An Excursion-Theoretic Approach to Stability ofDiscrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for L 1-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in L 1. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to ge

Predicted thresholds for natural vegetation cover to safeguard pollinator services in agricultural landscapes

This is the author accepted manuscriptThe conversion of natural vegetation into cultivated land can cause pollinator declines and thereby degrade pollination services to crops and wildflowers. The effect of landscape composition on pollinator abundance is well established, but its impact on pollination intensity and crop yield is not fully resolved. We therefore studied pollination of two crops in India, brinjal (Solanum melongena) and mustard (Brassica nigra), along a landscape-scale gradient in habitat transformation from forest-dominated natural vegetation to intensive cultivation. We quantified the pollination requirements (pollen receipt-seed set relationships) of the crops and the levels of pollen delivery by their principal pollinators, bees. Combining these with field surveys of pollinator abundance, we modelled the levels of pollination service to fields along the landscape gradient. Projected pollination services declined as the area occupied by natural vegetation decreased. We identified thresholds at which bee pollination no longer supported maximum seed set, which were landscapes with approximately one quarter (27 %) of nearby natural vegetation for brinjal fields and one fifth (18 %) for mustard. Our findings indicate that preserving or restoring the cover of natural habitats above these minimum thresholds could be a valuable strategy for maintaining pollinator abundance and safeguarding yield in these bee-pollinated crops.Darwin Initiativ

Selection of Non-Crop Plant Mixes Informed by Arthropod-Plant Network Analyses for Multiple Ecosystem Services Delivery towards Ecological Intensification of Agriculture

Ecological intensification (EI) of agriculture through the improvement of ecosystem service delivery has recently emerged as the alternative to the conventional intensification of agriculture that is widely considered unsustainable and has negative impacts on the environment. Although tropical agricultural landscapes are still heterogeneous, they are rapidly losing diversity due to agricultural intensification. Restoration of natural or semi-natural habitats, habitat diversity, and provision of multiple benefits have been identified as important targets for the transition to EI. Choosing the right plant mixes for the restoration of habitats that can offer multiple ecosystem service benefits is therefore crucial. The selection of candidate species for plant mixes is generally informed by studies focusing on a specific ecosystem service (e.g., pollination) and not based on the whole arthropod—non-crop plant interactions matrix. In this study, we try to identify non-crop plant mixes that would provide habitat for pollinators, act as refugia for natural pest predators, and also as a trap crop for potential crop pests by studying non-crop plants—arthropod interaction network. We have identified the non-crop plant species mixes by first identifying the connector species based on their centrality in the network and then by studying how their sequential exclusions affect the stability of the network

The last mile:Using local knowledge to identify barriers to sustainable grain legume production

Grain legumes (or pulses–annual leguminous crops that are harvested solely for their dried seeds such as lentils or chickpeas) are essential for sustainable cropping systems. They positively contribute to soil fertility and agricultural biodiversity and are a highly nutritious food source, yet they remain under-exploited across the world. In India–soon to be the world's most populous country and the world's largest importer, producer and consumer of pulses–they are substantially under-utilized and are the only major food group not to have increased in output since independence in 1947. Existing efforts to address low grain legume production have focused on the scientific and agronomic barriers, with little impact on productivity. In contrast, this project, using Tripura in India as a case study, recognizes the limits of imposing top-down solutions and instead focuses on the barriers to production as identified by the growers themselves. Working with 440 farmers from 19 non-tribal and 11 tribal villages in Tripura, NE India, we used facilitated discussion to identify their key barriers to pulse production, and facilitated pile sorting to identify the commonly consumed, grown and available pulses. Twenty-eight barriers to legume production were identified by farmers. The eight principal barriers were: insufficient water; lack of technical knowledge; unreliable seed supply; lack of processing units; soil fertility; financial constraints; limited fertilizer supply; and insufficient fencing material. These barriers are complex and overlapping and originate from system level failures to sufficiently prioritise grain legumes compared to cereals. However, recognizing the length of time it takes to address system level problems, in this paper we identify five immediately applicable mitigating strategies to help overcome the principle barriers identified here. Importantly, these will also create an improved environment to apply the technologically sophisticated grain legume R&D that has been carried out over the last 20 years but has yet to have a measurable impact on pulse production. Therefore understanding the wider socio-economic pathways to sustainable pulse production is essential to facilitate change on the ground. Our results, relevant to policy makers in India and around the world, demonstrate the value of listening to farmers when attempting to improve production, and emphasize the necessity of including the socio-economic systems surrounding pulse production, to complement the current emphasis on biological barriers

Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field

Collating and validating indigenous and local knowledge to apply multiple knowledge systems to an environmental challenge: A case-study of pollinators in India

There is an important role for indigenous and local knowledge in a Multiple Evidence Base to make decisions about the use of biodiversity and its management. This is important both to ensure that the knowledge base is complete (comprising both scientific and local knowledge) and to facilitate participation in the decision making process. We present a novel method to gather evidence in which we used a peer-to-peer validation process among farmers that we suggest is analogous to scientific peer review. We used a case-study approach to trial the process focussing on pollinator decline in India. Pollinator decline is a critical challenge for which there is a growing evidence base, however, this is not the case world–wide. In the state of Orissa, India, there are no validated scientific studies that record historical pollinator abundance, therefore local knowledge can contribute substantially and may indeed be the principle component of the available knowledge base. Our aim was to collate and validate local knowledge in preparation for integration with scientific knowledge from other regions, for the purpose of producing a Multiple Evidence Base to develop conservation strategies for pollinators. Farmers reported that vegetable crop yields were declining in many areas of Orissa and that the abundance of important insect crop pollinators has declined sharply across the study area in the last 10–25 years, particularly Apis cerana, Amegilla sp. and Xylocopa sp. Key pollinators for commonly grown crops were identified; both Apris cerana and Xylocopa sp. were ranked highly as pollinators by farmer participants. Crop yield declines were attributed to soil quality, water management, pests, climate change, overuse of chemical inputs and lack of agronomic expertise. Pollinator declines were attributed to the quantity and number of pesticides used. Farmers suggested that fewer pesticides, more natural habitat and the introduction of hives would support pollinator populations. This process of knowledge creation was supported by participants, which led to this paper being co-authored by both scientists and farmers

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for $L_1$-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in $L_1$. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.Comment: Revised. 17 pages. To appear in Applied Mathematics & Optimizatio

Wild insect diversity increases inter-annual stability in global crop pollinator communities.

While an increasing number of studies indicate that range, diversity and abundance of many wild pollinators has declined, the global area of pollinator-dependent crops has significantly increased over the last few decades. Crop pollination studies to date, have mainly focused on either identifying different guilds pollinating various crops, or on factors driving spatial changes and turnover observed in these communities. The mechanisms driving temporal stability for ecosystem functioning and services, however, remain poorly understood. Our study quantifies temporal variability observed in crop pollinators in 21 different crops across multiple years at a global scale. Using data from 43 studies from six continents, we show that (i) higher pollinator diversity confers greater inter-annual stability in pollinator communities, (ii) temporal variation observed in pollinator abundance is primarily driven by the three most dominant species, and (iii) crops in tropical regions demonstrate higher inter-annual variability in pollinator species richness than crops in temperate regions. We highlight the importance of recognising wild pollinator diversity in agricultural landscapes to stabilize pollinator persistence across years to protect both biodiversity and crop pollination services. Short-term agricultural management practices aimed at dominant species for stabilising pollination services need to be considered alongside longer-term conservation goals focussed on maintaining and facilitating biodiversity to confer ecological stability