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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
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##### 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]
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