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 and 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, Edward Arnold, London, (1956), 372-441.

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[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, J. Math. Anal. Appl., 373 (2011), 512-520. doi: 10.1016/j.jmaa.2010.08.001.

[4]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439.

[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, Appl. Math. Comput., 217 (2011), 7478-7487. doi: 10.1016/j.amc.2011.02.049.

[6]

M. De la Sen, The generalized Beverton-Holt equation and the control of populations, Appl. Math. Model., 32 (2008), 2312-2328. doi: 10.1016/j.apm.2007.09.007.

[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, Appl. Math. Comput., 215 (2009), 2616-2633. doi: 10.1016/j.amc.2009.09.003.

[8]

X. Ding and J. Jiang, Positive periodic solutions in delayed Gause-type predator-prey systems, J. Math. Anal. Appl., 339 (2008), 1220-1230. doi: 10.1016/j.jmaa.2007.07.079.

[9]

X. Ding and J. Jiang, Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses, Nonlinear Anal. Real World Appl., 10 (2009), 2819-2827. doi: 10.1016/j.nonrwa.2008.08.012.

[10]

X. Ding, B. Su and J. Hao, Positive periodic solutions for impulsive Gause-type predator-prey systems, Appl. Math. Comput., 218 (2012), 6785-6797. doi: 10.1016/j.amc.2011.12.046.

[11]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: http://dx.doi.org/10.1090/S0002-9947-97-01842-4.

[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, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021.

[13]

G. F. Gause, The struggle for existence, Williams and Wilkins, Baltimore, 1936

[14]

G. F. Gause, N. P. Smaragdova and A. A. Witt, Further studies of interaction between predator and prey, J. Animal Ecol., 5 (1936), 1-18.

[15]

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594. doi: 10.1002/cpa.3160471203.

[16]

K. Hasík, Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline, Math. Biosci., 164 (2000), 203-215. doi: 10.1016/S0025-5564(00)00007-9.

[17]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344 (2008), 217-230. doi: 10.1016/j.jmaa.2008.03.006.

[18]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. Real World Appl., 10 (2009), 2558-2573. doi: 10.1016/j.nonrwa.2008.05.012.

[19]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.

[20]

Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems, Appl. Anal., 29 (1988), 269-287. doi: 10.1080/00036818808839785.

[21]

Y. Kuang, On the location and period of limit cycles in Gause-type predator-prey systems, J. Math. Anal. Appl., 142 (1989), 130-143. doi: 10.1016/0022-247X(89)90170-4.

[22]

Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol., 28 (1990), 463-474. doi: 10.1007/BF00178329.

[23]

L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal., 40 (1991), 281-295. doi: 10.1080/00036819108840010.

[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, J. Differential Equations, 251 (2011), 2980-2986. doi: 10.1016/j.jde.2011.04.017.

[25]

G. Liu, W. Yan and J. Yan, Positive periodic solutions for a class of neutral delay Gause-type predator-prey system}, Nonlinear Anal., 71 (2009), 4438-4447. doi: MR2548674.

[26]

Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems, Proc. Amer. Math. Soc., 133 (2005), 3619-3626. doi: http://dx.doi.org/10.1090/S0002-9939-05-08026-3.

[27]

S. M. Moghadas and M. E. Alexander, Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response, Chaos Solitons Fractals, 23 (2005), 55-65. doi: 10.1016/j.chaos.2004.04.030.

[28]

H. Nie and J. Wu, Multiplicity and stability of a predator-prey model with non-monotonic conversion rate, Nonlinear Anal. Real World Appl., 10 (2009), 154-171. doi: 10.1016/j.nonrwa.2007.08.020.

[29]

C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.

[30]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472. doi: 10.1137/S0036139999361896.

[31]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1468-7.

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[33]

G. Sun, Z. Jin, Q. Liu and L. Li, Dynamical complexity of a spatial predator-prey model with migration, Ecological modelling, 219 (2008), 248-255. doi: 10.1016/j.ecolmodel.2008.08.009.

[34]

S. Tang, R. A. Cheke and Y. Xiao, Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations, Nonlinear Anal., 65 (2006), 2311-2341. doi: 10.1016/j.na.2006.02.049.

[35]

S. Tang and J. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonlinear Anal., 76 (2013), 165-180. doi: 10.1016/j.na.2012.08.013.

[36]

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

[37]

M. Wang, Nonlinear Elliptic equations, Science Press, Beijing, 2010 (in Chinese).

[38]

M. Wang, Nonlinear Parabolic Equation, Science Press, Beijing, 1993 (in Chinese).

[39]

W. Wang, Y. Lin, L. Zhang, F. Rao and Y. Tan, Complex patterns in a predator-prey model with self and cross-diffusion, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2006-2015. doi: 10.1016/j.cnsns.2010.08.035.

[40]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal. Ser. A: Theory Methods, 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[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, J. Math. Anal. Appl., 265 (2002), 148-162. doi: 10.1006/jmaa.2001.7701.

[42]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffsion, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. VI, Elsevier, Amsterdam, 2008, 411-501. doi: 10.1016/S1874-5733(08)80023-X.

[43]

G. Zhang and X. Wang, Positive solutions for a general Gause-type predator-prey model with monotonic functional response, Abstr. Appl. Anal., (2011), Art. ID 547060, 16 pp. doi: 10.1155/2011/547060.

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, Edward Arnold, London, (1956), 372-441.

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[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, J. Math. Anal. Appl., 373 (2011), 512-520. doi: 10.1016/j.jmaa.2010.08.001.

[4]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439.

[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, Appl. Math. Comput., 217 (2011), 7478-7487. doi: 10.1016/j.amc.2011.02.049.

[6]

M. De la Sen, The generalized Beverton-Holt equation and the control of populations, Appl. Math. Model., 32 (2008), 2312-2328. doi: 10.1016/j.apm.2007.09.007.

[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, Appl. Math. Comput., 215 (2009), 2616-2633. doi: 10.1016/j.amc.2009.09.003.

[8]

X. Ding and J. Jiang, Positive periodic solutions in delayed Gause-type predator-prey systems, J. Math. Anal. Appl., 339 (2008), 1220-1230. doi: 10.1016/j.jmaa.2007.07.079.

[9]

X. Ding and J. Jiang, Multiple periodic solutions in generalized Gause-type predator-prey systems with non-monotonic numerical responses, Nonlinear Anal. Real World Appl., 10 (2009), 2819-2827. doi: 10.1016/j.nonrwa.2008.08.012.

[10]

X. Ding, B. Su and J. Hao, Positive periodic solutions for impulsive Gause-type predator-prey systems, Appl. Math. Comput., 218 (2012), 6785-6797. doi: 10.1016/j.amc.2011.12.046.

[11]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: http://dx.doi.org/10.1090/S0002-9947-97-01842-4.

[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, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021.

[13]

G. F. Gause, The struggle for existence, Williams and Wilkins, Baltimore, 1936

[14]

G. F. Gause, N. P. Smaragdova and A. A. Witt, Further studies of interaction between predator and prey, J. Animal Ecol., 5 (1936), 1-18.

[15]

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594. doi: 10.1002/cpa.3160471203.

[16]

K. Hasík, Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline, Math. Biosci., 164 (2000), 203-215. doi: 10.1016/S0025-5564(00)00007-9.

[17]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344 (2008), 217-230. doi: 10.1016/j.jmaa.2008.03.006.

[18]

W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. Real World Appl., 10 (2009), 2558-2573. doi: 10.1016/j.nonrwa.2008.05.012.

[19]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.

[20]

Y. Kuang, Nonuniqueness of limit cycles of Gause-type predator-prey systems, Appl. Anal., 29 (1988), 269-287. doi: 10.1080/00036818808839785.

[21]

Y. Kuang, On the location and period of limit cycles in Gause-type predator-prey systems, J. Math. Anal. Appl., 142 (1989), 130-143. doi: 10.1016/0022-247X(89)90170-4.

[22]

Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol., 28 (1990), 463-474. doi: 10.1007/BF00178329.

[23]

L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal., 40 (1991), 281-295. doi: 10.1080/00036819108840010.

[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, J. Differential Equations, 251 (2011), 2980-2986. doi: 10.1016/j.jde.2011.04.017.

[25]

G. Liu, W. Yan and J. Yan, Positive periodic solutions for a class of neutral delay Gause-type predator-prey system}, Nonlinear Anal., 71 (2009), 4438-4447. doi: MR2548674.

[26]

Y. Liu, Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems, Proc. Amer. Math. Soc., 133 (2005), 3619-3626. doi: http://dx.doi.org/10.1090/S0002-9939-05-08026-3.

[27]

S. M. Moghadas and M. E. Alexander, Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response, Chaos Solitons Fractals, 23 (2005), 55-65. doi: 10.1016/j.chaos.2004.04.030.

[28]

H. Nie and J. Wu, Multiplicity and stability of a predator-prey model with non-monotonic conversion rate, Nonlinear Anal. Real World Appl., 10 (2009), 154-171. doi: 10.1016/j.nonrwa.2007.08.020.

[29]

C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.

[30]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472. doi: 10.1137/S0036139999361896.

[31]

H. H. Schaefer and M. P. Wolff, Topological Vector Spaces, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1468-7.

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[33]

G. Sun, Z. Jin, Q. Liu and L. Li, Dynamical complexity of a spatial predator-prey model with migration, Ecological modelling, 219 (2008), 248-255. doi: 10.1016/j.ecolmodel.2008.08.009.

[34]

S. Tang, R. A. Cheke and Y. Xiao, Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations, Nonlinear Anal., 65 (2006), 2311-2341. doi: 10.1016/j.na.2006.02.049.

[35]

S. Tang and J. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonlinear Anal., 76 (2013), 165-180. doi: 10.1016/j.na.2012.08.013.

[36]

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

[37]

M. Wang, Nonlinear Elliptic equations, Science Press, Beijing, 2010 (in Chinese).

[38]

M. Wang, Nonlinear Parabolic Equation, Science Press, Beijing, 1993 (in Chinese).

[39]

W. Wang, Y. Lin, L. Zhang, F. Rao and Y. Tan, Complex patterns in a predator-prey model with self and cross-diffusion, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2006-2015. doi: 10.1016/j.cnsns.2010.08.035.

[40]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal. Ser. A: Theory Methods, 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[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, J. Math. Anal. Appl., 265 (2002), 148-162. doi: 10.1006/jmaa.2001.7701.

[42]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffsion, in: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. VI, Elsevier, Amsterdam, 2008, 411-501. doi: 10.1016/S1874-5733(08)80023-X.

[43]

G. Zhang and X. Wang, Positive solutions for a general Gause-type predator-prey model with monotonic functional response, Abstr. Appl. Anal., (2011), Art. ID 547060, 16 pp. doi: 10.1155/2011/547060.

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