Fukazawa S. (1992): Spatial pattern in a chemical reaction system: prey and predator in the position-fixed limit.—J. Phys. Soc. Jpn. 61, No. 6, pp. 1891–1894.

[22] Waniewski J. and JędruchW. (1999): Individual based modeling and parameter estimation for a Lotka-Volterra system. — Math. Biosci. 157, pp. 23–36.

[23] Waniewski J. and Jędruch W. (2000): Spatial heterogenity and local oscillation phase drifts in individual-based simulations of a prey-predator system. — Int. J. Appl. Math. Comp. Sci., Vol. 10, No. 1, pp. 175–192.

[24] Wiegand T., Moloney K.A., Naves J. and Knauer F. (1999): Finding the missing link between landscape structure and population dynamics: A spatially explicit perspective. — Am. Nat., Vol. 154, No. 6, pp. 605–627.

[25] Wilson W.G., McCauley E. and de Roos A.M. (1995): Effect of dimensionality on Lotka-Volterra predator-prey dynamics: individual based simulation results. — Bull. Math. Biol., 57, No. 4, pp. 507–526.

[26] Wilson W.G., de Roos A.M. and McCauley E. (1993): Spatial instabilities within the diffusive Lotka-Volterra system: Individual-based simulation results.—Theor. Popul. Biol. 43, pp. 91–127.