# American Institute of Mathematical Sciences

## Positive steady states of a density-dependent predator-prey model with diffusion

 1 School of Mathematics and Statistics, Northwest Normal University Lanzhou, 730070, China School of Mathematical Science, Huaiyin Normal University Huaian, 223300, China 2 School of Mathematical Science, Huaiyin Normal University Huaian, 223300, China 3 School of Physical and Mathematical Sciences, Nanjing Tech University Nanjing, Jiangsu 211816, China 4 School of Mathematics and Statistics, Northwest Normal University Lanzhou, 730070, China

* Corresponding author

* Corresponding author

Received  March 2017 Revised  July 2017 Published  September 2017

Fund Project: This research was supported by the National Science Foundation of China (Grant number 61672013,11601179,11601226,11361055 and 61373005) and the Natural Science Foundation of Jiangsu Province of China (BK20140927)

In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.

Citation: Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017209
##### References:
 [1] P. A. Abrams, The fallacies of "ratio-dependent" predation, Ecology, 75 (1994), 1842-1850. [2] P. A. Abrams and L. R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology and Evolution 15 (2000), p337. [3] A. Ardito and P. Ricciardi, Lyapunov functions for a generalized gause-type model, Journal of Mathematical Biology, 33 (1995), 816-828. doi: 10.1007/BF00187283. [4] N. F. Britton, Essential Mathematical Biology Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd. , London, 2003. [5] Y. L. Cai, W. M. Wang and J. F. Wang, Dynamics of a diffusive predator-prey model with additive allee effect, International Journal of Biomathematics, 39 (2012), 105-115. [6] Y. L. Cai, M. Banerjee, Y. Kang and W. M. Wang, Spatiotemporal complexity in a predator--prey model with weak allee effects, Mathematical Biosciences and Engineering, 11 (2014), 1247-1274. doi: 10.3934/mbe.2014.11.1247. [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. [8] Y. H. Du, P. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620. doi: 10.1137/070684173. [9] A. Eigenwillig, Real root isolation for exact and approximate polynomials using descartes' rule of signs, 2008. [10] S. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, Natural Science Journal of Hainan Teacheres College, 2 (1989), 91-98. [11] S. M. Fu, Z. J. Wen and S. B. Cui, Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model, Nonlinear Analysis Real World Applications, 9 (2008), 272-289. doi: 10.1016/j.nonrwa.2006.10.003. [12] K. Fujii, Complexity-stability relationship of two-prey-one-predator species system model: Local and global stability, Journal of Theoretical Biology, 69 (1977), 613-623. doi: 10.1016/0022-5193(77)90370-8. [13] K. Hasík, On a predator-prey system of gause type, Journal of Mathematical Biology, 60 (2010), 59-74. doi: 10.1007/s00285-009-0257-8. [14] J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of turing patterns in the 1-d lengyel-epstein model, Journal of Dynamics and Differential Equations, 16 (2004), 297-320. doi: 10.1007/s10884-004-2782-x. [15] Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, Journal of Mathematical Biology, 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z. [16] W. Ko and K. Ryu, Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment, Journal of Mathematical Analysis and Applications, 327 (2007), 539-549. doi: 10.1016/j.jmaa.2006.04.077. [17] Y. Kuang, Rich dynamics of gause-type ratio-dependent predator-prey system, Fields Institute Communications, 21 (1999), 325-337. [18] Y. Kuang, Global stability of gause-type predator-prey systems, Journal of Mathematical Biology, 28 (1990), 463-474. doi: 10.1007/BF00178329. [19] S. B. Li, J. H. Wu and Y. Y. Dong, Turing patterns in a reaction-diffusion model with the degn-harrison reaction scheme, Journal of Differential Equations, 259 (2015), 1990-2029. doi: 10.1016/j.jde.2015.03.017. [20] X. Li, W. H. Jiang and J. P. Shi, Hopf bifurcation and turing instability in the reaction-diffusion holling-tanner predator-prey model, Journal of Applied Mathematics, 78 (2013), 287-306. doi: 10.1093/imamat/hxr050. [21] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [22] Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, Journal of Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559. [23] W. M. Ni and M. Tang, Turing patterns in the lengyel-epstein system for the cima reaction, Transactions of the American Mathematical Society, 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9. [24] W. M. Ni, Cross-diffusion and their spike-layer steady states, Notices of the American Mathematical Society, 45 (1998), 9-18. [25] L. Nirenberg, Topics in Nonlinear Functional Analysis Courant Institute of Mathematical Sciences, 1974. [26] P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, Journal of Differential Equations, 200 (2004), 245-273. doi: 10.1016/j.jde.2004.01.004. [27] R. Peng and M. X. Wang, Note on a ratio-dependent predator-prey system with diffusion, Nonlinear Analysis Real World Applications, 7 (2006), 1-11. doi: 10.1016/j.nonrwa.2004.11.008. [28] J. F. Savino and R. A. Stein, Predator-prey interaction between largemouth bass and bluegills as influenced by simulated, submersed vegetation, Transactions of the American Fisheries Society, 111 (1982), 255-266. [29] J. P. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531. [30] H. B. Shi and S. G. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, Journal of Applied Mathematics, 80 (2015), 1534-1568. doi: 10.1093/imamat/hxv006. [31] H. B. Shi, W. T. Li and G. Lin, Positive steady states of a diffusive predator-prey system with modified holling-tanner functional response, Nonlinear Analysis Real World Applications, 11 (2010), 3711-3721. doi: 10.1016/j.nonrwa.2010.02.001. [32] A. Sikder and A. B. Roy, Persistence of a generalized gause-type two prey-two predator pair linked by competition, Mathematical Biosciences, 122 (1994), 1-23. doi: 10.1016/0025-5564(94)90080-9. [33] I. Takagi, Point-condensation for a reaction-diffusion system, Journal of Differential Equations, 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1. [34] M. X. Wang, Stationary patterns of strongly coupled prey-predator models, Journal of Mathematical Analysis and Applications, 292 (2004), 484-505. doi: 10.1016/j.jmaa.2003.12.027. [35] M. X. Wang, Non-constant positive steady states of the sel'kov model, Journal of Differential Equations, 190 (2003), 600-620. doi: 10.1016/S0022-0396(02)00100-6. [36] Z. J. Wen and S. M. Fu, Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics, Journal of Computational and Applied Mathematics, 230 (2009), 34-43. doi: 10.1016/j.cam.2008.10.064. [37] X. Z. Zeng and Z. H. Liu, Non-constant positive steady states of a prey-predator system with cross-diffusions, Journal of Mathematical Analysis and Applications, 332 (2007), 989-1009. doi: 10.1016/j.jmaa.2006.10.075. [38] X. Z. Zeng and Z. H. Liu, Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion, Journal of Mathematical Analysis and Applications, 11 (2010), 372-390. doi: 10.1016/j.nonrwa.2008.11.010.

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##### References:
 [1] P. A. Abrams, The fallacies of "ratio-dependent" predation, Ecology, 75 (1994), 1842-1850. [2] P. A. Abrams and L. R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology and Evolution 15 (2000), p337. [3] A. Ardito and P. Ricciardi, Lyapunov functions for a generalized gause-type model, Journal of Mathematical Biology, 33 (1995), 816-828. doi: 10.1007/BF00187283. [4] N. F. Britton, Essential Mathematical Biology Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd. , London, 2003. [5] Y. L. Cai, W. M. Wang and J. F. Wang, Dynamics of a diffusive predator-prey model with additive allee effect, International Journal of Biomathematics, 39 (2012), 105-115. [6] Y. L. Cai, M. Banerjee, Y. Kang and W. M. Wang, Spatiotemporal complexity in a predator--prey model with weak allee effects, Mathematical Biosciences and Engineering, 11 (2014), 1247-1274. doi: 10.3934/mbe.2014.11.1247. [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. [8] Y. H. Du, P. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620. doi: 10.1137/070684173. [9] A. Eigenwillig, Real root isolation for exact and approximate polynomials using descartes' rule of signs, 2008. [10] S. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, Natural Science Journal of Hainan Teacheres College, 2 (1989), 91-98. [11] S. M. Fu, Z. J. Wen and S. B. Cui, Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model, Nonlinear Analysis Real World Applications, 9 (2008), 272-289. doi: 10.1016/j.nonrwa.2006.10.003. [12] K. Fujii, Complexity-stability relationship of two-prey-one-predator species system model: Local and global stability, Journal of Theoretical Biology, 69 (1977), 613-623. doi: 10.1016/0022-5193(77)90370-8. [13] K. Hasík, On a predator-prey system of gause type, Journal of Mathematical Biology, 60 (2010), 59-74. doi: 10.1007/s00285-009-0257-8. [14] J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of turing patterns in the 1-d lengyel-epstein model, Journal of Dynamics and Differential Equations, 16 (2004), 297-320. doi: 10.1007/s10884-004-2782-x. [15] Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, Journal of Mathematical Biology, 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z. [16] W. Ko and K. Ryu, Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment, Journal of Mathematical Analysis and Applications, 327 (2007), 539-549. doi: 10.1016/j.jmaa.2006.04.077. [17] Y. Kuang, Rich dynamics of gause-type ratio-dependent predator-prey system, Fields Institute Communications, 21 (1999), 325-337. [18] Y. Kuang, Global stability of gause-type predator-prey systems, Journal of Mathematical Biology, 28 (1990), 463-474. doi: 10.1007/BF00178329. [19] S. B. Li, J. H. Wu and Y. Y. Dong, Turing patterns in a reaction-diffusion model with the degn-harrison reaction scheme, Journal of Differential Equations, 259 (2015), 1990-2029. doi: 10.1016/j.jde.2015.03.017. [20] X. Li, W. H. Jiang and J. P. Shi, Hopf bifurcation and turing instability in the reaction-diffusion holling-tanner predator-prey model, Journal of Applied Mathematics, 78 (2013), 287-306. doi: 10.1093/imamat/hxr050. [21] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [22] Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, Journal of Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559. [23] W. M. Ni and M. Tang, Turing patterns in the lengyel-epstein system for the cima reaction, Transactions of the American Mathematical Society, 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9. [24] W. M. Ni, Cross-diffusion and their spike-layer steady states, Notices of the American Mathematical Society, 45 (1998), 9-18. [25] L. Nirenberg, Topics in Nonlinear Functional Analysis Courant Institute of Mathematical Sciences, 1974. [26] P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, Journal of Differential Equations, 200 (2004), 245-273. doi: 10.1016/j.jde.2004.01.004. [27] R. Peng and M. X. Wang, Note on a ratio-dependent predator-prey system with diffusion, Nonlinear Analysis Real World Applications, 7 (2006), 1-11. doi: 10.1016/j.nonrwa.2004.11.008. [28] J. F. Savino and R. A. Stein, Predator-prey interaction between largemouth bass and bluegills as influenced by simulated, submersed vegetation, Transactions of the American Fisheries Society, 111 (1982), 255-266. [29] J. P. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531. [30] H. B. Shi and S. G. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, Journal of Applied Mathematics, 80 (2015), 1534-1568. doi: 10.1093/imamat/hxv006. [31] H. B. Shi, W. T. Li and G. Lin, Positive steady states of a diffusive predator-prey system with modified holling-tanner functional response, Nonlinear Analysis Real World Applications, 11 (2010), 3711-3721. doi: 10.1016/j.nonrwa.2010.02.001. [32] A. Sikder and A. B. Roy, Persistence of a generalized gause-type two prey-two predator pair linked by competition, Mathematical Biosciences, 122 (1994), 1-23. doi: 10.1016/0025-5564(94)90080-9. [33] I. Takagi, Point-condensation for a reaction-diffusion system, Journal of Differential Equations, 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1. [34] M. X. Wang, Stationary patterns of strongly coupled prey-predator models, Journal of Mathematical Analysis and Applications, 292 (2004), 484-505. doi: 10.1016/j.jmaa.2003.12.027. [35] M. X. Wang, Non-constant positive steady states of the sel'kov model, Journal of Differential Equations, 190 (2003), 600-620. doi: 10.1016/S0022-0396(02)00100-6. [36] Z. J. Wen and S. M. Fu, Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics, Journal of Computational and Applied Mathematics, 230 (2009), 34-43. doi: 10.1016/j.cam.2008.10.064. [37] X. Z. Zeng and Z. H. Liu, Non-constant positive steady states of a prey-predator system with cross-diffusions, Journal of Mathematical Analysis and Applications, 332 (2007), 989-1009. doi: 10.1016/j.jmaa.2006.10.075. [38] X. Z. Zeng and Z. H. Liu, Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion, Journal of Mathematical Analysis and Applications, 11 (2010), 372-390. doi: 10.1016/j.nonrwa.2008.11.010.
Numerical simulations of the long time behavior of solution $(N(x, t), P(x, t))$ of model (5) with different values of $d_2$. (a) $d_2=0.6$; (b) $d_2=0.25$;
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