# American Institute of Mathematical Sciences

April  2018, 15(2): 393-406. doi: 10.3934/mbe.2018017

## Effect of rotational grazing on plant and animal production

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

* Corresponding author: Junping Shi.

Received  September 16, 2017 Accepted  April 23, 2017 Published  June 2017

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

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.

Citation: Mayee Chen, Junping Shi. Effect of rotational grazing on plant and animal production. Mathematical Biosciences & Engineering, 2018, 15 (2) : 393-406. doi: 10.3934/mbe.2018017
##### References:
 [1] How Much Feed Will My Cow Eat? Alberta Agricultural and Rural Development Edmonton, Alberta, 2003, Available from: http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/faq7811 Google Scholar [2] Raising Cattle for Beef Production and Beef Safety, Cattlemen's Beef Board and National Cattlemen's Beef Association, Centennial, Colorado, 2013, Available from: http://www.explorebeef.org/raisingbeef.aspx Google Scholar [3] Using the Animal Unit Month (AUM) Effectively, Alberta Agricultural and Rural Development, Edmonton, Alberta, 2001, Available from: http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/agdex1201 Google Scholar [4] L. I. Aniţa, S. Aniţa and V. Arnăutu, Global behavior for an age-dependent population model with logistic term and periodic vital rates, Appl. Math. Comput., 206 (2008), 368-379.  doi: 10.1016/j.amc.2008.09.016.  Google Scholar [5] L. I. Aniţa, S. Aniţa and V. 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Noy-Meir, Rotational grazing in a continuously growing pasture: A simple model, Agri. Systems., 1 (1976), 87-112.  doi: 10.1016/0308-521X(76)90009-3.  Google Scholar [21] E. B. Rayburn, Number and size of paddocks in a grazing system, West Virginia University Extension Service, Morgantown, West Virginia. 1992. Google Scholar [22] J. P. Ritten, W. M. Frasier, C. T. Bastian and S. T. Gray, Optimal rangeland stocking decisions under stochastic and climate-impacted weather, Amer. Jour. Agri. Econ., 92 (2010), 1242-1255.  doi: 10.1093/ajae/aaq052.  Google Scholar [23] A. Savory and D. P. Stanley, The Savory grazing method, Rangelands, 2 (1980), 234-237.   Google Scholar [24] N. F. Sayre, Viewpoint: The need for qualitative research to understand ranch management, Rang. Ecol. & Mana., 57 (2004), 668-674.   Google Scholar [25] M. Scheffer, Critical transitions in nature and society, Princeton University Press, Princeton, New Jersey, 2009. Google Scholar [26] M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar [27] R. Smith, G. Lacefield, R. Burris, D. Ditsch, B. Coleman, J. Lehmkuhler and J. Henning, Rotational grazing, University of Kentucky College of Agriculture; Lexington, Kentucky, 2011. Google Scholar [28] J. Sprinkle and D. Bailey, How many animals can I graze on my pasture?, The University of Arizona Cooperative Extension, Tucson, Arizona, 2004. Google Scholar

show all references

##### References:
 [1] How Much Feed Will My Cow Eat? Alberta Agricultural and Rural Development Edmonton, Alberta, 2003, Available from: http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/faq7811 Google Scholar [2] Raising Cattle for Beef Production and Beef Safety, Cattlemen's Beef Board and National Cattlemen's Beef Association, Centennial, Colorado, 2013, Available from: http://www.explorebeef.org/raisingbeef.aspx Google Scholar [3] Using the Animal Unit Month (AUM) Effectively, Alberta Agricultural and Rural Development, Edmonton, Alberta, 2001, Available from: http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/agdex1201 Google Scholar [4] L. I. Aniţa, S. Aniţa and V. Arnăutu, Global behavior for an age-dependent population model with logistic term and periodic vital rates, Appl. Math. Comput., 206 (2008), 368-379.  doi: 10.1016/j.amc.2008.09.016.  Google Scholar [5] L. I. Aniţa, S. Aniţa and V. Arnăutu, Optimal harvesting for periodic age-dependent population dynamics with logistic term, Appl. Math. Comput., 215 (2009), 2701-2715.  doi: 10.1016/j.amc.2009.09.010.  Google Scholar [6] S. K. Bamhart, Estimating available pasture forage, Iowa State University Extension College of Agriculture, Ames, Iowa, 2009. Google Scholar [7] S. Behringer and T. Upmann, Optimal harvesting of a spatial renewable resource, J. Econ. Dyn. Control., 42 (2014), 105-120.  doi: 10.1016/j.jedc.2014.03.008.  Google Scholar [8] A. O. Belyakov, A. A. Davydov and V. M. Veliov, Optimal cyclic exploitation of renewable resources, J. Dyn. Control Syst., 21 (2015), 475-494.  doi: 10.1007/s10883-015-9271-x.  Google Scholar [9] A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phenom., 9 (2014), 20-37.  doi: 10.1051/mmnp/20149403.  Google Scholar [10] Briske and Rotational grazing on rangelands: Reconciliation of perception, Rotational grazing on rangelands: Reconciliation of perception and experimental evidence, Rang. Ecol. & Mana., 61 (2008), 3-17.  doi: 10.2111/06-159R.1.  Google Scholar [11] L. Fu, T. Bo, G. Du and X. Zheng, Modeling the responses of grassland vegetation coverage to grazing disturbance in an alpine meadow, Ecol. Modelling, 247 (2012), 221-232.  doi: 10.1016/j.ecolmodel.2012.08.027.  Google Scholar [12] R. K. Heitschmidt, S. L. Dowhower and J. W. Walker, 14-vs. 42-paddock rotational grazing: Aboveground biomass dynamics, forage production, and harvest efficiency, Jour. Range Mana., 40 (1987), 216-223.  doi: 10.2307/3899082.  Google Scholar [13] C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar [14] C. Hurtado-Uria, D. Hennessy, L. Shalloo, R. Schulte, L. Delaby and D. O'Connor, Evaluation of three grass growth models to predict grass growth in Ireland, Jour. Agri. Sci., 151 (2013), 91-104.  doi: 10.1017/S0021859612000317.  Google Scholar [15] I. R. Johnson, T. E. Ameziane and J. H. M. Thornley, A model of grass growth, Annals of Botany, 51 (1983), 599-609.  doi: 10.1093/oxfordjournals.aob.a086506.  Google Scholar [16] R. Kallenbach, Calculating stocking rates of cows, High Plains Journal, 2010. Google Scholar [17] R. Lemus, Developing a grazing system, Mississippi State University Extension, Mississippi State, Mississippi, 2008. Google Scholar [18] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.  doi: 10.1038/269471a0.  Google Scholar [19] I. Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, The Journal of Ecology, 63 (1975), 459-481.  doi: 10.2307/2258730.  Google Scholar [20] I. Noy-Meir, Rotational grazing in a continuously growing pasture: A simple model, Agri. Systems., 1 (1976), 87-112.  doi: 10.1016/0308-521X(76)90009-3.  Google Scholar [21] E. B. Rayburn, Number and size of paddocks in a grazing system, West Virginia University Extension Service, Morgantown, West Virginia. 1992. Google Scholar [22] J. P. Ritten, W. M. Frasier, C. T. Bastian and S. T. Gray, Optimal rangeland stocking decisions under stochastic and climate-impacted weather, Amer. Jour. Agri. Econ., 92 (2010), 1242-1255.  doi: 10.1093/ajae/aaq052.  Google Scholar [23] A. Savory and D. P. Stanley, The Savory grazing method, Rangelands, 2 (1980), 234-237.   Google Scholar [24] N. F. Sayre, Viewpoint: The need for qualitative research to understand ranch management, Rang. Ecol. & Mana., 57 (2004), 668-674.   Google Scholar [25] M. Scheffer, Critical transitions in nature and society, Princeton University Press, Princeton, New Jersey, 2009. Google Scholar [26] M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar [27] R. Smith, G. Lacefield, R. Burris, D. Ditsch, B. Coleman, J. Lehmkuhler and J. Henning, Rotational grazing, University of Kentucky College of Agriculture; Lexington, Kentucky, 2011. Google Scholar [28] J. Sprinkle and D. Bailey, How many animals can I graze on my pasture?, The University of Arizona Cooperative Extension, Tucson, Arizona, 2004. Google Scholar
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
Illustration of continuous grazing (left), and rotational grazing (right).
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.
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.
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.
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.
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 $V_S^R(T)$.
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
 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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