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Article Contents

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

• * Corresponding author
• 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.

Mathematics Subject Classification: Primary: 34D23; Secondary: 97M60.

 Citation:

• Figure 1.  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

Figure 2.  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

Figure 3.  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 $(1500,1000)$

Figure 4.  Two stable equilibria $E_{0}$ and $E_{2}$ coexist. Here $r_{1} = 0.227$, $\alpha = 1000$, $p = 1.68$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$

Figure 5.  The outer closed curve $L$: $OE_{0}FGHO$ around $E_{2}$

Table 1.  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]

Figures(5)

Tables(1)