American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications
  • DCDS Home
  • This Issue
  • Next Article
    Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats
April  2015, 35(4): 1641-1663. doi: 10.3934/dcds.2015.35.1641

Avoidance behavior in intraguild predation communities: A cross-diffusion model

Daniel Ryan 1, and  Robert Stephen Cantrell 2,

1. 

National Institute for Mathematical and Biological Synthesis, University of Tennessee, Knoxville, TN 37996, United States
2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, United States

Received  July 2013 Revised  August 2014 Published  November 2014

A cross-diffusion model of an intraguild predation community in a two-dimensional bounded domain where the intraguild prey employs a fitness based avoidance strategy is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor is proved. It is shown that if the intraguild prey has positive fitness at any point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states where the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems. Additionally, the effects of fitness based avoidance on eigenvalues in more general systems are discussed.
Keywords: Cross-diffusion, avoidance, global existence., uniform persistence, intraguild predation.
Mathematics Subject Classification: 35K59, 35B40, 92D4.
Citation: Daniel Ryan, Robert Stephen Cantrell. Avoidance behavior in intraguild predation communities: A cross-diffusion model. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1641-1663. doi: 10.3934/dcds.2015.35.1641
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equaitons I: Abstract evolution equaitons, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9.

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250. doi: 10.1007/BF01215256.

[3]

H. Amann, Dynamic theory of quasilinear parabolic systems II: reaction-diffusion systems, Differential and Integral Equations, 3 (1990), 13-75.

[4]

M. Arim and P. A. Marquet, Intraguild predation: A widespread interaction related to species biology, Ecology Letters, 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x.

[5]

P. Amarasekare, Productivity, dispersal and the coexistence of intraguild predators and prey, Journal of Theoretical Biology, 243 (2006), 121-133. doi: 10.1016/j.jtbi.2006.06.007.

[6]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, The American Naturalist, 170 (2007), 819-831. doi: 10.1086/522837.

[7]

J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bulletin of the American Mathematical Society, 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3.

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296.

[9]

S. M. Durant, Living with the enemy: Avoidance of hyenas and lions by cheetahs in the Serengeti, Behavioral Ecology, 11 (2000), 624-632. doi: 10.1093/beheco/11.6.624.

[10]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395. doi: 10.1137/0520025.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, The American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.

[13]

T. Kimbrell, R. D. Holt and P. Lundberg, The influence of vigilance on intraguild predation, Journal of Theoretical Biology, 249 (2007), 218-234. doi: 10.1016/j.jtbi.2007.07.031.

[14]

D. Le, Cross diffusion systems on n spatial dimensional domains, Indiana University Mathematics Journal, 51 (2002), 625-643. doi: 10.1512/iumj.2002.51.2198.

[15]

G. Lieberman, Second Order Parabolic Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[16]

E. Lucas, D. Coderre and J. Brodeur, Selection of molting and pupation sites by Coleomegilla maculata (Coleoptera: Coccinellidae): avoidance of intraguild predation, Environmental Entomology, 29 (2000), 454-459. doi: 10.1603/0046-225X-29.3.454.

[17]

L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 13 (1959), 115-162.

[18]

T. Okuyama and R. L. Ruyle, Analysis of adaptive foraging in an intraguild predation system, Web Ecology, 4 (2003), 1-6. doi: 10.5194/we-4-1-2003.

[19]

F. Palomares and P. Ferreras, Spatial relationships between Iberian lynx and other carnivores in an area of southwestern Spain, Journal of Applied Ecology, 33 (1996), 5-13.

[20]

F. Palomares and T. M. Caro, Interspecific killing among mammalian carnivores, The American Naturalist, 153 (1999), 492-508. doi: 10.1086/303189.

[21]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities, Ph.D thesis, University of Miami, 2011.

[22]

F. Sergio, L. Marchesi and P. Pedrini, Spatial refugia and the coexistence of a diurnal raptor with its intraguild owl predator, Journal of Animal Ecology, 72 (2003), 232-245. doi: 10.1046/j.1365-2656.2003.00693.x.

[23]

F. Sergio, L. Marchesi, P. Pedrini and V. Penteriani, Coexistence of a generalist owl with its intraguild predator: distance-sensitive or habitat-mediated avoidance?, Animal Behaviour, 74 (2007), 1607-1616. doi: 10.1016/j.anbehav.2006.10.022.

[24]

R. Skeel and M. Berzins, A method for the spatial descretization of parabolic equations in one space variable, SIAM Journal on Scientific and Statistical Computing, 11 (1990), 1-32. doi: 10.1137/0911001.

[25]

R. Temam, Infinite Dimensional Dynamical Systems, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.

[26]

C. M. Thompson and E. M. Gese, Food webs and intraguild predation: Community interactions of a native mesocarnivore, Ecology, 88 (2007), 334-346.

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equaitons I: Abstract evolution equaitons, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9.

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250. doi: 10.1007/BF01215256.

[3]

H. Amann, Dynamic theory of quasilinear parabolic systems II: reaction-diffusion systems, Differential and Integral Equations, 3 (1990), 13-75.

[4]

M. Arim and P. A. Marquet, Intraguild predation: A widespread interaction related to species biology, Ecology Letters, 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x.

[5]

P. Amarasekare, Productivity, dispersal and the coexistence of intraguild predators and prey, Journal of Theoretical Biology, 243 (2006), 121-133. doi: 10.1016/j.jtbi.2006.06.007.

[6]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, The American Naturalist, 170 (2007), 819-831. doi: 10.1086/522837.

[7]

J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bulletin of the American Mathematical Society, 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3.

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296.

[9]

S. M. Durant, Living with the enemy: Avoidance of hyenas and lions by cheetahs in the Serengeti, Behavioral Ecology, 11 (2000), 624-632. doi: 10.1093/beheco/11.6.624.

[10]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395. doi: 10.1137/0520025.

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, The American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.

[13]

T. Kimbrell, R. D. Holt and P. Lundberg, The influence of vigilance on intraguild predation, Journal of Theoretical Biology, 249 (2007), 218-234. doi: 10.1016/j.jtbi.2007.07.031.

[14]

D. Le, Cross diffusion systems on n spatial dimensional domains, Indiana University Mathematics Journal, 51 (2002), 625-643. doi: 10.1512/iumj.2002.51.2198.

[15]

G. Lieberman, Second Order Parabolic Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[16]

E. Lucas, D. Coderre and J. Brodeur, Selection of molting and pupation sites by Coleomegilla maculata (Coleoptera: Coccinellidae): avoidance of intraguild predation, Environmental Entomology, 29 (2000), 454-459. doi: 10.1603/0046-225X-29.3.454.

[17]

L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 13 (1959), 115-162.

[18]

T. Okuyama and R. L. Ruyle, Analysis of adaptive foraging in an intraguild predation system, Web Ecology, 4 (2003), 1-6. doi: 10.5194/we-4-1-2003.

[19]

F. Palomares and P. Ferreras, Spatial relationships between Iberian lynx and other carnivores in an area of southwestern Spain, Journal of Applied Ecology, 33 (1996), 5-13.

[20]

F. Palomares and T. M. Caro, Interspecific killing among mammalian carnivores, The American Naturalist, 153 (1999), 492-508. doi: 10.1086/303189.

[21]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities, Ph.D thesis, University of Miami, 2011.

[22]

F. Sergio, L. Marchesi and P. Pedrini, Spatial refugia and the coexistence of a diurnal raptor with its intraguild owl predator, Journal of Animal Ecology, 72 (2003), 232-245. doi: 10.1046/j.1365-2656.2003.00693.x.

[23]

F. Sergio, L. Marchesi, P. Pedrini and V. Penteriani, Coexistence of a generalist owl with its intraguild predator: distance-sensitive or habitat-mediated avoidance?, Animal Behaviour, 74 (2007), 1607-1616. doi: 10.1016/j.anbehav.2006.10.022.

[24]

R. Skeel and M. Berzins, A method for the spatial descretization of parabolic equations in one space variable, SIAM Journal on Scientific and Statistical Computing, 11 (1990), 1-32. doi: 10.1137/0911001.

[25]

R. Temam, Infinite Dimensional Dynamical Systems, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.

[26]

C. M. Thompson and E. M. Gese, Food webs and intraguild predation: Community interactions of a native mesocarnivore, Ecology, 88 (2007), 334-346.

[1]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
[2]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[3]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[4]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
[5]

Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194
[6]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719
[7]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193
[8]

Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure and Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911
[9]

Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367
[10]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[11]

Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i
[12]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010
[13]

Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179
[14]

Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
[15]

Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147
[16]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[17]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[18]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[19]

Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745
[20]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (126)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy