Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise

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

doi:10.3934/mbe.2018034

Article Contents

2018, Volume 15, Issue 3: 765-773. Doi: 10.3934/mbe.2018034

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Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise

1.
Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China
2.
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada
3.
North-West Plateau Institute of Biology, the Chinese Academy of Sciences, Key Laboratory of Ecology Restoration in Cold Region in Qinghai Province, Xining, Qinghai 810001, China

* Corresponding author
* Corresponding author

Received: May 19, 2017

Revised: September 10, 2017

Published: June 2018

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Abstract

As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

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Mathematics Subject Classification: Primary: 34D23; Secondary: 92D25.

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Figure 1. The stability regions of equilibria in the $C-A$ plane.

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Figure 2. Bifurcation diagrams of Model (2) showing the globally stable equilibria only. Left: $C< \theta K$; Middle: $\theta K<C<K$; Right: $C>K$.

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Table 1. Variation of coordinates $x^{*}, y^{*}, z^{*}, u^{*}$ of $E_{5}$ and values of $A$, $\theta$ when one of the parameters $d_{1}, r, K, d_{3}, d_{4}, q$ increases (all other parameters are fixed).

Parameter	$d_{1}$	$r$	$K$	$d_{3}$	$d_{4}$	$q$
$x^{*}$	$-$	$-$	$-$	$\nearrow$	$-$	$-$
$y^{*}$	$-$	$-$	$-$	$-$	$\nearrow$	$\searrow$
$z^{*}$	$\searrow$	$-$	$-$	$-$	$\nearrow$	$\searrow$
$u^{*}$	$-$	$\nearrow$	$\nearrow$	$\searrow$	$\searrow$	$\nearrow$
$A$	$\nearrow$	$-$	$-$	$-$	$-$	$-$
$\theta$	$-$	$\nearrow$	$-$	$\searrow$	$-$	$-$

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References

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