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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. [2] H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Math. Z., 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] 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. [5] P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, Am. Nat., 170 (2007), 819-831. [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. [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. [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. [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs, Vol. 25, Amer. Math. Soc. , Providence, RI, 1988. [10] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [11] R. D. Holt and G. A. Polis, A theoretical modeling framework for intraguild predation, Am. Nat., 149 (1997), 745-764. [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. [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. [14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [15] 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. [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. [17] D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities Ph. D thesis, University of Miami, 2011. [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. [19] F. Sergio, L. 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. [20] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Vol. 118. Providence, RI, American Mathematical Society, 2011. [21] 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 equations Ⅰ: Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919. doi: 10.1016/0362-546X(88)90073-9. [2] H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Math. Z., 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] 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. [5] P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, Am. Nat., 170 (2007), 819-831. [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. [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. [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. [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems Math. Surveys and Monographs, Vol. 25, Amer. Math. Soc. , Providence, RI, 1988. [10] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [11] R. D. Holt and G. A. Polis, A theoretical modeling framework for intraguild predation, Am. Nat., 149 (1997), 745-764. [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. [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. [14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [15] 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. [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. [17] D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities Ph. D thesis, University of Miami, 2011. [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. [19] F. Sergio, L. 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. [20] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Vol. 118. Providence, RI, American Mathematical Society, 2011. [21] C. M. Thompson and E. M. Gese, Food webs and intraguild predation: Community interactions of a native mesocarnivore, Ecology, 88 (2007), 334-346.
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