American Institute of Mathematical Sciences

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April  2021, 26(4): 2085-2113. 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  April 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (4) : 2085-2113. 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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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. 

[9]

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

[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.

[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. 

[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.

[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.

[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.

[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. 

[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. 

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[32]

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

[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.
[34]

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

[35]

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

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

[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. 

[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.

[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. 

[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.

[42]

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

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

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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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. 

[9]

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

[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.

[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. 

[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.

[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.

[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.

[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. 

[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. 

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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. 

[32]

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

[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.
[34]

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

[35]

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

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

[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. 

[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.

[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. 

[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.

[42]

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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