Impact of different destocking strategies on the resilience of dry rangelands

Abstract

Half of the world's livestock live in (semi-)arid regions, where a large proportion of people rely on animal husbandry for their survival. However, overgrazing can lead to land degradation and subsequent socio-economic crises. Sustainable management of dry rangeland requires suitable stocking strategies and has been the subject of intense debate in the last decades. Our goal is to understand how variations in stocking strategies affect the resilience of dry rangelands. We describe rangeland dynamics through a simple mathematical model consisting of a system of coupled differential equations. In our model, livestock density is limited only by forage availability, which is itself limited by water availability. We model processes typical of dryland vegetation as a strong Allee effect, leading to bistability between a vegetated and a degraded state, even in the absence of herbivores. We study analytically the impact of varying the stocking density and the destocking adaptivity on the resilience of the system to the effects of drought. By using dynamical systems theory, we look at how different measures of resilience are affected by variations in destocking strategies. We find that the following: (1) Increasing stocking density decreases resilience, giving rise to an expected trade-off between productivity and resilience. (2) There exists a maximal sustainable livestock density above which the system can only be degraded. This carrying capacity is common to all strategies. (3) Higher adaptivity of the destocking rate to available forage makes the system more resilient: the more adaptive a system is, the bigger the losses of vegetation it can recover from, without affecting the long-term level of productivity. The first two results emphasize the need for suitable dry rangeland management strategies, to prevent degradation resulting from the conflict between profitability and sustainability. The third point offers a theoretical suggestion for such a strategy.

Keywords: adaptive management; consumer‐resource; dry rangeland; resilience.

FIGURE 1

The four extreme management strategies and how they relate to our control parameters $\delta$ and $\alpha$. Past research on optimal grazing strategies in dry rangelands typically opposed the poles II and IV, where II was associated with traditional pastoral systems and IV with commercial pasture management (Campbell et al., 2006). In this work, decreasing the parameter $\delta$ increases the stocking density and increasing $\alpha$ increases the stocking adaptivity.

FIGURE 2

Example of vegetation growth function satisfying the constraint a1. The function given here is of the shape $f\left(V\right)=\mathit{rV}\left(\frac{V}{A}-1\right)\left(1-\frac{V}{K}\right)$, where $r$ is a positive constant.

FIGURE 3

Qualitative 3D stability landscape of our two dimensional system. The separatrix (green continuous line) marks the limit between the basin of attraction of the sustainable productive steady state $\left(V_{\mathrm{eq}};H_{\mathrm{eq}}\right)$ and the basin of attraction of the degraded state (0;0). The position of the separatrix depends on the destocking strategy.

FIGURE 4

Graphical summary of the general stability analysis, as a phase plane. Note that the vertical management nullcline $\mathit{\gamma g}\left(V\right)-D_{\delta ;\alpha }\left(V\right)=0$ is not shown, as its position varies with $\delta$. Decreasing (resp. increasing) $\delta$ moves the management nullcline to the left (resp.right). The phase plane is partitioned into five areas (labeled 1–5) where the vertical management nullcline can lie, and for each of which the system displays qualitatively different behavior.

FIGURE 5

Bifurcation diagrams for vegetation density (top panel) and animal density (bottom panel): as the destocking parameter $\delta$ (horizontal axis) varies, the system undergoes transitions in its stability properties.

FIGURE 6

Comparing the resilience of a sustainable productive system to the effects of drought for different baseline destocking rates ${\delta }_{\mathrm{l}}$ and ${\delta }_{\mathrm{h}}$, where ${\delta }_{\mathrm{h}}<{\delta }_{\mathrm{l}}$ and with fixed adaptivity $\alpha =0$. Left panels (a, c, e) show the response to a weak perturbation while right panels (b, d, f) show the response to a stronger perturbation. Top panels (a,b) show the phase‐space while the middle and bottom (c, e, d, f) show the trajectories of the biomass variables over time.

FIGURE 7

Comparing the resilience of a sustainable productive system to the effects of drought for different degrees of adaptivity ${\alpha }_{\mathrm{c}}=0$ and ${\alpha }_{\mathrm{a}}=1$, where the baseline destocking rate $\delta ={\delta }_{\mathrm{h}}$ and hence the long‐term stocking density $H_{\mathrm{eq}}$ is fixed. Left panels (a, c, e) show the response to a weak perturbation while right panels (b, d, f) show the response to a stronger perturbation. Top panels (a,b) show the phase‐space while the middle and bottom (c, e, d, f) show the trajectories of the biomass variables over time.

FIGURE 8

Relation between the resilience and the productivity of a sustainable productive system, for different values of the management parameters $\delta$ and $\alpha$. On the horizontal axis, the long‐term stocking density $H_{\mathrm{eq}}\left(\delta \right)$ is a proxy for the rangeland's productivity. On the vertical axis, the resilience is measured through the maximal loss ratio, $\frac{V_{\mathrm{eq}}}{V_{\text{pmax}}}\left(\delta ;\alpha \right)$.