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

A boundary value problem for integrodifference population models with cyclic kernels

Jon Jacobsen 1, and  Taylor McAdam 2,

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.
Keywords: trinomial coefficients., Integrodifference, principal eigenvalue, cyclic kernels, population persistence.
Mathematics Subject Classification: Primary: 45C05, 92B05; Secondary: 34B0.
Citation: Jon Jacobsen, Taylor McAdam. A boundary value problem for integrodifference population models with cyclic kernels. Discrete & 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.  Google Scholar

[2]

R. Dirzo and P. H. Raven, Global state of biodiversity and loss, Annual Review of Environment and Resources, 28 (2003), 137-167. Google Scholar

[3]

N.-E. Fahssi, Polynomial triangles revisited. arXiv:1202.0228v7 [math.CO], (2012). Google Scholar

[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.  Google Scholar

[5]

M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems, no. 13 in Monographs in Population Biology, Princeton University Press, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

N. J. A. Sloane, Online Encyclopedia of Integer Sequences,, 2013., ().   Google Scholar

[15]

C. Tudge, Last Animals at the Zoo: How Mass Extinction Can Be Stopped, Island Press, Washington, D.C., 1992. Google Scholar

[16]

R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

show all references

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.  Google Scholar

[2]

R. Dirzo and P. H. Raven, Global state of biodiversity and loss, Annual Review of Environment and Resources, 28 (2003), 137-167. Google Scholar

[3]

N.-E. Fahssi, Polynomial triangles revisited. arXiv:1202.0228v7 [math.CO], (2012). Google Scholar

[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.  Google Scholar

[5]

M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems, no. 13 in Monographs in Population Biology, Princeton University Press, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

N. J. A. Sloane, Online Encyclopedia of Integer Sequences,, 2013., ().   Google Scholar

[15]

C. Tudge, Last Animals at the Zoo: How Mass Extinction Can Be Stopped, Island Press, Washington, D.C., 1992. Google Scholar

[16]

R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

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