A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing

Paulo Amorim; Bruno Telch; Luis M. Villada; Paulo Amorim; Bruno Telch; Luis M. Villada

doi:10.3934/mbe.2019257

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5114-5145. doi: 10.3934/mbe.2019257

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A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing

1.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, Cidade Universitária - Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ - Brasil
2.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile and CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received: 04 January 2019 Accepted: 16 May 2019 Published: 05 June 2019

In this paper, we propose and analyze a reaction-diffusion model for predator-prey interaction, featuring both prey and predator taxis mediated by nonlocal sensing. Both predator and prey densities are governed by parabolic equations. The prey and predator detect each other indirectly by means of odor or visibility fields, modeled by elliptic equations. We provide uniform estimates in Lebesgue spaces which lead to boundedness and the global well-posedness for the system. Numerical experiments are presented and discussed, allowing us to showcase the dynamical properties of the solutions.
- mathematical biology,
- predator-prey,
- mechanistic models,
- reaction-diffusion equations,
- numerical simulations,
- ecology,
- chemotaxis,
- animal movement
Citation: Paulo Amorim, Bruno Telch, Luis M. Villada. A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5114-5145. doi: 10.3934/mbe.2019257

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Abstract

In this paper, we propose and analyze a reaction-diffusion model for predator-prey interaction, featuring both prey and predator taxis mediated by nonlocal sensing. Both predator and prey densities are governed by parabolic equations. The prey and predator detect each other indirectly by means of odor or visibility fields, modeled by elliptic equations. We provide uniform estimates in Lebesgue spaces which lead to boundedness and the global well-posedness for the system. Numerical experiments are presented and discussed, allowing us to showcase the dynamical properties of the solutions.

References

[1]	B. Ainseba, M. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal- Real., 9 (2008), 2086–2105.
[2]	A. Chakraborty, M. Singh, D. Lucy, et.al., Predator-prey model with prey-taxis and diffusion, Math. Comput. Model., 46 (2007), 482–498.
[3]	T. Goudon, B. Nkonga, M. Rascle, et. al., Self-organized populations interacting under pursuit- evasion dynamics, Physica D: Nonlinear Phenomena, 304–305 (2015), 1–22.
[4]	T. Goudon and L. Urrutia, Analysis of kinetic and macroscopic models of pursuit-evasion dynamics, Commun. Math. Sci., 14 (2016), 2253–2286.
[5]	X. He and S. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator- prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73–77.
[6]	H. Jin and Z. Wang, Global stability of prey-taxis systems, J. Differ. Equations, 262 (2017), 1257– 1290.
[7]	J. M. Lee, T. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis. Bull. Math. Biol., 70 (2008), 654.
[8]	J. M. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551–573.
[9]	Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis Nonlinear Anal. Real World Appl., 11 (2010), 2056–2064.
[10]	Y. Tyutyunov, L. Titova and R. Arditi, A Minimal Model of Pursuit-Evasion in a Predator-Prey System.Math. Model. Nat. Pheno., 2 (2007), 122–134.
[11]	Ke Wang, Qi Wang and Feng Yu, Stationary and time-periodic patterns of two-predator and one- prey systems with prey-taxis, Discrete Contin. Dyn. S., 37 (2016), 505–543.
[12]	X. Wang, W. Wang and G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis,Math. Method. Appl. Sci., 38 (2015), 431–443.
[13]	S. Wu, J. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis, J. Differ. Equations, 260, (2016).
[14]	T.Xiang, Global dynamicsfora diffusive predator-preymodelwith prey-taxisand classicalLotka– Volterra kinetics, Nonlinear Anal. Real World Appl., 39 (2018), 278–299.
[15]	Y. Tao and M. Winkler. Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. S., 20 (2015), 3165–3183.
[16]	R. Alonso, P. Amorim and T. Goudon, Analysis of a chemotaxis system modeling ant foraging, Math. Models Methods Appl. Sci., 26 (2016),1785–1824.
[17]	E. De Giorgi, Sulla differenciabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Acccad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43.
[18]	E. Keller and L. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415.
[19]	E. Keller and L. Segel, Model for Chemotaxis, J. theor. Biol., 30 (1971), 225–234.
[20]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183–217.
[21]	B. Perthame, Transport Equations in Biology. Birkhäuser Verlag, Basel - Boston - Berlin.
[22]	L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi- geostrophic equation, Ann. Math., 171 (2010), 1903–1930.
[23]	L. Caffarelli and A. Vasseur, The De Giorgi method for nonlocal fluid dynamics, in Nonlinear Partial Differential Equations, Advanced Courses in Mathematics–CRM Barcelona (Birhäuser, 2012), 1–38.
[24]	B. Perthame and A. Vasseur, Regularization in Keller–Segel type systems and the De Giorgi method, Commun. Math. Sci., 10 (2012), 463–476.
[25]	H. Brézis, Analyse Fonctionnelle, Théorie et Applications (Masson, 1987).
[26]	A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (208), 169–205.
[27]	H. Holden, K. H. Karlsen and N. H. Risebro, On uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations, Electron. J. Differential Equations, 46 (2003), 1–31.
[28]	R. Bürger, R. Ruiz-Baier and K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media, J. Sci. Comput., 43 (2010), 261–290.

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