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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 & Continuous Dynamical Systems - A, 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, 12 (1988), 895. doi: 10.1016/0362-546X(88)90073-9. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (2003). doi: 10.1002/0470871296. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

G. Lieberman, Second Order Parabolic Equations,, World Scientific, (1996). doi: 10.1142/3302. Google Scholar

[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. doi: 10.1603/0046-225X-29.3.454. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[21]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities,, Ph.D thesis, (2011). Google Scholar

[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. doi: 10.1046/j.1365-2656.2003.00693.x. Google Scholar

[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. doi: 10.1016/j.anbehav.2006.10.022. Google Scholar

[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. doi: 10.1137/0911001. Google Scholar

[25]

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

[26]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (2003). doi: 10.1002/0470871296. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

G. Lieberman, Second Order Parabolic Equations,, World Scientific, (1996). doi: 10.1142/3302. Google Scholar

[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. doi: 10.1603/0046-225X-29.3.454. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[21]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities,, Ph.D thesis, (2011). Google Scholar

[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. doi: 10.1046/j.1365-2656.2003.00693.x. Google Scholar

[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. doi: 10.1016/j.anbehav.2006.10.022. Google Scholar

[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. doi: 10.1137/0911001. Google Scholar

[25]

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

[26]

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

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