Analytical and numerical tools for diffusion-based movement models.

I present a general diffusion-based modeling framework for the analysis of animal movements in heterogeneous landscapes, including terms representing advection, mortality, and edge-mediated behavior. I use adjoint operator theory to develop mathematical machinery for the assessment of a number of biologically relevant quantities, such as occupancy times, hitting probabilities, quasi-stationary distributions, the backwards equation, and conditional probability densities. I derive finite-element approximations, which can be used to obtain numerical solutions in domains which do not allow for an analytical treatment. As an example, I model the movements of the butterfly Melitaea cinxia in an island consisting of a set of habitat patches and the intervening matrix habitat. I illustrate the behavior of the model and the mathematical theory by examining the effects of a hypothetical movement barrier and advection caused by prevailing wind conditions.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  T. Ricketts.  The Matrix Matters: Effective Isolation in Fragmented Landscapes , 2001, The American Naturalist.

[3]  Séverine Vuilleumier,et al.  Animal dispersal modelling: handling landscape features and related animal choices , 2006 .

[4]  S. Aviron,et al.  When is landscape matrix important for determining animal fluxes between resource patches , 2005 .

[5]  J. Gamarra,et al.  Metapopulation Ecology , 2007 .

[6]  Hal Caswell,et al.  On the Estimation of Dispersal Kernels from Individual Mark-Recapture Data , 2005, Environmental and Ecological Statistics.

[7]  Miguel Delibes,et al.  Effects of Matrix Heterogeneity on Animal Dispersal: From Individual Behavior to Metapopulation‐Level Parameters , 2004, The American Naturalist.

[8]  H. Reuter,et al.  Dispersal of carabid beetles—emergence of distribution patterns , 2005 .

[9]  Benjamin M Bolker,et al.  Effects of Landscape Corridors on Seed Dispersal by Birds , 2005, Science.

[10]  R. Arditi,et al.  Clustering due to Acceleration in the Response to Population Gradient: A Simple Self‐Organization Model , 2004, The American Naturalist.

[11]  Helen T. Murphy,et al.  Context and connectivity in plant metapopulations and landscape mosaics: does the matrix matter? , 2004 .

[12]  R. Hood,et al.  An individual-based numerical model of medusa swimming behavior , 2006 .

[13]  Eric J. Gustafson,et al.  Simulating dispersal of reintroduced species within heterogeneous landscapes , 2004 .

[14]  Nicolas Perrin,et al.  Effects of cognitive abilities on metapopulation connectivity , 2006 .

[15]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[16]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[17]  Oscar E. Gaggiotti,et al.  Ecology, genetics, and evolution of metapopulations , 2004 .

[18]  J. Vielliard,et al.  Effects of structural and functional connectivity and patch size on the abundance of seven Atlantic Forest bird species , 2005 .

[19]  Nicolas Schtickzelle,et al.  Behavioural responses to habitat patch boundaries restrict dispersal and generate emigration-patch area relationships in fragmented landscapes. , 2003, The Journal of animal ecology.

[20]  Volker Grimm,et al.  Individual-based modelling and ecological theory: synthesis of a workshop , 1999 .

[21]  Simone K. Heinz,et al.  Connectivity in Heterogeneous Landscapes: Analyzing the Effect of Topography , 2005, Landscape Ecology.

[22]  Otso Ovaskainen,et al.  The quasistationary distribution of the stochastic logistic model , 2001, Journal of Applied Probability.

[23]  Elja Arjas,et al.  Bayesian methods for analyzing movements in heterogeneous landscapes from mark-recapture data. , 2008, Ecology.

[24]  Otso Ovaskainen,et al.  HABITAT-SPECIFIC MOVEMENT PARAMETERS ESTIMATED USING MARK–RECAPTURE DATA AND A DIFFUSION MODEL , 2004 .

[25]  Stig Larsson,et al.  Partial differential equations with numerical methods , 2003, Texts in applied mathematics.

[26]  C. Cooper,et al.  Landscape patterns and dispersal success: simulated population dynamics in the brown treecreeper , 2002 .

[27]  Kyle J. Haynes,et al.  Interpatch movement and edge effects: the role of behavioral responses to the landscape matrix , 2006 .

[28]  M. Baguette,et al.  Gene flow and functional connectivity in the natterjack toad , 2006, Molecular ecology.

[29]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[30]  Lenore Fahrig,et al.  How does landscape structure influence landscape connectivity , 2002 .

[31]  R. O'Neill,et al.  Landscape Ecology Explained@@@Landscape Ecology in Theory and Practice: Pattern and Process , 2001 .

[32]  L. Fahrig,et al.  Connectivity is a vital element of landscape structure , 1993 .

[33]  John Crank,et al.  The Mathematics Of Diffusion , 1956 .

[34]  N. Schtickzelle,et al.  Quantifying functional connectivity: experimental evidence for patch-specific resistance in the Natterjack toad (Bufo calamita) , 2004, Landscape Ecology.

[35]  S. Levin,et al.  Diffusion and Ecological Problems: Modern Perspectives , 2013 .

[36]  G. Barton.  The Mathematics of Diffusion 2nd edn , 1975 .

[37]  Atte Moilanen,et al.  ESTIMATING THE PARAMETERS OF SURVIVAL AND MIGRATION OF INDIVIDUALS IN METAPOPULATIONS , 2000 .

[38]  Monica G. Turner,et al.  Landscape connectivity and population distributions in heterogeneous environments , 1997 .

[39]  R. O’Hara,et al.  Consequences of the spatial configuration of resources for the distribution and dynamics of the endangered Parnassius apollo butterfly , 2006 .

[40]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[41]  J. Morales,et al.  The effects of plant distribution and frugivore density on the scale and shape of dispersal kernels. , 2006, Ecology.

[42]  Otso Ovaskainen,et al.  Biased movement at a boundary and conditional occupancy times for diffusion processes , 2003, Journal of Applied Probability.

[43]  S. Redner.  A guide to first-passage processes , 2001 .

[44]  P. Turchin.  Quantitative analysis of movement : measuring and modeling population redistribution in animals and plants , 1998 .

[45]  J. Z. Zhu,et al.  The finite element method , 1977 .

[46]  Uta Berger,et al.  Pattern-Oriented Modeling of Agent-Based Complex Systems: Lessons from Ecology , 2005, Science.

[47]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[48]  T. J. Roper,et al.  Nonrandom movement behavior at habitat boundaries in two butterfly species: implications for dispersal. , 2006, Ecology.

[49]  Otso Ovaskainen,et al.  From Individual Behavior to Metapopulation Dynamics: Unifying the Patchy Population and Classic Metapopulation Models , 2004, The American Naturalist.

[50]  Otso Ovaskainen,et al.  Dispersal-related life-history trade-offs in a butterfly metapopulation. , 2006, The Journal of animal ecology.

[51]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

[52]  Marc Bélisle,et al.  MEASURING LANDSCAPE CONNECTIVITY: THE CHALLENGE OF BEHAVIORAL LANDSCAPE ECOLOGY , 2005 .

[53]  J. Roughgarden,et al.  The Impact of Directed versus Random Movement on Population Dynamics and Biodiversity Patterns , 2005, The American Naturalist.

[54]  Yuan Lou,et al.  Does movement toward better environments always benefit a population , 2003 .

[55]  E. Zauderer,et al.  Partial Differential Equations of Applied Mathematics: Zauderer/Partial , 2006 .

[56]  M A Lewis,et al.  Persistence, spread and the drift paradox. , 2005, Theoretical population biology.

[57]  Elizabeth E. Crone,et al.  EDGE-MEDIATED DISPERSAL BEHAVIOR IN A PRAIRIE BUTTERFLY , 2001 .

[58]  Otso Ovaskainen,et al.  Space and stochasticity in population dynamics , 2006, Proceedings of the National Academy of Sciences.

[59]  Brett J. Goodwin,et al.  Is landscape connectivity a dependent or independent variable? , 2003, Landscape Ecology.