A boundary value problem for integrodifference population models with cyclic kernels

Jon Jacobsen; Taylor McAdam

doi:10.3934/dcdsb.2014.19.3191

Article Contents

2014, Volume 19, Issue 10: 3191-3207. Doi: 10.3934/dcdsb.2014.19.3191

This issue Previous Article Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage Next Article Persistence and extinction of diffusing populations with two sexes and short reproductive season

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
2.
Department of Mathematics, University of Texas, Austin, TX 78712

Received: July 2013

Revised: December 2013

Published: December 2014

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 45C05, 92B05; Secondary: 34B05.

Citation:

Related Papers

Cited by

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.