A boundary value problem for integrodifference population models with cyclic kernels

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

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

Abstract
Related Papers

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. 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. Google Scholar
[3]	N.-E. Fahssi, Polynomial triangles revisited., , (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. doi: 10.1007/BF00276367. Google Scholar
[5]	M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems,, no. 13 in Monographs in Population Biology, (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. 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. 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. doi: 10.1016/0025-5564(86)90069-6. Google Scholar
[10]	M. A. Krasnosel'skii, Positive Solutions of Operator Equations,, P. Noordhoff Ltd., (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. 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. 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, (1991), 239. 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, (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. 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. 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. Google Scholar
[3]	N.-E. Fahssi, Polynomial triangles revisited., , (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. doi: 10.1007/BF00276367. Google Scholar
[5]	M. P. Hassel, The Dynamics of Arthropod Predator-Prey Systems,, no. 13 in Monographs in Population Biology, (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. 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. 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. doi: 10.1016/0025-5564(86)90069-6. Google Scholar
[10]	M. A. Krasnosel'skii, Positive Solutions of Operator Equations,, P. Noordhoff Ltd., (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. 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. 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, (1991), 239. 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, (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. Google Scholar

[1]	J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[2]	Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
[3]	Getachew K. Befekadu, Panos J. Antsaklis. On noncooperative $n$-player principal eigenvalue games. Journal of Dynamics & Games, 2015, 2 (1) : 51-63. doi: 10.3934/jdg.2015.2.51
[4]	Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
[5]	Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[6]	Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[7]	Étienne Bernard, Marie Doumic, Pierre Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. Kinetic & Related Models, 2019, 12 (3) : 551-571. doi: 10.3934/krm.2019022
[8]	Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741
[9]	Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004
[10]	V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44.
[11]	Yangyang Xu, Wotao Yin, Stanley Osher. Learning circulant sensing kernels. Inverse Problems & Imaging, 2014, 8 (3) : 901-923. doi: 10.3934/ipi.2014.8.901
[12]	Wenxian Shen, Xiaoxia Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1665-1696. doi: 10.3934/dcds.2015.35.1665
[13]	Peng Zhong, Suzanne Lenhart. Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations. Evolution Equations & Control Theory, 2013, 2 (4) : 749-769. doi: 10.3934/eect.2013.2.749
[14]	Peng Zhong, Suzanne Lenhart. Optimal control of integrodifference equations with growth-harvesting-dispersal order. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2281-2298. doi: 10.3934/dcdsb.2012.17.2281
[15]	Marco V. Martinez, Suzanne Lenhart, K. A. Jane White. Optimal control of integrodifference equations in a pest-pathogen system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1759-1783. doi: 10.3934/dcdsb.2015.20.1759
[16]	Daniel Balagué, José A. Carrillo, Yao Yao. Confinement for repulsive-attractive kernels. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1227-1248. doi: 10.3934/dcdsb.2014.19.1227
[17]	Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551
[18]	Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399
[19]	Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
[20]	Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences