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

Hanwu Liu; Lin Wang; Fengqin Zhang; Qiuying Li; Huakun Zhou

doi:10.3934/dcdsb.2019028

Article Contents

2019, Volume 24, Issue 9: 4739-4753. Doi: 10.3934/dcdsb.2019028

This issue Previous Article Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping Next Article Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

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
^* Corresponding author

Received: August 31, 2017

Revised: April 30, 2018

Early access: February 2019

Published: July 2019

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Abstract

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.

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

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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

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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

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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) $

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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 $

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

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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]

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