doi: 10.3934/dcdsb.2022015
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On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation

Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 28503, Republic of Korea

* Corresponding author: kowl@cju.ac.kr (Wonlyul Ko)

Received  August 2021 Revised  October 2021 Early access February 2022

The current paper provides a qualitative study of a diffusive predator-prey system with a hunting-cooperation functional response under homogeneous Neumann boundary conditions. The considered functional response explains a behavioral mechanism in which predators search for and hunt their prey cooperatively. We investigate the global attractor for nonnegative time-dependent solutions to the system and the nonpersistence leading to the extinction of the predator species in the system. Moreover, we study the local stability at all feasible nonnegative constant steady states and the occurrence of Hopf bifurcation (i.e., the existence of time-periodic orbits) in the system. Finally, we study the existence and nonexistence of nonconstant positive steady states in the system, and the limiting behavior of the positive steady states according to the diffusion rate. Through this study, we observe interesting results, such as bistability, extinction of the predator species induced by a large consumption rate, and stationary patterns in the system with hunting cooperation in predators.

Citation: Kimun Ryu, Wonlyul Ko. On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022015
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

R. Arditi and L. R. Ginzburg, Couplng in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. 

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[4]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

[5]

R. S. CantrellC. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 533-559.  doi: 10.1017/S0308210500025877.

[6]

W. Chen and M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44.  doi: 10.1016/j.mcm.2005.05.013.

[7]

K.-S. ChengS.-B. Hsu and S.-S. Lin, Some results on global stability of a predator-prey system, J. Math. Biology, 12 (1981), 115-126.  doi: 10.1007/BF00275207.

[8]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Population Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.

[9]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. 

[10]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895.

[11]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.

[12]

H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508.  doi: 10.1016/S0092-8240(86)90004-2.

[13]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[14]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011.

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. 
[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations: Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York, 1981.

[19]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 46 (1995), 1-60.  doi: 10.4039/entm9745fv.

[20]

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.

[21]

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

[22]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[23]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[24]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1973.

[25]

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

[26]

P. Y. H. Pang and M. Wang, Nonconstant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion, Proc. London math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[28]

L. Přibylová and L. Berec, Predator interference and stability of predator-prey dynamics, J. Math. Biol., 71 (2015), 301-323.  doi: 10.1007/s00285-014-0820-9.

[29]

S. Ruan and D. Xiao, Gobal analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.

[30]

K. Ryu and W. Ko, Asymptotic behavior of positive solutions to a predator-prey elliptic system with strong hunting cooperation in predators, Phys. A, 531 (2019), 121726, 8 pp. doi: 10.1016/j.physa.2019.121726.

[31]

K. RyuW. Ko and M. Haque, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynam., 94 (2018), 1639-1656. 

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

[33]

D. Song, C. Li and Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal. Real World Appl., 54 (2020), 103106, 24 pp. doi: 10.1016/j.nonrwa.2020.103106.

[34]

M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13-22.  doi: 10.1016/j.jtbi.2017.02.002.

[35]

H.-Y. Wang, S. Guo and S. Z. Li, Stationary solutions of advective Lotka-Volterra models with a weak Allee effect and large diffusion, Nonlinear Anal. Real World Appl., 56 (2020), 103171, 23 pp. doi: 10.1016/j.nonrwa.2020.103171.

[36]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[37]

D. Wu and M. Zhao, Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Phys. A, 515 (2019), 299-309.  doi: 10.1016/j.physa.2018.09.176.

[38]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[2]

R. Arditi and L. R. Ginzburg, Couplng in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. 

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.

[4]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

[5]

R. S. CantrellC. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 533-559.  doi: 10.1017/S0308210500025877.

[6]

W. Chen and M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44.  doi: 10.1016/j.mcm.2005.05.013.

[7]

K.-S. ChengS.-B. Hsu and S.-S. Lin, Some results on global stability of a predator-prey system, J. Math. Biology, 12 (1981), 115-126.  doi: 10.1007/BF00275207.

[8]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Population Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.

[9]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. 

[10]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895.

[11]

Y. DuR. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.

[12]

H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508.  doi: 10.1016/S0092-8240(86)90004-2.

[13]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[14]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011.

[17] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. 
[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations: Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York, 1981.

[19]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 46 (1995), 1-60.  doi: 10.4039/entm9745fv.

[20]

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.

[21]

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

[22]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[23]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[24]

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Science, New York, 1973.

[25]

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

[26]

P. Y. H. Pang and M. Wang, Nonconstant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion, Proc. London math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[28]

L. Přibylová and L. Berec, Predator interference and stability of predator-prey dynamics, J. Math. Biol., 71 (2015), 301-323.  doi: 10.1007/s00285-014-0820-9.

[29]

S. Ruan and D. Xiao, Gobal analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.  doi: 10.1137/S0036139999361896.

[30]

K. Ryu and W. Ko, Asymptotic behavior of positive solutions to a predator-prey elliptic system with strong hunting cooperation in predators, Phys. A, 531 (2019), 121726, 8 pp. doi: 10.1016/j.physa.2019.121726.

[31]

K. RyuW. Ko and M. Haque, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynam., 94 (2018), 1639-1656. 

[32]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

[33]

D. Song, C. Li and Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal. Real World Appl., 54 (2020), 103106, 24 pp. doi: 10.1016/j.nonrwa.2020.103106.

[34]

M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13-22.  doi: 10.1016/j.jtbi.2017.02.002.

[35]

H.-Y. Wang, S. Guo and S. Z. Li, Stationary solutions of advective Lotka-Volterra models with a weak Allee effect and large diffusion, Nonlinear Anal. Real World Appl., 56 (2020), 103171, 23 pp. doi: 10.1016/j.nonrwa.2020.103171.

[36]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[37]

D. Wu and M. Zhao, Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Phys. A, 515 (2019), 299-309.  doi: 10.1016/j.physa.2018.09.176.

[38]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

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