American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020081

A spatially heterogeneous predator-prey model

Julián López-Gómez , and  Eduardo Muñoz-Hernández

Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, 28040, Spain

* Corresponding author

This paper is dedicated to Sze-Bi Hsu, with admiration for his pioneering mathematical work and our deepest gratitude for his friendship. At the occasion of his 70th anniversary

Received  April 2019 Published  January 2020

Fund Project: This paper has been supported by the IMI of the Complutense University of Madrid and the Ministry of Science and Universities of Spain under Research Grant PGC2018-097104-B-100. The second author, ORCID: 0000-0003-1184-6231, has been also supported by contract CT42/18-CT43/18 of Complutense University of Madrid.

This paper introduces a spatially heterogeneous diffusive predator–prey model unifying the classical Lotka–Volterra and Holling–Tanner ones through a prey saturation coefficient, $ m(x) $, which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka–Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.

Keywords: Predator-prey systems, Lotka–Volterra and Holling–Tanner kinetics, prey saturation effects, unifying Lotka–Volterra and Holling–Tanner kinetics, coexistence states, uniqueness of coexistence states.
Mathematics Subject Classification: Primary: 35J57, 35Q92, 35A16.
Citation: Julián López-Gómez, Eduardo Muñoz-Hernández. A spatially heterogeneous predator-prey model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020081
References:
[1]

D. Aleja, I. Antón and J. López-Gómez, Global structure of the periodic-positive solutions for a general class of periodic-parabolic logistic equations with indefinite weights, Submitted. Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Maths., 47 (1994), 72-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.  doi: 10.1006/jdeq.2000.4003.  Google Scholar

[5]

A. Casal, C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.   Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[8]

E. N. Dancer, J. López-Gómez and R. Ortega, On the spectrum of some linear noncooperative elliptic systems with radial symmetry, Diff. Int. Eqns., 8 (1995), 515-523.   Google Scholar

[9]

D. Daners and J. López-Gómez, Global dynamics of generalized logistic equations, Adv. Nonl. Studies, 18 (2018), 217-236.   Google Scholar

[10]

Y. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[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.   Google Scholar

[12]

Y. Du and Y. Lou, $S$-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Diff. Eqns., 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.  Google Scholar

[13]

Y. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Eqns., 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[14]

Y. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[15]

X. Feng, Y. Song and X. An, Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod–Haldane functional response, Open Math., 16 (2018), 623-635.   Google Scholar

[16]

S. Fernández-Rincón and J. López-Gómez, The singular perturbation problem for a class of generalized logistic equations under non-classical mixed boundary conditions, Adv. Nonl. Studies, 19 (2019), 1-27.   Google Scholar

[17]

J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1995), 295-319.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[18]

H. Freedman, Deterministic Mathematical Models in Population Biology, Marcel and Dekker, New York, 1980.  Google Scholar

[19]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonl. Anal., 72 (2010), 1632-1646.  doi: 10.1016/j.na.2009.09.003.  Google Scholar

[20]

G. Guo and J. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194.  doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar

[21]

S. B. Hsu, On global stability of a predator-prey system, Math. Biosc., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

[22]

S. B. Hsu and T. W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[23]

Y. Jia, J. Wu and H. K. Xu, Spatial pattern in an ecosystem of phytoplankton-nutrient from remote sensing, J. Math. Anal. Appl., 402 (2013), 23-34.  doi: 10.1016/j.jmaa.2012.12.071.  Google Scholar

[24]

H. Jiang and L. Wang, Analysis of steady-state for variable territory model with limited self-limitation, Acta Appl. Math., 148 (2017), 103-120.  doi: 10.1007/s10440-016-0080-3.  Google Scholar

[25]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Diff. Eqns., 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001.  Google Scholar

[26]

W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonl. Anal. RWA, 8 (2007), 769-786.  doi: 10.1016/j.nonrwa.2006.03.003.  Google Scholar

[27]

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

[28]

S. Li, J. Wu and Y. Dong, Effects of degeneracy and response in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483.  doi: 10.1088/1361-6544/aaa2de.  Google Scholar

[29]

S. Li, J. Wu and Y. Dong, Effects of degeneracy in a diffusion predator-prey model with Holling type-Ⅱ functional response, Nonl. Anal. RWA, 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.  Google Scholar

[30]

J. López-Gómez, Positive periodic solutions of Lotka–Volterra reaction-diffusion systems, Diff. Int. Eqns., 5 (1992), 55-72.   Google Scholar

[31]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation. Application to the search of positive solutions for general Lotka-Volterra Reaction-Diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.   Google Scholar

[32]

J. López-Gómez, A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.   Google Scholar

[33] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton, FL, 2001.  doi: 10.1201/9781420035506.  Google Scholar
[34]

J. López-Gómez, Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.   Google Scholar

[35]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific, Singapore, 2013. doi: 10.1142/8664.  Google Scholar

[36] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016.   Google Scholar
[37]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly elliptic systems and some applications, Diff. Int. Eqns, 7 (1994), 383-398.   Google Scholar

[38]

J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71.   Google Scholar

[39]

J. López-Gómez and R. M. Pardo, Coexistence regions in Lotka-Volterra systems with diffusion, Nonl. Anal., 19 (1992), 11-28.  doi: 10.1016/0362-546X(92)90027-C.  Google Scholar

[40]

J. López-Gómez and R. M. Pardo, The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031.   Google Scholar

[41]

J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonl. Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.  Google Scholar

[42]

J. López-Gómez, A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.   Google Scholar

[43] R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974.   Google Scholar
[44]

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

[45]

P. Y. H. Pang and M. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Royal Soc. Edinburgh, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[46]

P. Y. H. Pang and M. Wang, Non-constant positive steady-states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.  Google Scholar

[47]

R. Peng, M. X. Wang and W. Y. Chen, Positive steady-states of a predator-prey model with diffusion and non-monotonic conversion rate, Acta Math. Sinica, 23 (2007), 749-760.  doi: 10.1007/s10114-005-0789-9.  Google Scholar

[48]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Diff. Eqns., 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.  Google Scholar

[49]

M. X. Wang and Q. Wu, Positive solutions fo a predator-prey model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.  doi: 10.1016/j.jmaa.2008.04.054.  Google Scholar

[50]

H. Yuan, J. Wu, Y. Jia and H. Nie, Coexistence states of a predator-prey model with cross-diffusion, Nonl. Anal. RWA, 41 (2018), 179-203.  doi: 10.1016/j.nonrwa.2017.10.009.  Google Scholar

[51]

X. Zeng, W. Zeng and L. Liu, Effects of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626.  doi: 10.1016/j.jmaa.2018.02.060.  Google Scholar

[52]

J. Zhou and C. Mu, Coexistence states of a Holling type-Ⅱ predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.  doi: 10.1016/j.jmaa.2010.04.001.  Google Scholar

[53]

J. Zhou and C. Mu, Coexistence of a diffusive predator-prey model with Holling-type Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.  doi: 10.1016/j.jmaa.2011.07.027.  Google Scholar

[54]

J. Zhou and J. P. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator-prey model with Holling type-Ⅱ functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.  doi: 10.1016/j.jmaa.2013.03.064.  Google Scholar

show all references

References:
[1]

D. Aleja, I. Antón and J. López-Gómez, Global structure of the periodic-positive solutions for a general class of periodic-parabolic logistic equations with indefinite weights, Submitted. Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Maths., 47 (1994), 72-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.  doi: 10.1006/jdeq.2000.4003.  Google Scholar

[5]

A. Casal, C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.   Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[8]

E. N. Dancer, J. López-Gómez and R. Ortega, On the spectrum of some linear noncooperative elliptic systems with radial symmetry, Diff. Int. Eqns., 8 (1995), 515-523.   Google Scholar

[9]

D. Daners and J. López-Gómez, Global dynamics of generalized logistic equations, Adv. Nonl. Studies, 18 (2018), 217-236.   Google Scholar

[10]

Y. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[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.   Google Scholar

[12]

Y. Du and Y. Lou, $S$-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Diff. Eqns., 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.  Google Scholar

[13]

Y. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Eqns., 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[14]

Y. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[15]

X. Feng, Y. Song and X. An, Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod–Haldane functional response, Open Math., 16 (2018), 623-635.   Google Scholar

[16]

S. Fernández-Rincón and J. López-Gómez, The singular perturbation problem for a class of generalized logistic equations under non-classical mixed boundary conditions, Adv. Nonl. Studies, 19 (2019), 1-27.   Google Scholar

[17]

J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1995), 295-319.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[18]

H. Freedman, Deterministic Mathematical Models in Population Biology, Marcel and Dekker, New York, 1980.  Google Scholar

[19]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonl. Anal., 72 (2010), 1632-1646.  doi: 10.1016/j.na.2009.09.003.  Google Scholar

[20]

G. Guo and J. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194.  doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar

[21]

S. B. Hsu, On global stability of a predator-prey system, Math. Biosc., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

[22]

S. B. Hsu and T. W. Hwang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[23]

Y. Jia, J. Wu and H. K. Xu, Spatial pattern in an ecosystem of phytoplankton-nutrient from remote sensing, J. Math. Anal. Appl., 402 (2013), 23-34.  doi: 10.1016/j.jmaa.2012.12.071.  Google Scholar

[24]

H. Jiang and L. Wang, Analysis of steady-state for variable territory model with limited self-limitation, Acta Appl. Math., 148 (2017), 103-120.  doi: 10.1007/s10440-016-0080-3.  Google Scholar

[25]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Diff. Eqns., 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001.  Google Scholar

[26]

W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonl. Anal. RWA, 8 (2007), 769-786.  doi: 10.1016/j.nonrwa.2006.03.003.  Google Scholar

[27]

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

[28]

S. Li, J. Wu and Y. Dong, Effects of degeneracy and response in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483.  doi: 10.1088/1361-6544/aaa2de.  Google Scholar

[29]

S. Li, J. Wu and Y. Dong, Effects of degeneracy in a diffusion predator-prey model with Holling type-Ⅱ functional response, Nonl. Anal. RWA, 43 (2018), 78-95.  doi: 10.1016/j.nonrwa.2018.02.003.  Google Scholar

[30]

J. López-Gómez, Positive periodic solutions of Lotka–Volterra reaction-diffusion systems, Diff. Int. Eqns., 5 (1992), 55-72.   Google Scholar

[31]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation. Application to the search of positive solutions for general Lotka-Volterra Reaction-Diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.   Google Scholar

[32]

J. López-Gómez, A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.   Google Scholar

[33] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton, FL, 2001.  doi: 10.1201/9781420035506.  Google Scholar
[34]

J. López-Gómez, Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.   Google Scholar

[35]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific, Singapore, 2013. doi: 10.1142/8664.  Google Scholar

[36] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016.   Google Scholar
[37]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly elliptic systems and some applications, Diff. Int. Eqns, 7 (1994), 383-398.   Google Scholar

[38]

J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71.   Google Scholar

[39]

J. López-Gómez and R. M. Pardo, Coexistence regions in Lotka-Volterra systems with diffusion, Nonl. Anal., 19 (1992), 11-28.  doi: 10.1016/0362-546X(92)90027-C.  Google Scholar

[40]

J. López-Gómez and R. M. Pardo, The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031.   Google Scholar

[41]

J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonl. Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.  Google Scholar

[42]

J. López-Gómez, A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.   Google Scholar

[43] R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974.   Google Scholar
[44]

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

[45]

P. Y. H. Pang and M. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Royal Soc. Edinburgh, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.  Google Scholar

[46]

P. Y. H. Pang and M. Wang, Non-constant positive steady-states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.  Google Scholar

[47]

R. Peng, M. X. Wang and W. Y. Chen, Positive steady-states of a predator-prey model with diffusion and non-monotonic conversion rate, Acta Math. Sinica, 23 (2007), 749-760.  doi: 10.1007/s10114-005-0789-9.  Google Scholar

[48]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Diff. Eqns., 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.  Google Scholar

[49]

M. X. Wang and Q. Wu, Positive solutions fo a predator-prey model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.  doi: 10.1016/j.jmaa.2008.04.054.  Google Scholar

[50]

H. Yuan, J. Wu, Y. Jia and H. Nie, Coexistence states of a predator-prey model with cross-diffusion, Nonl. Anal. RWA, 41 (2018), 179-203.  doi: 10.1016/j.nonrwa.2017.10.009.  Google Scholar

[51]

X. Zeng, W. Zeng and L. Liu, Effects of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626.  doi: 10.1016/j.jmaa.2018.02.060.  Google Scholar

[52]

J. Zhou and C. Mu, Coexistence states of a Holling type-Ⅱ predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.  doi: 10.1016/j.jmaa.2010.04.001.  Google Scholar

[53]

J. Zhou and C. Mu, Coexistence of a diffusive predator-prey model with Holling-type Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.  doi: 10.1016/j.jmaa.2011.07.027.  Google Scholar

[54]

J. Zhou and J. P. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator-prey model with Holling type-Ⅱ functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.  doi: 10.1016/j.jmaa.2013.03.064.  Google Scholar

Figure 1.  Stability of the semitrivial solutions
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Figure 2.  Stability of $ (0, 0) $
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Figure 3.  An admissible component $ \mathfrak{C} $
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Figure 4.  By Theorem 5.1, (5) admits a coexistence state for each $ (\lambda, \mu) $ in the green region, while it cannot admit a coexistence state in the white one. Within the yellow region, (5) might admit, or not, a coexistence state as it will become apparent later
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Figure 5.  The coexistence wedges of Figure 4 when $ \mathrm{Int\, }m^{-1}(0)\neq \emptyset $
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