• Previous Article
    Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity
  • DCDS-B Home
  • This Issue
  • Next Article
    Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$
December  2017, 22(10): 3653-3661. doi: 10.3934/dcdsb.2017145

A PDE model of intraguild predation with cross-diffusion

1. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA

3. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author

Received  November 2016 Revised  January 2017 Published  April 2017

Fund Project: RSC is partially supportted by Beijing Municipal Foreign Expert Bureau, and NSF grants DMS-11-18623 and DMS-15-14752. XC is supported by the Research Funds of Renmin University of China (15XNLF21), and by China Postdoctoral Science Foundation (2016M591319). KYL is partially supported by NSF grant DMS-1411476. TX is supported by NSF of China (No. 11601516,11571364 and 11571363) and research Funds of Renmin University of China (No. 15XNLF10)

This note concerns a quasilinear parabolic system modeling an intraguild predation community in a focal habitat in $\mathbb{R}^n$, $n ≥ 2$. In this system the intraguild prey employs a fitness-based dispersal strategy whereby the intraguild prey moves away from a locale when predation risk is high enough to render the locale undesirable for resource acquisition. The system modifies the model considered in Ryan and Cantrell (2015) by adding an element of mutual interference among predators to the functional response terms in the model, thereby switching from Holling Ⅱ forms to Beddington-DeAngelis forms. We show that the resulting system can be realized as a semi-dynamical system with a global attractor for any $n ≥ 2$. In contrast, the orginal model was restricted to two dimensional spatial habitats. The permanance of the intraguild prey then follows as in Ryan and Cantrell by means of the Acyclicity Theorem of Persistence Theory.

Citation: Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations Ⅰ: Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9. Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Math. Z., 202 (1989), 219-250. 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), 13-75. Google Scholar

[4]

P. Amarasekare, Productivity, dispersal and the coexistence of intraguild predators and prey, J. Theor. Ecol., 243 (2006), 121-133. doi: 10.1016/j.jtbi.2006.06.007. Google Scholar

[5]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, Am. Nat., 170 (2007), 819-831. Google Scholar

[6]

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

[7]

J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. Google Scholar

[8]

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. Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs, Vol. 25, Amer. Math. Soc. , Providence, RI, 1988. Google Scholar

[10]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. Google Scholar

[11]

R. D. Holt and G. A. Polis, A theoretical modeling framework for intraguild predation, Am. Nat., 149 (1997), 745-764. Google Scholar

[12]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, Vol. 23, Amer. Math. Soc. , 1988. Google Scholar

[13]

D. Le, Cross diffusion systems on n dimensional spatial domains, Indiana Univ. Math. J., 51 (2002), 625-643. doi: 10.1512/iumj.2002.51.2198. Google Scholar

[14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar
[15]

E. LucasD. 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. Google Scholar

[16]

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

[17]

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

[18]

D. Ryan and R. S. Cantrell, Avoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete Contin. Dyn. Syst. A, 35 (2015), 1641-1663. doi: 10.3934/dcds.2015.35.1641. Google Scholar

[19]

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

[20]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Vol. 118. Providence, RI, American Mathematical Society, 2011. Google Scholar

[21]

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

show all references

References:
[1]

H. Amann, Dynamic theory of quasilinear parabolic equations Ⅰ: Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9. Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Math. Z., 202 (1989), 219-250. 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), 13-75. Google Scholar

[4]

P. Amarasekare, Productivity, dispersal and the coexistence of intraguild predators and prey, J. Theor. Ecol., 243 (2006), 121-133. doi: 10.1016/j.jtbi.2006.06.007. Google Scholar

[5]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, Am. Nat., 170 (2007), 819-831. Google Scholar

[6]

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

[7]

J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 (1971), 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. Google Scholar

[8]

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. Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs, Vol. 25, Amer. Math. Soc. , Providence, RI, 1988. Google Scholar

[10]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. Google Scholar

[11]

R. D. Holt and G. A. Polis, A theoretical modeling framework for intraguild predation, Am. Nat., 149 (1997), 745-764. Google Scholar

[12]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, Vol. 23, Amer. Math. Soc. , 1988. Google Scholar

[13]

D. Le, Cross diffusion systems on n dimensional spatial domains, Indiana Univ. Math. J., 51 (2002), 625-643. doi: 10.1512/iumj.2002.51.2198. Google Scholar

[14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar
[15]

E. LucasD. 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. Google Scholar

[16]

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

[17]

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

[18]

D. Ryan and R. S. Cantrell, Avoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete Contin. Dyn. Syst. A, 35 (2015), 1641-1663. doi: 10.3934/dcds.2015.35.1641. Google Scholar

[19]

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

[20]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Vol. 118. Providence, RI, American Mathematical Society, 2011. Google Scholar

[21]

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

[1]

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

[2]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. 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 & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133

[17]

Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163

[18]

Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063

[19]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152

[20]

Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (29)
  • HTML views (7)
  • Cited by (0)

[Back to Top]