April  2022, 27(4): 2189-2219. doi: 10.3934/dcdsb.2021129

A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

2. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author: Hongmei Cheng

Received  January 2021 Published  April 2022 Early access  April 2021

Fund Project: The second author is supported by NSFC grant 11701341. The third author is supported by NSFC grant 11771044

This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region $ [0,h_0] $. We first show that the problem has a unique solution defined for all $ t>0 $. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria.

Citation: Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 466 (1975), 5-49. 

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[4]

P. W. Bates and G. Zhou, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[5]

H. BerestyckiJ. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.

[6]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media., 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[7]

J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.

[8]

J.-F. CaoW.-T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[9]

H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579.  doi: 10.1515/fca-2015-0035.

[10]

H. Cheng and R. Yuan, Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response, Taiwanese J. Math., 20 (2016), 801-822.  doi: 10.11650/tjm.20.2016.6284.

[11]

H. Cheng and R. Yuan, Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3007-3022.  doi: 10.3934/dcdsb.2017160.

[12]

H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 5433-5454.  doi: 10.3934/dcds.2017236.

[13]

H. Cheng and R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion, Appl. Math. Comput., 338 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.049.

[14]

H. Cheng and S. Niu, A free boundary problem of some Lesile-Gower predator-prey model with higher dimensional environment, submitted, (2019).

[15]

W. DingR. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.

[16]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[17]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.

[18]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[19]

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

[20]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.

[21]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[22]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[23]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.

[24]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[25]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.

[26]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[27]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492. 

[28]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[29]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.

[30]

Y. LiuZ GuoM. El Smaily and L. Wang, A Leslie-Gower predator-prey model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2063-2084.  doi: 10.3934/dcdss.2019133.

[31]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.

[32]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.

[33]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17(1987), 241–280. doi: 10.32917/hmj/1206130066.

[34]

S. Niu and H. Cheng, A free boundary problem for a Leslie-Gower predator-prey model in higher dimensions and heterogeneous environment, American Journal of Applied Mathematics, 8 (2020), 284–292.

[35]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wave fronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[36]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.

[37]

N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8.

[38]

N. Sun, A time-periodic reaction-diffusion-advection equation with a free boundary and sign-changing coefficients, Nonlinear Anal. Real World Appl., 51 (2020), 102952, 28 pp. doi: 10.1016/j.nonrwa.2019.06.002.

[39]

N. Sun and X. Han, Asymptotic behavior of solutions of a reaction-diffusion model with a protection zone and a free boundary, Appl. Math. Lett., 107 (2020), 106470, 7 pp. doi: 10.1016/j.aml.2020.106470.

[40]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch. Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988.

[41]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.

[42]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[43]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[44]

B. Yan and C. An, The sign-changing solutions for a class of nonlocal elliptic problem in an annulus, Topol. Methods Nonlinear Anal, 55 (2020), 1-18.  doi: 10.12775/tmna.2019.081.

[45]

R. Yang and J. Wei, The effect of delay on a diffusive predator-prey system with modified Leslie-Gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7.

[46]

Y. Zhang and M. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

[47]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[48]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[49]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.

[50]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 466 (1975), 5-49. 

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[4]

P. W. Bates and G. Zhou, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.

[5]

H. BerestyckiJ. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.

[6]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media., 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[7]

J.-F. CaoY. DuF. Li and W.-T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.

[8]

J.-F. CaoW.-T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[9]

H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579.  doi: 10.1515/fca-2015-0035.

[10]

H. Cheng and R. Yuan, Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response, Taiwanese J. Math., 20 (2016), 801-822.  doi: 10.11650/tjm.20.2016.6284.

[11]

H. Cheng and R. Yuan, Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3007-3022.  doi: 10.3934/dcdsb.2017160.

[12]

H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 5433-5454.  doi: 10.3934/dcds.2017236.

[13]

H. Cheng and R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion, Appl. Math. Comput., 338 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.049.

[14]

H. Cheng and S. Niu, A free boundary problem of some Lesile-Gower predator-prey model with higher dimensional environment, submitted, (2019).

[15]

W. DingR. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.

[16]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[17]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.

[18]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[19]

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

[20]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.

[21]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[22]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[23]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.

[24]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[25]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.

[26]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[27]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492. 

[28]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[29]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.

[30]

Y. LiuZ GuoM. El Smaily and L. Wang, A Leslie-Gower predator-prey model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2063-2084.  doi: 10.3934/dcdss.2019133.

[31]

M. MimuraY. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186.  doi: 10.1007/BF03167042.

[32]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.

[33]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17(1987), 241–280. doi: 10.32917/hmj/1206130066.

[34]

S. Niu and H. Cheng, A free boundary problem for a Leslie-Gower predator-prey model in higher dimensions and heterogeneous environment, American Journal of Applied Mathematics, 8 (2020), 284–292.

[35]

S. PanW.-T. Li and G. Lin, Existence and stability of traveling wave fronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.  doi: 10.1016/j.na.2009.12.008.

[36]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.

[37]

N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8.

[38]

N. Sun, A time-periodic reaction-diffusion-advection equation with a free boundary and sign-changing coefficients, Nonlinear Anal. Real World Appl., 51 (2020), 102952, 28 pp. doi: 10.1016/j.nonrwa.2019.06.002.

[39]

N. Sun and X. Han, Asymptotic behavior of solutions of a reaction-diffusion model with a protection zone and a free boundary, Appl. Math. Lett., 107 (2020), 106470, 7 pp. doi: 10.1016/j.aml.2020.106470.

[40]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch. Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988.

[41]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.

[42]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[43]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.

[44]

B. Yan and C. An, The sign-changing solutions for a class of nonlocal elliptic problem in an annulus, Topol. Methods Nonlinear Anal, 55 (2020), 1-18.  doi: 10.12775/tmna.2019.081.

[45]

R. Yang and J. Wei, The effect of delay on a diffusive predator-prey system with modified Leslie-Gower functional response, Bull. Malays. Math. Sci. Soc., 40 (2017), 51-73.  doi: 10.1007/s40840-015-0261-7.

[46]

Y. Zhang and M. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.

[47]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[48]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[49]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.

[50]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.

[1]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240

[2]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121

[3]

Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133

[4]

Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245

[5]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128

[6]

Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340

[7]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[8]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189

[9]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations and Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

[10]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501

[11]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317

[12]

Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693

[13]

Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069

[14]

Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146

[15]

Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133

[16]

Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236

[17]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[18]

Yanfei Du, Ben Niu, Junjie Wei. A predator-prey model with cooperative hunting in the predator and group defense in the prey. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021298

[19]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[20]

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

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (472)
  • HTML views (387)
  • Cited by (0)

Other articles
by authors

[Back to Top]