# zbMATH — the first resource for mathematics

Existence and asymptotic behavior of solutions for a predator-prey system with a nonlinear growth rate. (English) Zbl 1387.35365
Summary: The paper is concerned with a predator-prey diffusive system subject to homogeneous Neumann boundary conditions, where the growth rate $$(\frac{\alpha}{1+\beta v})$$ of the predator population is nonlinear. We study the existence of equilibrium solutions and the long-term behavior of the solutions. The main tools used here include the super-sub solution method, the bifurcation theory and linearization method.

##### MSC:
 35K57 Reaction-diffusion equations 35B35 Stability in context of PDEs 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Beverton, R.J.H.; Holt, S.J.; Graham, M. (ed.), The theory of fishing, 372-441, (1956), London [2] Crandall, M.G.; Rabinowitz, P.H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340, (1971) · Zbl 0219.46015 [3] Sen, M., The generalized beverton-Holt equation and the control of populations, Appl. Math. Model., 32, 2312-2328, (2008) · Zbl 1156.39301 [4] Sen, M.; Alonso-Quesada, S., Control issues for the beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases, Appl. Math. Comput., 215, 2616-2633, (2009) · Zbl 1179.92069 [5] López-Gómez, J.: Spectral Theory and Nonlinear Functional Analysis. Research Notes in Mathematics, vol. 426. CRC Press, Boca Raton (2001) · Zbl 0978.47048 [6] López-Gómez, J.; Molina-Meyer, M., Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas, J. Differ. Equ., 209, 416-441, (2005) · Zbl 1066.47063 [7] Liu, P.; Shi, J.; Wang, Y., Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251, 573-600, (2007) · Zbl 1139.47042 [8] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ., 131, 79-131, (1996) · Zbl 0867.35032 [9] Pao, C.V.: Nonlinear Parabolic and Ellitic Equations. Plenum, New York (1992) · Zbl 0780.35044 [10] Pao, C.V., Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26, 1889-1903, (1996) · Zbl 0853.35056 [11] Potier-Ferry, M., The linearization principle for the stability of solutions of quasilinear parabolic equations. I, Arch. Ration. Mech. Anal., 77, 301-320, (1981) · Zbl 0497.35006 [12] Smoller, J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer, New York (1994) · Zbl 0807.35002 [13] Tang, S.; Cheke, R.A.; Xiao, Y., Optimal impulsive harvesting on non-autonomous beverton-Holt difference equations, Nonlinear Anal., 65, 2311-2341, (2006) · Zbl 1119.39011 [14] Yang, W.; Wu, J.; Nie, H., Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14, 1183-1204, (2015) · Zbl 1422.35045 [15] Yang, Z.P.; Pao, C.V., Positive solutions and dynamics of some reaction diffusion models in HIV transmission, Nonlinear Anal., 35, 323-341, (1999) · Zbl 0914.92022
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.