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Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate. (English) Zbl 1422.35045
Summary: The paper is concerned with a predator-prey diffusive dynamics subject to homogeneous Dirichlet boundary conditions, in which the growth rate of the the predator is nonlinear. Taking \(m\) as the main parameter, we show the existence, stability and exact number of positive solution when \(m\) is large, and some numerical simulations are done to complement the analytical results. The main tools used here include the fixed point index theory, the super-sub solution method, the bifurcation theory and the perturbation technique.

MSC:
35J57 Boundary value problems for second-order elliptic systems
35K57 Reaction-diffusion equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B09 Positive solutions to PDEs
35B35 Stability in context of PDEs
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
92D25 Population dynamics (general)
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[1] R. J. Beverton, The theory of fishing, in: M. Graham (Ed.), Sea Fisheries;, Their Investigation in the United Kingdom, 372, (1956)
[2] J. Blat, Global bifurcation of positive solutions in some systems of elliptic equations,, SIAM J. Math. Anal., 17, 1339, (1986) · Zbl 0613.35008
[3] J. L. Bravo, Existence of a polycycle in non-Lipschitz Gause-type predator-prey models,, J. Math. Anal. Appl., 373, 512, (2011) · Zbl 1216.34038
[4] A. Casal, Existence and uniqueness of coexistence states for a predator-prey model with diffusion,, Differential Integral Equations, 7, 411, (1994) · Zbl 0823.35050
[5] B. Dai, Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses,, Appl. Math. Comput., 217, 7478, (2011) · Zbl 1221.92077
[6] M. De la Sen, The generalized Beverton-Holt equation and the control of populations,, Appl. Math. Model., 32, 2312, (2008) · Zbl 1156.39301
[7] M. De la Sen, 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, (2009) · Zbl 1179.92069
[8] X. Ding, Positive periodic solutions in delayed Gause-type predator-prey systems,, J. Math. Anal. Appl., 339, 1220, (2008) · Zbl 1137.34033
[9] X. Ding, Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses,, Nonlinear Anal. Real World Appl., 10, 2819, (2009) · Zbl 1179.34090
[10] X. Ding, Positive periodic solutions for impulsive Gause-type predator-prey systems,, Appl. Math. Comput., 218, 6785, (2012) · Zbl 1239.92079
[11] Y. Du, Some uniqueness and exact multiplicity results for a predator-prey model,, Trans. Amer. Math. Soc., 349, 2443, (1997) · Zbl 0965.35041
[12] R. M. Etoua, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III,, J. Differential Equations, 249, 2316, (2010) · Zbl 1217.34080
[13] G. F. Gause, <em>The struggle for existence</em>,, Williams and Wilkins, (1936)
[14] G. F. Gause, Further studies of interaction between predator and prey,, J. Animal Ecol., 5, 1, (1936)
[15] C. Gui, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model,, Comm. Pure Appl. Math., 47, 1571, (1994) · Zbl 0829.92015
[16] K. Has\'\ik, Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline,, Math. Biosci., 164, 203, (2000) · Zbl 0952.92021
[17] W. Ko, A qualitative study on general Gause-type predator-prey models with constant diffusion rates,, J. Math. Anal. Appl., 344, 217, (2008) · Zbl 1144.35029
[18] W. Ko, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,, Nonlinear Anal. Real World Appl., 10, 2558, (2009) · Zbl 1163.35339
[19] Y. Kuang, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, Math. Biosci., 88, 67, (1988) · Zbl 0642.92016
[20] Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems,, Appl. Anal., 29, 269, (1988) · Zbl 0629.34036
[21] Y. Kuang, On the location and period of limit cycles in Gause-type predator-prey systems,, J. Math. Anal. Appl., 142, 130, (1989) · Zbl 0675.92017
[22] Y. Kuang, Global stability of Gause-type predator-prey systems,, J. Math. Biol., 28, 463, (1990) · Zbl 0742.92022
[23] L. Li, On positive solutions of general nonlinear elliptic symbiotic interacting systems,, Appl. Anal., 40, 281, (1991) · Zbl 0757.35023
[24] S. Laurin, Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III,, J. Differential Equations, 251, 2980, (2011) · Zbl 1236.34059
[25] G. Liu, Positive periodic solutions for a class of neutral delay Gause-type predator-prey system · Zbl 1181.34089
[26] Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems,, Proc. Amer. Math. Soc., 133, 3619, (2005) · Zbl 1077.34056
[27] S. M. Moghadas, Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response,, Chaos Solitons Fractals, 23, 55, (2005) · Zbl 1058.92049
[28] H. Nie, Multiplicity and stability of a predator-prey model with non-monotonic conversion rate,, Nonlinear Anal. Real World Appl., 10, 154, (2009) · Zbl 1154.35386
[29] C. V. Pao, <em>Nonlinear parabolic and elliptic equations</em>,, Plenum Press, (1992) · Zbl 0777.35001
[30] S. Ruan, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM J. Appl. Math., 61, 1445 · Zbl 0986.34045
[31] H. H. Schaefer, <em>Topological Vector Spaces</em>,, 2\(^{nd}\) edition, (1999) · Zbl 0983.46002
[32] J. Smoller, <em>Shock Waves and Reaction-Diffusion Equations</em>,, 2\(^{nd}\) edition, (1994) · Zbl 0807.35002
[33] G. Sun, Dynamical complexity of a spatial predator-prey model with migration,, Ecological modelling, 219, 248, (2008)
[34] S. Tang, Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations,, Nonlinear Anal., 65, 2311, (2006) · Zbl 1119.39011
[35] S. Tang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge,, Nonlinear Anal., 76, 165, (2013) · Zbl 1256.34037
[36] P. Turchin, <em>Complex Population Dynamics: A Theoretical/Empirical Synthesis</em>,, Princeton University Press, (2003) · Zbl 1062.92077
[37] M. Wang, <em>Nonlinear Elliptic equations</em>,, Science Press, (2010)
[38] M. Wang, <em>Nonlinear Parabolic Equation</em>,, Science Press, (1993)
[39] W. Wang, Complex patterns in a predator-prey model with self and cross-diffusion,, Commun. Nonlinear Sci. Numer. Simul., 16, 2006, (2011) · Zbl 1221.35423
[40] J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal. Ser. A: Theory Methods, 39, 817, (2000) · Zbl 0940.35114
[41] R. Xu, Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks,, J. Math. Anal. Appl., 265, 148, (2002) · Zbl 1013.34074
[42] Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffsion, in: <em>Handbook of Differential Equations: Stationary Partial Differential Equations</em>, Vol. VI,, Elsevier, 411, (2008)
[43] G. Zhang, Positive solutions for a general Gause-type predator-prey model with monotonic functional response,, Abstr. Appl. Anal., (2011) · Zbl 1226.92072
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