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A spatial model of range-dependent succession

Part of: Special processes Mathematical biology in general

Published online by Cambridge University Press:  14 July 2016

Stephen M. Krone  and
Claudia Neuhauser
Stephen M. Krone*
Affiliation:
University of Idaho
Claudia Neuhauser*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Mathematics, University of Idaho, Moscow, ID 83844, USA. Email address: [email protected]
∗∗ Postal address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
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Abstract

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.

Keywords

Interacting particle systemssuccessionrange-dependent dispersalcoexistencespatial patternsTuring instabilities

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 92B05: General biology and biomathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1044 - 1060
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

[1] Andjel, E., and Schinazi, R. (1996). A complete convergence theorem for an epidemic model. J. Appl. Prob. 33, 741–748.CrossRefGoogle Scholar
[2] Berg, J. van den, Grimmett, G., and Schinazi, R. (1998). Dependent random graphs and spatial epidemics. Ann. Appl. Prob. 8, 317–336.Google Scholar
[3] Besicovitch, A. S. (1927). On the fundamental geometrical properties of linearly measurable sets of points. Math. Ann. 98, 442–464.Google Scholar
[4] Bramson, M., and Neuhauser, C. (1997). Coexistence for a catalytic surface reaction model. Ann. Appl. Prob. 7, 565–614.CrossRefGoogle Scholar
[5] Cox, T., and Durrett, R. (1988). Limit theorems for the spread of epidemics and forest fires. Stoch. Proc. Appl. 30, 171–191.CrossRefGoogle Scholar
[6] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.CrossRefGoogle Scholar
[7] Durrett, R. (1995). Ten lectures on particle systems. In Ecole d'Eté de Probabilités de Saint-Flour (Lecture Notes in Math. 1608). Springer, Berlin, pp. 97–201.Google Scholar
[8] Durrett, R., and Levin, S. A. (1998). Spatial aspects of interspecific competition. Theoret. Popn Biol. 53, 30–43.CrossRefGoogle ScholarPubMed
[9] Durrett, R., and Neuhauser, C. (1991). Epidemics with recovery in d=2. Ann. Appl. Prob. 1, 189–206.CrossRefGoogle Scholar
[10] Durrett, R., and Neuhauser, C. (1994). Particle systems and reaction-diffusion equations. Ann. Prob. 22, 289–333.Google Scholar
[11] Durrett, R., and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Prob. 3, 1047–1066.CrossRefGoogle Scholar
[12] Jeger, M. (1990). Mathematical analysis and modeling of spatial aspects of plant disease epidemics. In Epidemics of Plant Diseases, ed. Kranz, J., 2nd edn. Springer, New York.Google Scholar
[13] Murray, J. D. (1989). Mathematical Biology. Springer, New York.CrossRefGoogle Scholar
[14] Neuhauser, C. (1994). A long range sexual reproduction process. Stoch. Proc. Appl. 53, 193–220.CrossRefGoogle Scholar
[15] Oliver, C., and Larson, B. (1990). Forest Stand Dynamics. McGraw-Hill, New York.Google Scholar
[16] Penrose, M. (1996). Spatial epidemics with large finite range. J. Appl. Prob. 33, 933–939.CrossRefGoogle Scholar
[17] Roberts, D., and Boothroyd, C. (1984). Fundamentals of Plant Pathology, 2nd edn. W. H. Freeman and Co., San Francisco.Google Scholar
[18] Stipes, R. J., and Campana, R. J. (eds) (1981). Compendium of Elm Diseases. American Phytopathological Society, St Paul, MN.Google Scholar
[19] Tilman, D. (1988). Plant Strategies and the Dynamics and Structure of Plant Communities. Princeton University Press.Google Scholar
[20] Zhang, Y. (1993). A shape theorem for epidemics and forest fires with finite range interactions. Ann. Prob. 21, 1755–1781.CrossRefGoogle Scholar