Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction

Abstract. Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competition mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equilibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed.

[1]  Chris Cosner,et al.  On the effects of spatial heterogeneity on the persistence of interacting species , 1998 .

[2]  Adele Manes,et al.  Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine , 1973 .

[3]  Chris Cosner,et al.  Ecological models, permanence and spatial heterogeneity , 1996 .

[4]  C. Packer,et al.  A canine distemper virus epidemic in Serengeti lions (Panthera leo) , 1996, Nature.

[5]  Song-Sun Lin,et al.  On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function , 1980 .

[6]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[7]  Avner Friedman,et al.  Partial differential equations , 1969 .

[8]  D. Macdonald.  Dangerous liaisons and disease , 1996, Nature.

[9]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[10]  John H. Lawton,et al.  The Ecological Consequences of Shared Natural Enemies , 1994 .

[11]  Michael Begon,et al.  Disease and Community Structure: The Importance of Host Self-Regulation in a Host-Host-Pathogen Model , 1992, The American Naturalist.

[12]  A. Dobson,et al.  INFECTIOUS DISEASE AND INTENSIVE MANAGEMENT: POPULATION DYNAMICS, THREATENED HOSTS, AND THEIR PARASITES , 1993 .

[13]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  Chris Cosner,et al.  Diffusive logistic equations with indefinite weights: population models in disrupted environments , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  Anne Matthews.  Where the Buffalo Roam , 1992 .

[16]  R. Speare,et al.  Epidemic Disease and the Catastrophic Decline of Australian Rain Forest Frogs , 1996 .

[17]  C. Cosner,et al.  Stable Coexistence States in the Volterra–Lotka Competition Model with Diffusion , 1984 .

[18]  Chris Cosner,et al.  Practical persistence in ecological models via comparison methods , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[20]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[21]  S. Carroll,et al.  Birds, bugs and blood: avian parasitism and conservation. , 1995, Trends in ecology & evolution.

[22]  M. Meyer,et al.  On the Origin of Brucellosis in Bison of Yellowstone National Park: A Review , 1994 .

[23]  Chris Cosner,et al.  How Habitat Edges Change Species Interactions , 1999, The American Naturalist.

[24]  O. Schmitz,et al.  PARASITE-MEDIATED COMPETITION IN DEER AND MOOSE: HOW STRONG IS THE EFFECT OF MENINGEAL WORM ON MOOSE? , 1994 .

[25]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[26]  M. Wilson,et al.  Microclimate-dependent survival of unfed adult Ixodes scapularis (Acari:Ixodidae) in nature: life cycle and study design implications. , 1996, Journal of medical entomology.

[27]  Stefan Senn,et al.  On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics , 1983 .

[28]  Peter Hess,et al.  On some linear and nonlinear eigenvalue problems with an indefinite weight function , 1980 .

[29]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[30]  Chris Cosner,et al.  Should a Park Be an Island? , 1993, SIAM J. Appl. Math..

[31]  Chris Cosner,et al.  Diffusive logistic equations with indefinite weights: population models in disrupted environments II , 1991 .

[32]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[33]  R. Holt,et al.  Apparent Competition and Enemy-Free Space in Insect Host-Parasitoid Communities , 1993, The American Naturalist.