May  2015, 14(3): 1183-1204. doi: 10.3934/cpaa.2015.14.1183

Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate

1. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shannxi 710062, China

2. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119

3. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062

Received  August 2014 Revised  January 2015 Published  March 2015

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.
Citation: Wen-Bin Yang, Jianhua Wu, Hua Nie. Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1183-1204. doi: 10.3934/cpaa.2015.14.1183
References:
[1]

R. J. Beverton and S. J. Holt, The theory of fishing, in: M. Graham (Ed.), Sea Fisheries;, Their Investigation in the United Kingdom, (1956), 372.   Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations,, \emph{SIAM J. Math. Anal.}, 17 (1986), 1339.  doi: 10.1137/0517094.  Google Scholar

[3]

J. L. Bravo, M. Fernández, M. Gámez, B. Granados and A. Tineo, Existence of a polycycle in non-Lipschitz Gause-type predator-prey models,, \emph{J. Math. Anal. Appl.}, 373 (2011), 512.  doi: 10.1016/j.jmaa.2010.08.001.  Google Scholar

[4]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion,, \emph{Differential Integral Equations}, 7 (1994), 411.   Google Scholar

[5]

B. Dai, Y. Li and Z. Luo, Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses,, \emph{Appl. Math. Comput.}, 217 (2011), 7478.  doi: 10.1016/j.amc.2011.02.049.  Google Scholar

[6]

M. De la Sen, The generalized Beverton-Holt equation and the control of populations,, \emph{Appl. Math. Model.}, 32 (2008), 2312.  doi: 10.1016/j.apm.2007.09.007.  Google Scholar

[7]

M. De la Sen and S. Alonso-Quesada, Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases,, \emph{Appl. Math. Comput.}, 215 (2009), 2616.  doi: 10.1016/j.amc.2009.09.003.  Google Scholar

[8]

X. Ding and J. Jiang, Positive periodic solutions in delayed Gause-type predator-prey systems,, \emph{J. Math. Anal. Appl.}, 339 (2008), 1220.  doi: 10.1016/j.jmaa.2007.07.079.  Google Scholar

[9]

X. Ding and J. Jiang, Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2819.  doi: 10.1016/j.nonrwa.2008.08.012.  Google Scholar

[10]

X. Ding, B. Su and J. Hao, Positive periodic solutions for impulsive Gause-type predator-prey systems,, \emph{Appl. Math. Comput.}, 218 (2012), 6785.  doi: 10.1016/j.amc.2011.12.046.  Google Scholar

[11]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model,, \emph{Trans. Amer. Math. Soc.}, 349 (1997), 2443.  doi: http://dx.doi.org/10.1090/S0002-9947-97-01842-4.  Google Scholar

[12]

R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III,, \emph{J. Differential Equations}, 249 (2010), 2316.  doi: 10.1016/j.jde.2010.06.021.  Google Scholar

[13]

G. F. Gause, The struggle for existence,, Williams and Wilkins, (1936).   Google Scholar

[14]

G. F. Gause, N. P. Smaragdova and A. A. Witt, Further studies of interaction between predator and prey,, \emph{J. Animal Ecol.}, 5 (1936), 1.   Google Scholar

[15]

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 1571.  doi: 10.1002/cpa.3160471203.  Google Scholar

[16]

K. Hasík, Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline,, \emph{Math. Biosci.}, 164 (2000), 203.  doi: 10.1016/S0025-5564(00)00007-9.  Google Scholar

[17]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates,, \emph{J. Math. Anal. Appl.}, 344 (2008), 217.  doi: 10.1016/j.jmaa.2008.03.006.  Google Scholar

[18]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2558.  doi: 10.1016/j.nonrwa.2008.05.012.  Google Scholar

[19]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[20]

Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems,, \emph{Appl. Anal.}, 29 (1988), 269.  doi: 10.1080/00036818808839785.  Google Scholar

[21]

Y. Kuang, On the location and period of limit cycles in Gause-type predator-prey systems,, \emph{J. Math. Anal. Appl.}, 142 (1989), 130.  doi: 10.1016/0022-247X(89)90170-4.  Google Scholar

[22]

Y. Kuang, Global stability of Gause-type predator-prey systems,, \emph{J. Math. Biol.}, 28 (1990), 463.  doi: 10.1007/BF00178329.  Google Scholar

[23]

L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems,, \emph{Appl. Anal.}, 40 (1991), 281.  doi: 10.1080/00036819108840010.  Google Scholar

[24]

S. Laurin and C. Rousseau, Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III,, \emph{J. Differential Equations}, 251 (2011), 2980.  doi: 10.1016/j.jde.2011.04.017.  Google Scholar

[25]

G. Liu, W. Yan and J. Yan, Positive periodic solutions for a class of neutral delay Gause-type predator-prey system},, \emph{Nonlinear Anal.}, 71 (2009), 4438.  doi: MR2548674.  Google Scholar

[26]

Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems,, \emph{Proc. Amer. Math. Soc.}, 133 (2005), 3619.  doi: http://dx.doi.org/10.1090/S0002-9939-05-08026-3.  Google Scholar

[27]

S. M. Moghadas and M. E. Alexander, Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response,, \emph{Chaos Solitons Fractals}, 23 (2005), 55.  doi: 10.1016/j.chaos.2004.04.030.  Google Scholar

[28]

H. Nie and J. Wu, Multiplicity and stability of a predator-prey model with non-monotonic conversion rate,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 154.  doi: 10.1016/j.nonrwa.2007.08.020.  Google Scholar

[29]

C. V. Pao, Nonlinear parabolic and elliptic equations,, Plenum Press, (1992).   Google Scholar

[30]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, \emph{SIAM J. Appl. Math.}, 61 (): 1445.  doi: 10.1137/S0036139999361896.  Google Scholar

[31]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2$^{nd}$ edition, (1999).  doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2$^{nd}$ edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[33]

G. Sun, Z. Jin, Q. Liu and L. Li, Dynamical complexity of a spatial predator-prey model with migration,, \emph{Ecological modelling}, 219 (2008), 248.  doi: 10.1016/j.ecolmodel.2008.08.009.  Google Scholar

[34]

S. Tang, R. A. Cheke and Y. Xiao, Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations,, \emph{Nonlinear Anal.}, 65 (2006), 2311.  doi: 10.1016/j.na.2006.02.049.  Google Scholar

[35]

S. Tang and J. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge,, \emph{Nonlinear Anal.}, 76 (2013), 165.  doi: 10.1016/j.na.2012.08.013.  Google Scholar

[36]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis,, Princeton University Press, (2003).   Google Scholar

[37]

M. Wang, Nonlinear Elliptic equations,, Science Press, (2010).   Google Scholar

[38]

M. Wang, Nonlinear Parabolic Equation,, Science Press, (1993).   Google Scholar

[39]

W. Wang, Y. Lin, L. Zhang, F. Rao and Y. Tan, Complex patterns in a predator-prey model with self and cross-diffusion,, \emph{Commun. Nonlinear Sci. Numer. Simul.}, 16 (2011), 2006.  doi: 10.1016/j.cnsns.2010.08.035.  Google Scholar

[40]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, \emph{Nonlinear Anal. Ser. A: Theory Methods}, 39 (2000), 817.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[41]

R. Xu and M. A. J. Chaplain, Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks,, \emph{J. Math. Anal. Appl.}, 265 (2002), 148.  doi: 10.1006/jmaa.2001.7701.  Google Scholar

[42]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffsion, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. VI,, Elsevier, (2008), 411.  doi: 10.1016/S1874-5733(08)80023-X.  Google Scholar

[43]

G. Zhang and X. Wang, Positive solutions for a general Gause-type predator-prey model with monotonic functional response,, \emph{Abstr. Appl. Anal.}, (2011).  doi: 10.1155/2011/547060.  Google Scholar

show all references

References:
[1]

R. J. Beverton and S. J. Holt, The theory of fishing, in: M. Graham (Ed.), Sea Fisheries;, Their Investigation in the United Kingdom, (1956), 372.   Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations,, \emph{SIAM J. Math. Anal.}, 17 (1986), 1339.  doi: 10.1137/0517094.  Google Scholar

[3]

J. L. Bravo, M. Fernández, M. Gámez, B. Granados and A. Tineo, Existence of a polycycle in non-Lipschitz Gause-type predator-prey models,, \emph{J. Math. Anal. Appl.}, 373 (2011), 512.  doi: 10.1016/j.jmaa.2010.08.001.  Google Scholar

[4]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion,, \emph{Differential Integral Equations}, 7 (1994), 411.   Google Scholar

[5]

B. Dai, Y. Li and Z. Luo, Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses,, \emph{Appl. Math. Comput.}, 217 (2011), 7478.  doi: 10.1016/j.amc.2011.02.049.  Google Scholar

[6]

M. De la Sen, The generalized Beverton-Holt equation and the control of populations,, \emph{Appl. Math. Model.}, 32 (2008), 2312.  doi: 10.1016/j.apm.2007.09.007.  Google Scholar

[7]

M. De la Sen and S. Alonso-Quesada, Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases,, \emph{Appl. Math. Comput.}, 215 (2009), 2616.  doi: 10.1016/j.amc.2009.09.003.  Google Scholar

[8]

X. Ding and J. Jiang, Positive periodic solutions in delayed Gause-type predator-prey systems,, \emph{J. Math. Anal. Appl.}, 339 (2008), 1220.  doi: 10.1016/j.jmaa.2007.07.079.  Google Scholar

[9]

X. Ding and J. Jiang, Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2819.  doi: 10.1016/j.nonrwa.2008.08.012.  Google Scholar

[10]

X. Ding, B. Su and J. Hao, Positive periodic solutions for impulsive Gause-type predator-prey systems,, \emph{Appl. Math. Comput.}, 218 (2012), 6785.  doi: 10.1016/j.amc.2011.12.046.  Google Scholar

[11]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model,, \emph{Trans. Amer. Math. Soc.}, 349 (1997), 2443.  doi: http://dx.doi.org/10.1090/S0002-9947-97-01842-4.  Google Scholar

[12]

R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III,, \emph{J. Differential Equations}, 249 (2010), 2316.  doi: 10.1016/j.jde.2010.06.021.  Google Scholar

[13]

G. F. Gause, The struggle for existence,, Williams and Wilkins, (1936).   Google Scholar

[14]

G. F. Gause, N. P. Smaragdova and A. A. Witt, Further studies of interaction between predator and prey,, \emph{J. Animal Ecol.}, 5 (1936), 1.   Google Scholar

[15]

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 1571.  doi: 10.1002/cpa.3160471203.  Google Scholar

[16]

K. Hasík, Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline,, \emph{Math. Biosci.}, 164 (2000), 203.  doi: 10.1016/S0025-5564(00)00007-9.  Google Scholar

[17]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates,, \emph{J. Math. Anal. Appl.}, 344 (2008), 217.  doi: 10.1016/j.jmaa.2008.03.006.  Google Scholar

[18]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 2558.  doi: 10.1016/j.nonrwa.2008.05.012.  Google Scholar

[19]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[20]

Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems,, \emph{Appl. Anal.}, 29 (1988), 269.  doi: 10.1080/00036818808839785.  Google Scholar

[21]

Y. Kuang, On the location and period of limit cycles in Gause-type predator-prey systems,, \emph{J. Math. Anal. Appl.}, 142 (1989), 130.  doi: 10.1016/0022-247X(89)90170-4.  Google Scholar

[22]

Y. Kuang, Global stability of Gause-type predator-prey systems,, \emph{J. Math. Biol.}, 28 (1990), 463.  doi: 10.1007/BF00178329.  Google Scholar

[23]

L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems,, \emph{Appl. Anal.}, 40 (1991), 281.  doi: 10.1080/00036819108840010.  Google Scholar

[24]

S. Laurin and C. Rousseau, Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III,, \emph{J. Differential Equations}, 251 (2011), 2980.  doi: 10.1016/j.jde.2011.04.017.  Google Scholar

[25]

G. Liu, W. Yan and J. Yan, Positive periodic solutions for a class of neutral delay Gause-type predator-prey system},, \emph{Nonlinear Anal.}, 71 (2009), 4438.  doi: MR2548674.  Google Scholar

[26]

Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems,, \emph{Proc. Amer. Math. Soc.}, 133 (2005), 3619.  doi: http://dx.doi.org/10.1090/S0002-9939-05-08026-3.  Google Scholar

[27]

S. M. Moghadas and M. E. Alexander, Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response,, \emph{Chaos Solitons Fractals}, 23 (2005), 55.  doi: 10.1016/j.chaos.2004.04.030.  Google Scholar

[28]

H. Nie and J. Wu, Multiplicity and stability of a predator-prey model with non-monotonic conversion rate,, \emph{Nonlinear Anal. Real World Appl.}, 10 (2009), 154.  doi: 10.1016/j.nonrwa.2007.08.020.  Google Scholar

[29]

C. V. Pao, Nonlinear parabolic and elliptic equations,, Plenum Press, (1992).   Google Scholar

[30]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, \emph{SIAM J. Appl. Math.}, 61 (): 1445.  doi: 10.1137/S0036139999361896.  Google Scholar

[31]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2$^{nd}$ edition, (1999).  doi: 10.1007/978-1-4612-1468-7.  Google Scholar

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2$^{nd}$ edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[33]

G. Sun, Z. Jin, Q. Liu and L. Li, Dynamical complexity of a spatial predator-prey model with migration,, \emph{Ecological modelling}, 219 (2008), 248.  doi: 10.1016/j.ecolmodel.2008.08.009.  Google Scholar

[34]

S. Tang, R. A. Cheke and Y. Xiao, Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations,, \emph{Nonlinear Anal.}, 65 (2006), 2311.  doi: 10.1016/j.na.2006.02.049.  Google Scholar

[35]

S. Tang and J. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge,, \emph{Nonlinear Anal.}, 76 (2013), 165.  doi: 10.1016/j.na.2012.08.013.  Google Scholar

[36]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis,, Princeton University Press, (2003).   Google Scholar

[37]

M. Wang, Nonlinear Elliptic equations,, Science Press, (2010).   Google Scholar

[38]

M. Wang, Nonlinear Parabolic Equation,, Science Press, (1993).   Google Scholar

[39]

W. Wang, Y. Lin, L. Zhang, F. Rao and Y. Tan, Complex patterns in a predator-prey model with self and cross-diffusion,, \emph{Commun. Nonlinear Sci. Numer. Simul.}, 16 (2011), 2006.  doi: 10.1016/j.cnsns.2010.08.035.  Google Scholar

[40]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, \emph{Nonlinear Anal. Ser. A: Theory Methods}, 39 (2000), 817.  doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[41]

R. Xu and M. A. J. Chaplain, Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks,, \emph{J. Math. Anal. Appl.}, 265 (2002), 148.  doi: 10.1006/jmaa.2001.7701.  Google Scholar

[42]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffsion, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. VI,, Elsevier, (2008), 411.  doi: 10.1016/S1874-5733(08)80023-X.  Google Scholar

[43]

G. Zhang and X. Wang, Positive solutions for a general Gause-type predator-prey model with monotonic functional response,, \emph{Abstr. Appl. Anal.}, (2011).  doi: 10.1155/2011/547060.  Google Scholar

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