Effect of rotational grazing on plant and animal production

Mayee Chen; Junping Shi

doi:10.3934/mbe.2018017

Article Contents

2018, Volume 15, Issue 2: 393-406. Doi: 10.3934/mbe.2018017

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Effect of rotational grazing on plant and animal production

Mayee Chen^1, , and
Junping Shi^2, ,

1.
Jamestown High School, Williamsburg, VA 23185, USA,
2.
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Mayee Chen, mayeefchen@gmail.com ;
* Corresponding author: Junping Shi.
* Corresponding author: Junping Shi.

Received: September 15, 2017

Accepted: April 22, 2017

Published: March 2018

The second author is supported by NSF grant DMS-1313243.

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Abstract

It is a common understanding that rotational cattle grazing provides better yields than continuous grazing, but a quantitative analysis is lacking in agricultural literature. In rotational grazing, cattle periodically move among paddocks in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. We construct a differential equation model of vegetation grazing on a fixed area to show that production yields and stockpiled forage are greater for rotational grazing than continuous grazing. Our results show that both the number of cattle per acre and stockpiled forage increase for many rotational configurations.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Left: Growth rate of the grass and consumption rate by the cattle for continuous grazing. Here the growth rate $G(V)$ and the grazing rate $H\cdot c(V)$ are defined as in (2.1) and (2.2), with parameter values given as in Table 1 and $H=1.06$, $0.6$ and $0.2$ respectively. Right: A forage ($V$) versus cattle ($H$) bifurcation diagram for the continuous grazing system

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Figure 2. Illustration of continuous grazing (left), and rotational grazing (right).

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Figure 3. Amount of forage in a sustainable rotational configuration where $3$ out of $4$ paddocks are grazed. Here (2.7) and (2.8) are used for integration, $(n,m,T)=(4,3,10)$, $H=1.3$ and $T_{total}=365$ days.

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Figure 4. Amount of forage in a sustainable rotational configuration where $3$ out of $4$ paddocks are grazed. Here (2.7) and (2.8) are used for integration, $(n,m,T)=(4,3,10)$, $H=1.28$ and $T_{total}=3650$ days.

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Figure 5. Maximum $H$ for different paddock configurations and $T$. Here the horizontal axis is the rotation period $T$, the vertical axis is the maximum sustainable cattle number $H_{max}^R(T)$, and the legend shows $m: n$ (the number of paddocks grazed versus the number of total paddock). The horizontal line is $1.0631$ head of cattle per acre, which is from continuous grazing. Here $T_{total}=365$ is used.

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Figure 6. Maximum $V$ for different paddock configurations and $T$. Here the horizontal axis is the rotation period $T$, the vertical axis is the forage amount $V_S^R(T)$ when achieving the maximum sustainable cattle number $H_{max}^R(T)$, and the key is $m: n$ (the number of paddocks grazed versus the number of total paddock). The horizontal line is the forage amount when achieving the maximum sustainable cattle number $H_{max}$ for continuous grazing.

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Figure 7. Cattle and stockpiled forage plotted against the grazing ratio for a $15$-day rotation period. Here the horizontal axis is the grazing ratio of the rotational scheme, and the vertical axis is the maximum sustainable cattle number $H_{max}^R(T)$ and associated forage amount $V_S^R(T)$.

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Table 1. Table of variables and parameters in the equations.

Variable	Meaning	Units
$t$	time	days
$V_j(t)$	grass biomass in paddock $j$	pounds/acre
Parameter	Meaning	Units	Value	Reference
$V_{max}$	grass carrying capacity	pounds/acre	$2400$	[21]
$g_{max}$	maximum growth rate per capita rate per capita	day$^{-1}$	$0.05625$	[14]
$c_{max}$	maximum consumption rate per head of cattle	pounds/(acre$\cdot$day)	$35$	[1,2]
$K$	half-saturation value	pounds/acre	$120$
$H_j$	number of cattle per acre in paddock $j$	cattle/acre

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