# American Institute of Mathematical Sciences

December  2014, 19(10): 3191-3207. doi: 10.3934/dcdsb.2014.19.3191

## A boundary value problem for integrodifference population models with cyclic kernels

 1 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States 2 Department of Mathematics, University of Texas, Austin, TX 78712, United States

Received  July 2013 Revised  December 2013 Published  October 2014

The population dynamics of species with separate growth and dispersal stages can be modeled by a discrete-time, continuous-space integrodifference equation. Many authors have considered the case where the model parameters remain fixed over time, however real environments are constantly in flux. We develop a framework for analyzing the population dynamics when the dispersal parameters change over time in a cyclic fashion. In particular, for the case of $N$ cyclic dispersal kernels modeling movement in the presence of unidirectional flow, we derive a $2N^{th}$-order boundary value problem that can be used to study the linear stability of the associated integrodifference model.
Citation: Jon Jacobsen, Taylor McAdam. A boundary value problem for integrodifference population models with cyclic kernels. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3191-3207. doi: 10.3934/dcdsb.2014.19.3191
##### References:
 [1] C. J. Collins, C. I. Fraser, A. Ashcroft and J. M. Waters, Asymmetric dispersal of southern bull-kelp (Durvillaea antarctica) adults in coastal New Zealand: Testing and oceanographic hypothesis, Molecular Ecology, 19 (2010), 4572-4580. doi: 10.1111/j.1365-294X.2010.04842.x. [2] R. Dirzo and P. H. Raven, Global state of biodiversity and loss, Annual Review of Environment and Resources, 28 (2003), 137-167. [3] N.-E. Fahssi, Polynomial triangles revisited. arXiv:1202.0228v7 [math.CO], (2012). [4] D. P. Hardin, P. Takáč and G. F. Webb, Asymptotic properties of a continuous-space discrete-time population model in a random environment, Journal of Mathematical Biology, 26 (1988), 361-374. doi: 10.1007/BF00276367. [5] M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems, no. 13 in Monographs in Population Biology, Princeton University Press, 1978. [6] C. M. Herrera, P. Jordano, J. Guitian and A. Traveset, Annual variability in seed production by woody plants and the masting concept: Reassessment of principles and relationship to pollination and seed dispersal, The American Naturalist, 152 (1998), 576-594. doi: 10.1086/286191. [7] H. F. Howe and J. Smallwood, Ecology of seed dispersal, Annual Review of Ecology and Systematics, 13 (1982), 201-228. doi: 10.1146/annurev.es.13.110182.001221. [8] J. Jacobsen, Y. Jin, and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments, preprint. doi: 10.1007/s00285-014-0774-y. [9] M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Mathematical Biosciences, 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. [10] M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. [11] F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Review, 47 (2005), 749-772. doi: 10.1137/050636152. [12] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern-formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. doi: 10.1006/tpbi.1995.1020. [13] D. Pearce, An economic approach to saving the tropical forests, in Economic Policy Towards the Environment, (ed. D. Helm), Blackwell Publishers, 1991, 239-262. [14] N. J. A. Sloane, Online Encyclopedia of Integer Sequences, 2013. [15] C. Tudge, Last Animals at the Zoo: How Mass Extinction Can Be Stopped, Island Press, Washington, D.C., 1992. [16] R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137.

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##### References:
 [1] C. J. Collins, C. I. Fraser, A. Ashcroft and J. M. Waters, Asymmetric dispersal of southern bull-kelp (Durvillaea antarctica) adults in coastal New Zealand: Testing and oceanographic hypothesis, Molecular Ecology, 19 (2010), 4572-4580. doi: 10.1111/j.1365-294X.2010.04842.x. [2] R. Dirzo and P. H. Raven, Global state of biodiversity and loss, Annual Review of Environment and Resources, 28 (2003), 137-167. [3] N.-E. Fahssi, Polynomial triangles revisited. arXiv:1202.0228v7 [math.CO], (2012). [4] D. P. Hardin, P. Takáč and G. F. Webb, Asymptotic properties of a continuous-space discrete-time population model in a random environment, Journal of Mathematical Biology, 26 (1988), 361-374. doi: 10.1007/BF00276367. [5] M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems, no. 13 in Monographs in Population Biology, Princeton University Press, 1978. [6] C. M. Herrera, P. Jordano, J. Guitian and A. Traveset, Annual variability in seed production by woody plants and the masting concept: Reassessment of principles and relationship to pollination and seed dispersal, The American Naturalist, 152 (1998), 576-594. doi: 10.1086/286191. [7] H. F. Howe and J. Smallwood, Ecology of seed dispersal, Annual Review of Ecology and Systematics, 13 (1982), 201-228. doi: 10.1146/annurev.es.13.110182.001221. [8] J. Jacobsen, Y. Jin, and M. A. Lewis, Integrodifference models for persistence in temporally varying river environments, preprint. doi: 10.1007/s00285-014-0774-y. [9] M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Mathematical Biosciences, 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. [10] M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. [11] F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Review, 47 (2005), 749-772. doi: 10.1137/050636152. [12] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern-formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. doi: 10.1006/tpbi.1995.1020. [13] D. Pearce, An economic approach to saving the tropical forests, in Economic Policy Towards the Environment, (ed. D. Helm), Blackwell Publishers, 1991, 239-262. [14] N. J. A. Sloane, Online Encyclopedia of Integer Sequences, 2013. [15] C. Tudge, Last Animals at the Zoo: How Mass Extinction Can Be Stopped, Island Press, Washington, D.C., 1992. [16] R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137.
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