Acta mathematica scientia, Series B >
PERSISTENCE AND THE GLOBAL DYNAMICS OF THE POSITIVE SOLUTIONS FOR A RATIODEPENDENT PREDATOR-PREY SYSTEM WITH A CROWDING TERM IN THE PREY EQUATION
Received date: 2015-04-13
Revised date: 2015-11-20
Online published: 2016-06-25
Supported by
This work was supported by the National Natural Science Foundation of China (11271120, 11426099) and the Project of Hunan Natural Science Foundation of China (13JJ3085).
This paper deals with the global dynamical behaviors of the positive solutions for a parabolic type ratio-dependent predator-prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrated some special dynamical behaviors of the positive solutions of this system which the persistence of the coexistence of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we can obtain that under some conditions, the unique positive steady state solution of the system is globally asymptotically stable.
Xianzhong ZENG , Yonggeng GU . PERSISTENCE AND THE GLOBAL DYNAMICS OF THE POSITIVE SOLUTIONS FOR A RATIODEPENDENT PREDATOR-PREY SYSTEM WITH A CROWDING TERM IN THE PREY EQUATION[J]. Acta mathematica scientia, Series B, 2016 , 36(3) : 689 -703 . DOI: 10.1016/S0252-9602(16)30032-7
[1] Arino O, Mikram J, Chattopadhyay J. Infection on prey population may act as a biological control in ratio-dependent predator-prey model. Nonlinearity, 2004, 17: 1101-1116
[2] Arditi R, Ginzburg L R. Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol, 1989, 139: 311-326
[3] Amann H, López-Gómez J. A priori bounds and multiple solutions for superlinear indefinite elliptic prob-lems. J Differential Equations, 1998, 146: 336-374
[4] Berestycki H, Nirenberg L, Varadhan S R S. The principle eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm Pure Appl Math, 1994, 47(1): 47-92
[5] Brézis H, Oswald L. Remarks on sublinear elliptic equations. Nonlinear Analysis: TMA, 1986, 10: 55-64
[6] Cantrell R S, Cosner C. Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Proc Roy Soc Edinburgh Sect A, 1989, 112(3/4): 293-318
[7] Cirstea F C, Radulescu V. Existence and uniqueness of blow-up solutions for a class of logistic equations. Comm Comtemp Math, 2002, 4: 559-586
[8] Cirstea F C, Radulescu V. Uniqueness of the blow-up boundary solution of logistic equations with absorb-tion. C R Acad Sci Paris Ser I, 2002, 335: 447-452
[9] Cirstea F C, Radulescu V. Asymptotics of the blow-up boundary solution of logistic equations with ab-sorption. C R Acad Sci Paris Ser I, 2003, 336: 231-236
[10] Cano-Casanova S, López-Gómez J. Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. J Differential Equations, 2002, 178(1): 123-211
[11] Du Y. Effects of a degeneracy in the competition model. II. Perturbation and dynamical behaviour. J Differential Equations, 2002, 181(1): 133-164
[12] Du Y. Realization of prescribed patterns in the competition model. J Differential Equations, 2003, 193(1): 147-179
[13] Du Y. Spatial patterns for population models in a heterogeneous environment. Taiwanese J Math, 2004, 8(2): 155-182
[14] Du Y. Bifurcation and Related Topics in Elliptic Problems//Stationary partial differential equations. Vol II. Handb Differ Equ. Amsterdam: Elsevier/North-Holland, 2005: 127-209
[15] Du Y, Shi J. Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans Amer Math Soc, 2007, 359(9): 4557-4593
[16] Du Y, Hsu S B. A diffusive predator-prey model in heterogeneous environment. J Differential Equations, 2004, 203(2): 331-364
[17] Du Y, Shi J. A diffusive predator-prey model with a protection zone. J Differential Equations, 2006, 229(1): 63-91
[18] Du Y, Huang Q. Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J Math Anal, 1999, 31(1): 1-18
[19] Du Y. Asymptotic behavior and uniqueness results for boundary blow-up solutions. Differential and Integral Equations, 2004, 17: 819-834
[20] Freedman H I. Deterministic Mathematical Method in Population Ecology. New York: Dekker, 1980
[21] Fraile J M, Koch P, López-Gómez J, Merino S. Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation. J Differential Equations, 1996, 127: 295-319
[22] Gu Y G, Zeng X Z. Existence of positive stationary solutions for a prey-predator model with the third boundary value to the prey (in Chinese). Acta Mathematica Scientia, 2007, 27A(2): 248-262
[23] García-Melián J, Letelier-Albornoz R, Sabina de Lis J C. Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up. Proc Amer Math Soc, 2001, 129: 3593-3602
[24] García-Melián J, Gómez-Renasco R, López-Gómez J, Sabina de Lis J C. Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs. Arch Ration Mech Anal, 1998, 145(3): 261-289
[25] Mainul H. Ratio-Dependent Predator-Prey Models of Interacting Populations. Bull Math Biology, 2009, 71: 430-452
[26] Hsu S B, Hwang T W, Kuang Y. Rich dynamics of ratio-dependent one prey two predators model. J Math Biol, 2001, 43: 377-396
[27] Hsu S B, Hwang T W, Kuang Y. Global analysis of the Michaelis Menten type ratio-dependent predator-prey system. J Math Biol, 2001, 42: 489-506
[28] López-Gómez J, Sabina de Lis J C. Coexistence states and global attractivity for some convective diffusive competing species models. Trans Amer Math Soc, 1995, 347: 3797-3833
[29] López-Gómez J. Permanence under strong competition. WSSIAA, 1995, 4: 473-488
[30] López-Gómez J. Coexistence and metacoexistence states in competing species models. Houston J Math, 2003, 29(2): 485-538
[31] López-Gómez J, Molina-Meyer M. Superlinear indefinite systems: Beyond Lotka-Volterra models. J Differ-ential Equations, 2006, 221(2): 343-411
[32] López-Gómez J. Spectral Theory and Nonlinear Functional Analysis. Boca Raton: Chapman and Hall/CRC, 2001
[33] López-Gómez J, Sabina de Lis J C. First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs. J Differential Equations, 1998, 148: 47-64
[34] López-Gómez J. Dynamics of parabolic equations. From classical solutions to metasolutions. Differential and Integral Equations, 2003, 16: 813-828
[35] López-Gómez J. The boundary blow-up rate of large solutions. J Differential Equations, 2003, 195(1): 25-45
[36] López-Gómez J. Optimal uniqueness theorems and exact blow-up rates of large solutions. J Differential Equations, 2006, 224(2): 385-439
[37] Oeda K. Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone. J Differential Equations, 2011, 250(10): 3988-4009
[38] Kuto K. Stability of steady-state solutions to a prey-predator system with cross-diffusion. J Differential Equations, 2004, 197: 293-314
[39] Kuto K, Yamada Y. Multiple coexistence states for a prey-predator system with cross-diffusion. J Differ-ential Equations, 2004, 197: 315-348
[40] Ouyang T. On the positive solutions of semilinear equations Δu+λu-hup=0 on the compact manifolds. Trans Amer Math Soc, 1992, 331(2): 503-527
[41] Pang P Y H, Wang M X. Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc Roy Soc Edinburgh Sect A, 2003, 133: 919-942
[42] Peng R, Wang M X. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model. Appl Math Lett, 2007, 20(6): 664-670
[43] Peng R. Qualitative analysis on a diffusive and ratio-dependent predator-prey model. IMA J Appl Math, 2013, 78(3): 566-586
[44] Ye Q X, Li Z Y. An Introduction to Reaction-Diffusion Equations. Beijing: Science press, 1994(in Chinese).
[45] Wang M X. Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent func-tional responses and diffusion. Physica D, 2004, 196: 172-192
[46] Yamada Y. Global solutions for quasilinear parabolic systems with cross-diffusion effects. Nonlinear Anal: TMA, 1995, 24(9): 1395-1412
[47] Wang Y X, Li W T. Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system. Nonlinear Analysis: Real World Applications, 2013, 14(2): 1235-1246
[48] Wang Y X, Li W T. Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone. Nonlinear Analysis: Real World Applications, 2013, 14(1): 224-245
[49] Zeng X Z. Non-constant positive steady states of a prey-predator system with cross-diffusions. J Math Anal Appl, 2007, 332: 989-1009
[50] Zeng X Z. A ratio-dependent predator-prey model with diffusion. Nonlinear Analysis: Real World Appli-cations, 2007, 8(2): 1062-1078
[51] Zeng X Z, Liu Z H. Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion. Nonlinear Analysis: Real World Applications, 2010, 11(1): 372-390
[52] Zeng X Z, Zhang J C, Gu Y G. Uniqueness and stability of positive steady state solutions for a ratio-dependent predator-prey system with a crowding term in the prey equation. Nonlinear Analysis: Real World Applications, 2015, 24: 163-174
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