Patterns and rules for sensitivity and elasticity in population projection matrices

Sensitivity and elasticity analysis of population projection matrices (PPMs) are established tools in the analysis of structured populations, allowing comparison of the contributions made by different demographic rates to population growth. In some commonly used structures of PPM, however, there are mathematically inevitable patterns in the relative sensitivity and elasticity of certain demographic rates. We take a simulation approach to investigate these mathematical constraints for a range of PPM models. Our results challenge some previously proposed constraints on sensitivity and elasticity. We also identify constraints beyond those which have already been proven mathematically, and promote them as candidates for future mathematical proof. A general theme among these rules is that changes to the demographic rates of older or larger individuals have less impact on population growth than do equivalent changes among younger or smaller individuals. However, the validity of these rules in each case depends on the choice between sensitivity and elasticity, the growth rate of the population and the PPM structure used. If the structured population conforms perfectly to the assumptions of the PPM used to model it, the rules we describe represent biological reality, allowing us to prioritise management strategies in the absence of detailed demographic data. Conversely, if the model is a poor fit to the population (specifically; if demographic rates within stages are heterogeneous) such analyses could lead to inappropriate management prescriptions. Our results emphasise the importance of choosing a structured population model which fits the demographics of the population

A comparative study of herbage intake, ingestive behaviour and diet selection, and effects of condensed tannins upon body and wool growth in lambs grazing Yorkshire fog (Holcus lanatus) and annual ryegrass (Lolium multiflorum) dominant swards

An experiment was carried out from August to early November 1994 to examine differences in diet selection, herbage intake, grazing behaviour and animal performance between weaned lambs rotationally grazing swards of annual ryegrass (Lolium multiflorum)/white clover (Trifolium repens) and Yorkshire fog (Holcus lanatus)/T. repens with or without Lotus corniculatus. There were four replicate groups of six lambs per treatment. The effects of condensed tannins (CT) on lamb production were assessed by twice-daily oral administration of 10g polyethylene glycol (PEG; molecular weight 4000) to half the lambs on each sward. The Lotus content of all swards was very low, and results are presented here for main sward comparisons meaned over lotus treatments. Overall mean estimates of pre-grazing herbage mass and sward surface height for the annual ryegrass and Yorkshire fog swards respectively, were 5820 v. 4360 +/- 190 kg DM/ha (P , P < 0.01) and liveweight gain (141 v. 120 +/- 4.3 g per lamb per day, P < 0.01), although differences in carcass weight (17.9 v. 18.2 +/- 0.3 kg) and FEC transformed values (9.6 v. 11.0 +/- 06 eggs/g fresh faeces) were not significant. The effects of CT on animal performance were greater in Yorkshire fog swards. CT had no significant effects on diet selection, herbage intake and grazing behaviour patterns

Transients drive the demographic dynamics of plant populations in variable environments

The dynamics of structured plant populations in variable environments can be decomposed into the ‘asymptotic’ growth contributed by vital rates, and ‘transient’ growth caused by deviation from stable stage structure. We apply this framework to a large, global data base of longitudinal studies of projection matrix models for plant populations. We ask, what is the relative contribution of transient boom and bust to the dynamic trajectories of plant populations in stochastic environments? Is this contribution patterned by phylogeny, growth form or the number of life stages per population and per species? We show that transients contribute nearly 50% or more to the resulting trajectories, depending on whether transient and stable contributions are partitioned according to their absolute or net contribution to population dynamics. Both transient contributions and asymptotic contributions are influenced heavily by the number of life stages modelled. We discuss whether the drivers of transients should be considered real ecological phenomena, or artefacts of study design and modelling strategy. We find no evidence for phylogenetic signal in the contribution of transients to stochastic growth, nor clear patterns related to growth form. We find a surprising tendency for plant populations to boom rather than bust in response to temporal changes in vital rates and that stochastic growth rates increase with increasing tendency to boom. Synthesis. Transient dynamics contribute significantly to stochastic population dynamics but are often overlooked in ecological and evolutionary studies that employ stochastic analyses. Better understanding of transient responses to fluctuating population structure will yield better management strategies for plant populations, and better grasp of evolutionary dynamics in the real world

A note on the eigenvectors of perturbed matrices with applications to linear positive systems

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms — acting much like Lyapunov functions — for linear positive systems, which mayhelp estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples

Positive state controllability of positive linear systems

Controllability of positive systems by positive inputs arises naturally in applications where both external and internal variables must remain positive for all time. In many applications, particularly in population biology, the need for positive inputs is often overly restrictive. Relaxing this requirement, the notion of positive state controllability of positive systems is introduced. A connection between positive state controllability and positive input controllability of a related system is established and used to obtain Kalman-like controllability criteria. In doing so we aim to encourage further study in this underdeveloped area

Evaluating the environmental dimension of material efficiency strategies relating to the circular economy.

Material efficiency is a key element of new thinking to address the challenges of reducing impacts on the environment and of resource scarcity, whilst at the same time meeting service and functionality demands on materials. Directly related to material efficiency is the concept of the Circular Economy, which is based on the principle of optimising the utility embodied in materials and products through the life-cycle. Although materials such as steel, on account of high recycling rates at end-of-life, are amongst the most ‘circular’ of manufactured materials, significant opportunities for greater material efficiency exist, which are yet to be widely implemented. Life Cycle Assessment (LCA) is commonly used to assess the environmental benefits of recovering and recycling materials through the manufacturing supply chain and at end-of-life. Using an example taken from renewable energy generation, this paper explores the correlation between product circularity and the environmental case for strategies designed to improve material efficiency. An LCA-based methodology for accounting for the recovery and reuse of materials from the supply chain and at end-of-life is used as the basis for calculating the carbon footprint benefits of five material efficiency scenarios. The results are compared with a number of proposed material circularity indicators. Two conclusions from this exercise are that (i) LCA methodologies based around end-of-life approaches are well placed for quantifying the environmental benefits of material efficiency and circular economy strategies and (ii) when applying indicators relating to the circularity of materials these should also be supported by LCA-based studies.N/

Robust set-point regulation for ecological models with multiple management goals

Population managers will often have to deal with problems of meeting multiple goals, for example, keeping at specific levels both the total population and population abundances in given stage-classes of a stratified population. In control engineering, such set-point regulation problems are commonly tackled using multi-input, multi-output proportional and integral (PI) feedback controllers. Building on our recent results for population management with single goals, we develop a PI control approach in a context of multi-objective population management. We show that robust set-point regulation is achieved by using a modified PI controller with saturation and anti-windup elements, both described in the paper, and illustrate the theory with examples. Our results apply more generally to linear control systems with positive state variables, including a class of infinite-dimensional systems, and thus have broader appeal