# American Institute of Mathematical Sciences

• Previous Article
Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models
• DCDS-B Home
• This Issue
• Next Article
Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping
September  2019, 24(9): 4739-4753. doi: 10.3934/dcdsb.2019028

## Dynamics of a predator-prey model with state-dependent carrying capacity

 1 Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China 2 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B5A3, Canada 3 North-West Plateau Institute of Biology, Key Laboratory of Ecology Restoration in Cold Region in Qinghai Province, the Chinese Academy of Sciences, Xining, Qinghai 810001, China

* Corresponding author

Received  September 2017 Revised  May 2018 Published  February 2019

Vegetation and plateau pika are two key species in alpine meadow ecosystems on the Tibetan Plateau. It is frequently observed on the field that plateau pika reduces the carrying capacity of vegetation and the mortality of plateau pika increases along with the increasing height of vegetation. This motivates us to propose and study a predator-prey model with state-dependent carrying capacity. Theoretical analysis and numerical simulations show that the model exhibits complex dynamics including the occurrence of saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, and the coexistence of two stable equilibria.

Citation: Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028
##### References:

show all references

##### References:
A bifurcation diagram of Model (1) with $r_{1} = 0.227$, $\alpha = 4000$, $\mu = 0.001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$, for $p$ varying from $6$ to $12$. The solid curves represent stable equilibria, while the dotted curves represent unstable equilibria
A bifurcation diagram of Model (1) with $r_{1} = 0.227$, $\alpha = 1000$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$ and $p$ varying from $1.2$ to $2.2$. The solid curves represent stable equilibria or limit cycle, while the dotted curves represent unstable equilibria
Two solution trajectories of (1) are converging to a stable limit cycle. Parameter values used ere $r_{1} = 0.227$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$, $\alpha = 1000$ and $p = 2$; Two sets of initial conditions are $(3500,1000)$ and $(1500,1000)$
Two stable equilibria $E_{0}$ and $E_{2}$ coexist. Here $r_{1} = 0.227$, $\alpha = 1000$, $p = 1.68$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$
The outer closed curve $L$: $OE_{0}FGHO$ around $E_{2}$
The default values of parameters in simulations
 Parameter Value Reference $r_{1}$ $0.227\;year^{-1}$ [8] $K$ $4000\;kg$ [1,8,9,22] $q$ $0.36\;kg\cdot head^{-1}\cdot year^{-1}$ [23] $r_{2}$ $0.95\;year^{-1}$ [19,20] $p$ $1\sim12\;year^{-1}$ [13,14,17,24] $\lambda$ $0.32256\;kg\cdot head^{-1}$ [10]
 Parameter Value Reference $r_{1}$ $0.227\;year^{-1}$ [8] $K$ $4000\;kg$ [1,8,9,22] $q$ $0.36\;kg\cdot head^{-1}\cdot year^{-1}$ [23] $r_{2}$ $0.95\;year^{-1}$ [19,20] $p$ $1\sim12\;year^{-1}$ [13,14,17,24] $\lambda$ $0.32256\;kg\cdot head^{-1}$ [10]
 [1] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [2] Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101 [3] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020259 [4] Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141 [5] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [6] Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035 [7] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [8] Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020117 [9] 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 [10] Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265 [11] Julián López-Gómez, Eduardo Muñoz-Hernández. A spatially heterogeneous predator-prey model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020081 [12] Yu-Shuo Chen, Jong-Shenq Guo, Masahiko Shimojo. Recent developments on a singular predator-prey model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020040 [13] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [14] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020130 [15] Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020132 [16] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [17] Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021 [18] Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082 [19] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [20] Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

2018 Impact Factor: 1.008