Grazing management - shoot and crown biomass

Introduction

A major challenge in managing grazed pastures lies in reconciling conflicting objectives: increasing short-term profits while preserving the long-term viability of the rangeland. Increasing the stocking rate (number of animals per hectare) boosts short-term returns but can lead to virtually irreversible environmental degradation if the resulting grazing pressure is excessive. Conversely, maintaining an excessively low stocking rate can result in significant lost revenue.

In the article

Anderies, J., Janssen, M. & Walker, B. Grazing Management, Resilience, and the Dynamics of a Fire-driven Rangeland System. Ecosystems 5, 23–44 (2002). https://doi.org/10.1007/s10021-001-0053-9

The authors use the following model to account for the interactions between grass growth and grazing pressure:

$$
\left\{\begin{array}{ll}
c' &= r_cs-\delta_cc\\
s'&= c(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws}(\frac{w}{w^*})^\beta)-\gamma_gs\\
w'&=r_ww(1-\frac{w}{w^*})
\end{array}\right.
$$

Regarding the grass, $c(t)$ and $s(t)$ represent the biomass of the crown (roots plus growing points) and the shoot part, respectively, at time $t$. The parameter $s^*$ corresponds to the maximum shoot biomass per unit area, while $\gamma_g\leq 1$ represents the fraction of the shoot part removed by the grazing pressure exerted by the animals present. The crown grows at a rate $r_cs$ in the presence of shoot parts and declines at a rate $\delta_c$.
$a_c$ corresponds to the growth rate of shoot biomass per unit of crown biomass, and $r_s$ to the growth rate of shoot biomass per unit of crown biomass and per unit of shoot biomass.

Grass growth is also affected by the presence of shrubs: $w(t)$ represents shrub biomass, $r_w$ the intrinsic growth rate of shrubs, $w^*$ the maximum shrub biomass per unit area, and $w_s (=w/w^*)$ models the competitive effect of shrubs on the grass. The parameter $\beta$ measures the relative non-linearity of this effect.

The pasture manager's viability problem

The objective of rangeland management is to identify intervention strategies (adjustments to stocking rates) that reconcile two constraints: providing sufficient feed for the animals while preserving grass quality.

In the article

The authors study this viability problem using a simplified grass growth model, assuming that the maximum shoot biomass value, $s^*$, is 1 and that the impact of the shrub population is negligible—that is: 

$$
s^* = 1 \text{ et } \alpha_{ws} = 0.$$

The possible intervention on the system involves varying the grazing pressure—a variation assumed to be bounded—since purchasing or selling animals takes time :

$$
\gamma'(t) = u(t) \in [u_{min},u_{max}].
$$

The dynamics of the system are thus described by:

$$
\left\{\begin{array}{ll}
c' &= r_cs-\delta_cc\\
s'&= \frac{r_c}{\delta_c}s(a_c+r_ss)(1-s)-\gamma_gs\\
\gamma_g'&= u \in [u_{min},u_{max}]
\end{array}\right.
$$

The constraint regarding grass quality is modeled by a shoot biomass value exceeding a threshold, $s_{min}$, to ensure regrowth. The constraint regarding livestock feed is modeled by a grazing pressure that also exceeds a threshold, $\underline{g}$, over time. The set of constraints can thus be written as:

$$
K(s_{min},\underline{g}):=R^+\times [s_{min},+\infty[\times  [\underline{g},1]
$$

Solving this problem means finding the set of states $(s, \gamma_g)$ from which it is possible to feed the herd while maintaining high-quality grass by adjusting the number of animals—in other words, the viability kernel within the framework of viability theory.

Résultats

In the article cited above, the parameter values ​​are as follows: $r_c=1$, $\delta_c=1$, $a_c=0.1$,  $r_s=3$,  $s^*=1$, $u_{min} = -0.05$, $u_{max} = 0.05$, $s_{min} = 0.1$, $\underline{g}=0.65$.

We reproduce below the approximation of the viability kernel associated with these parameter values, generated by the ViabLab software using a 201×201×201 point grid:

As expected, starting from a higher grazing pressure results in a more restricted set of shoot and crown biomass combinations for which the system is viable.

This restriction is more pronounced when the capacity to reduce this pressure is low—that is, when $u_{min}$ approaches 0.