A reaction-diffusion system modeling predator-prey with prey-taxis. (English) Zbl 1156.35404

Summary: We are concerned with a system of nonlinear partial differential equations modeling the Lotka-Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation.


35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
Full Text: DOI


[1] Ainseba, B.E.; Heiser, F.; Langlais, M., A mathematical analysis of a predator – prey system in a highly heterogeneous environment, J. differential integr. equations, 15, 4, 85-404, (2002) · Zbl 1011.35075
[2] Aly, S.; Farkas, M., Prey – predator in patchy environment with cross diffusion, Differential equations dyn. syst., 13, 3-4, 311-321, (2005) · Zbl 1129.34031
[3] Bendahmane, M.; Karlsen, K.H.; Urbano, J.M., On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. models methods appl. sci., 17, 5, 783-804, (2007) · Zbl 1133.35061
[4] M. Farkas, Two ways of modelling cross-diffusion, in: Proceedings of the Second World Congress of Nonlinear Analysts, Part 2, Athens, 1996, Nonlinear Anal. 30 (2) (1997) 1225-1233. · Zbl 0893.35049
[5] Hillen, T.; Painter, K., Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. appl. math., 26, 280-301, (2001) · Zbl 0998.92006
[6] O.A. Ladyzhenskaya, V. Solonnikov, N. Ural’ceva, Linear and quasi-linear equations of parabolic type, Trans. AMS, vol. 23, Providence, RI, 1968.
[7] Levin, S.A., A more functional response to predator prey stability, Am. nat., 108, 207-228, (1977)
[8] Levin, S.A.; Segel, L.A., Hypothesis for origin of planktonic patchiness, Nature, 259, 659, (1976)
[9] Mimura, M.; Kawasaki, K., Spatial segregation in competitive interaction – diffusion equations, J. math. biol., 9, 49-64, (1980) · Zbl 0425.92010
[10] Mimura, M.; Murray, J.D., On a diffusion prey – predator model which exhibits patchiness, J. theor. biol., 75, 249-252, (1978)
[11] Mimura, M.; Yamaguti, M., Pattern formation in interacting and diffusive systems in population biology, Adv. biophys., 15, 19-65, (1982)
[12] Okubo, A., Diffusion and ecological problems: mathematical models, biomathematics, vol. 10, (1980), Springer Berlin
[13] Okubo, A.; Chiang, H.C., An analysis of the kinematics of swarming of anarete pritchardi kim (diptera: cecidomyiidae), Res. popul. ecol. (Kyoto), 16, 1-42, (1974)
[14] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. mat. pura appl. ser. (4), 146, 65-96, (1987) · Zbl 0629.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.